Pi
You are encouraged to solve this task according to the task description, using any language you may know.
Create a program to continually calculate and output the next decimal digit of (pi).
The program should continue forever (until it is aborted by the user) calculating and outputting each decimal digit in succession.
The output should be a decimal sequence beginning 3.14159265 ...
Note: this task is about calculating pi. For information on built-in pi constants see Real constants and functions.
Related Task Arithmetic-geometric mean/Calculate Pi
11l
V ndigits = 0
V q = BigInt(1)
V r = BigInt(0)
V t = q
V k = q
V n = BigInt(3)
V l = n
V first = 1B
L ndigits < 1'000
I 4 * q + r - t < n * t
print(n, end' ‘’)
ndigits++
I ndigits % 70 == 0
print()
I first
first = 0B
print(‘.’, end' ‘’)
V nr = 10 * (r - n * t)
n = ((10 * (3 * q + r)) I/ t) - 10 * n
q *= 10
r = nr
E
V nr = (2 * q + r) * l
V nn = (q * (7 * k + 2) + r * l) I/ (t * l)
q *= k
t *= l
l += 2
k++
n = nn
r = nr- Output:
3.141592653589793238462643383279502884197169399375105820974944592307816 4062862089986280348253421170679821480865132823066470938446095505822317 2535940812848111745028410270193852110555964462294895493038196442881097 5665933446128475648233786783165271201909145648566923460348610454326648 2133936072602491412737245870066063155881748815209209628292540917153643 6789259036001133053054882046652138414695194151160943305727036575959195 3092186117381932611793105118548074462379962749567351885752724891227938 1830119491298336733624406566430860213949463952247371907021798609437027 7053921717629317675238467481846766940513200056812714526356082778577134 2757789609173637178721468440901224953430146549585371050792279689258923 5420199561121290219608640344181598136297747713099605187072113499999983 7297804995105973173281609631859502445945534690830264252230825334468503 5261931188171010003137838752886587533208381420617177669147303598253490 4287554687311595628638823537875937519577818577805321712268066130019278 76611195909216420198
360 Assembly
The program uses one ASSIST macro (XPRNT) to keep the code as short as possible.
* Spigot algorithm do the digits of PI 02/07/2016
PISPIG CSECT
USING PISPIG,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) prolog
ST R13,4(R15) "
ST R15,8(R13) "
LR R13,R15 "
SR R0,R0 0
ST R0,MORE more=0
LA R6,1 i=1
LOOPI1 C R6,=A(NBUF) do i=1 to hbound(buf)
BH ELOOPI1 "
SR R9,R9 karray=0
L R7,=A(NVECT) j=hbound(vect)
LR R1,R7 j
SLA R1,2 .
LA R10,VECT-4(R1) r10=@vect(j)
LOOPJ EQU * do j=hbound(vect) to 1 by -1
L R5,=F'100000' 100000
M R4,0(R10) *vect(j)
LR R2,R5 r2=100000*vect(j)
LR R5,R9 karray
MR R4,R7 karray*j
AR R2,R5 r2+karray*j
LR R11,R2 n=100000*vect(j)+karray*j
LR R3,R7 j
SLA R3,1 2*j
BCTR R3,0 2*j-1)
LR R4,R11 n
SRDA R4,32 .
DR R4,R3 n/(2*j-1)
LR R9,R5 karray=n/(2*j-1)
LR R5,R9 karray
MR R4,R3 karray*(2*j-1)
LR R1,R11 n
SR R1,R5 n-karray*(2*j-1)
ST R1,0(R10) vect(j)=n-karray*(2*j-1)
SH R10,=H'4' r10=@vect(j)
BCT R7,LOOPJ end do j
LR R4,R9 karray
SRDA R4,32 .
D R4,=F'100000' karray/100000
LR R11,R5 k=karray/100000
L R2,MORE more
AR R2,R11 +k
LR R1,R6 i
SLA R1,2 .
ST R2,BUF-4(R1) buf(i)=more+k
LR R5,R11 k
M R4,=F'100000' *100000
LR R1,R9 karray
SR R1,R5 -k*100000
ST R1,MORE more=karray-k*100000
LA R6,1(R6) i=i+1
B LOOPI1 end do i
ELOOPI1 L R1,BUF buf(1)
CVD R1,PACKED convert buf(1) to packed decimal
OI PACKED+7,X'0F' prepare unpack
UNPK PG(1),PACKED packed decimal to zoned printable
MVI PG+1,C'.' output '.'
XPRNT PG,80 print buffer
MVC PG,=CL80' ' clear buffer
LA R3,PG pgi=0
LA R6,2 i=2
LOOPI2 C R6,=A(NBUF) do i=2 to hbound(buf)
BH ELOOPI2 "
MVC 0(1,R3),=C' ' output ' '
LA R3,1(R3) pgi=pgi+1
LR R1,R6 i
SLA R1,2 .
L R2,BUF-4(R1) buf(i)
CVD R2,PACKED convert v to packed decimal
OI PACKED+7,X'0F' prepare unpack
UNPK XDEC,PACKED packed decimal to zoned printable
MVC 0(5,R3),XDEC+7 output buf(i) with 5 decimals
LA R3,5(R3) pgi=pgi+5
LR R4,R6 i
BCTR R4,0 i-1
SRDA R4,32 .
D R4,=F'10' (i-1)/10
LTR R4,R4 if (i-1)//10=0
BNZ NOSKIP then
XPRNT PG,80 print buffer
LA R3,PG pgi=0
MVC PG,=CL80' ' clear buffer
NOSKIP LA R6,1(R6) i=i+1
B LOOPI2 end do i
ELOOPI2 L R13,4(0,R13) epilog
LM R14,R12,12(R13) "
XR R15,R15 "
BR R14 exit
LTORG
MORE DS F more
PACKED DS 0D,PL8 packed decimal
PG DC CL80' ' buffer
XDEC DS CL12 temp
BUF DC (NBUF)F'0' buf(nbuf)
VECT DC (NVECT)F'2' vect(nvect) init 2
YREGS
NBUF EQU 201 number of 5 decimals
NVECT EQU 3350 nvect=ceil(nbuf*50/3)
END PISPIG- Output:
3. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 50244 59455 34690 83026 42522 30825 33446 85035 26193 11881 71010 00313 78387 52886 58753 32083 81420 61717 76691 47303 59825 34904 28755 46873 11595 62863 88235 37875 93751 95778 18577 80532 17122 68066 13001 92787 66111 95909 21642 01989
AArch64 Assembly
/* ARM assembly AARCH64 Raspberry PI 3B */
/* program calculPi64.s */
/* for algorithm see a french site : http://www.yann-ollivier.org/pi/pi.php */
/************************************/
/* Constantes */
/************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
.equ NBVAL, 8400 //(2400 decimales) or 16800 (4800 décimales) or 33600 (9600 decimales)
.equ BASE, 10000
//.include "../ficmacros32.inc" // for debugging developper
/************************************/
/* Initialized data */
/************************************/
.data
szCarriageReturn: .asciz "\n"
szMessStart: .asciz "Program 64 bits start.\n"
szNBZero1: .asciz "0"
szNBZero2: .asciz "00"
szNBZero3: .asciz "000"
.align 2
/************************************/
/* UnInitialized data */
/************************************/
.bss
sZoneConv: .skip 24
ibuffer: .skip 4 * (NBVAL+1)
/************************************/
/* code section */
/************************************/
.text
.global main
main: // entry of program
ldr x0,qAdrszMessStart
bl affichageMess
ldr x7,iA
mov x0,x7
mov x1,#5
udiv x2,x0,x1 // A / 5
mov x1,#0
ldr x8,qAdribuffer
ldr x9,iC // C
1: // init start array
str w2,[x8,x1, lsl #2] // store A/5
add x1,x1,#1
cmp x1,x9
blt 1b
mov x10,#0 // E
2: // begin loop 1
mov x4,#0 // D
lsl x5,x9,#1 // G
mov x6,x9 // B
3: // loop 2
ldr w1,[x8,x6, lsl #2]
mul x1,x7,x1 //
add x4,x4,x1 // new D
sub x5,x5,#1 // g = g -1
mov x0,x4 // D
mov x1,x5 // G
udiv x2,x0,x1
msub x3,x1,x2,x0
str w3,[x8,x6, lsl #2] // D modulo G
mov x4,x2
sub x5,x5,#1 // G=G-1
subs x6,x6,#1 // B=B-1
ble 4f // end loop 2 ?
mul x4,x6,x4 // no compute new D
b 3b // and loop 2
4:
mov x0,x4 // D
mov x1,x7 // A
udiv x2,x0,x1
msub x3,x1,x2,x0
add x0,x2,x10 // D/A + E
bl afficherDecimale
mov x10,x3 // E=D modulo A
subs x9,x9,#14
bgt 2b // end loop 1 ?
100: // standard end of the program
mov x0, #0 // return code
mov x8,EXIT
svc 0 // perform the system call
qAdrsZoneConv: .quad sZoneConv
qAdrszCarriageReturn: .quad szCarriageReturn
qAdrszMessStart: .quad szMessStart
qAdribuffer: .quad ibuffer
iA: .quad BASE
iC: .quad NBVAL
/******************************************************************/
/* Display decimales */
/******************************************************************/
/* x0 contains decimale */
afficherDecimale:
stp x4,lr,[sp,-16]! // save registers
mov x4,x0
ldr x1,ival1
cmp x0,x1
bgt 3f
cmp x0,#99
ble 1f
ldr x0,qAdrszNBZero1 // display 1 zero
bl affichageMess
mov x0,x4
b 3f
1:
cmp x0,#9
ble 2f
ldr x0,qAdrszNBZero2 // display 2 zeroes
bl affichageMess
mov x0,x4
b 3f
2:
ldr x0,qAdrszNBZero3 // display 3 zeroes
bl affichageMess
mov x0,x4
3: // conversion and display
ldr x1,qAdrsZoneConv
bl conversion10 // decimal conversion
mov x2,#0
strb w2,[x1,x0]
ldr x0,qAdrsZoneConv
bl affichageMess
100:
ldp x4,lr,[sp],16 // restaur registers
ret
ival1: .quad 999
qAdrszNBZero1: .quad szNBZero1
qAdrszNBZero2: .quad szNBZero2
qAdrszNBZero3: .quad szNBZero3
/***************************************************/
/* ROUTINES INCLUDE */
/***************************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeARM64.inc"- Output:
Program 64 bits start. 314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349042875546873115956286388235378759375195778185778053217122680661300192787661119590921642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151557485724245415069595082953311686172785588907509838175463746493931925506040092770167113900984882401285836160356370766010471018194295559619894676783744944825537977472684710404753464620804668425906949129331367702898915210475216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992458631503028618297455570674983850549458858692699569092721079750930295532116534498720275596023648066549911988183479775356636980742654252786255181841757467289097777279380008164706001614524919217321721477235014144197356854816136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179049460165346680498862723279178608578438382796797668145410095388378636095068006422512520511739298489608412848862694560424196528502221066118630674427862203919494504712371378696095636437191728746776465757396241389086583264599581339047802759009946576407895126946839835259570982582262052248940772671947826848260147699090264013639443745530506820349625245174939965143142980919065925093722169646151570985838741059788595977297549893016175392846813826868386894277415599185592524595395943104997252468084598727364469584865383673622262609912460805124388439045124413654976278079771569143599770012961608944169486855584840635342207222582848864815845602850
Ada
uses same algorithm as Go solution, from http://web.comlab.ox.ac.uk/people/jeremy.gibbons/publications/spigot.pdf
- pi_digits.adb
with Ada.Command_Line;
with Ada.Text_IO;
with GNU_Multiple_Precision.Big_Integers;
with GNU_Multiple_Precision.Big_Rationals;
use GNU_Multiple_Precision;
procedure Pi_Digits is
type Int is mod 2 ** 64;
package Int_To_Big is new Big_Integers.Modular_Conversions (Int);
-- constants
Zero : constant Big_Integer := Int_To_Big.To_Big_Integer (0);
One : constant Big_Integer := Int_To_Big.To_Big_Integer (1);
Two : constant Big_Integer := Int_To_Big.To_Big_Integer (2);
Three : constant Big_Integer := Int_To_Big.To_Big_Integer (3);
Four : constant Big_Integer := Int_To_Big.To_Big_Integer (4);
Ten : constant Big_Integer := Int_To_Big.To_Big_Integer (10);
-- type LFT = (Integer, Integer, Integer, Integer
type LFT is record
Q, R, S, T : Big_Integer;
end record;
-- extr :: LFT -> Integer -> Rational
function Extr (T : LFT; X : Big_Integer) return Big_Rational is
use Big_Integers;
Result : Big_Rational;
begin
-- extr (q,r,s,t) x = ((fromInteger q) * x + (fromInteger r)) /
-- ((fromInteger s) * x + (fromInteger t))
Big_Rationals.Set_Numerator (Item => Result,
New_Value => T.Q * X + T.R,
Canonicalize => False);
Big_Rationals.Set_Denominator (Item => Result,
New_Value => T.S * X + T.T);
return Result;
end Extr;
-- unit :: LFT
function Unit return LFT is
begin
-- unit = (1,0,0,1)
return LFT'(Q => One, R => Zero, S => Zero, T => One);
end Unit;
-- comp :: LFT -> LFT -> LFT
function Comp (T1, T2 : LFT) return LFT is
use Big_Integers;
begin
-- comp (q,r,s,t) (u,v,w,x) = (q*u+r*w,q*v+r*x,s*u+t*w,s*v+t*x)
return LFT'(Q => T1.Q * T2.Q + T1.R * T2.S,
R => T1.Q * T2.R + T1.R * T2.T,
S => T1.S * T2.Q + T1.T * T2.S,
T => T1.S * T2.R + T1.T * T2.T);
end Comp;
-- lfts = [(k, 4*k+2, 0, 2*k+1) | k<-[1..]
K : Big_Integer := Zero;
function LFTS return LFT is
use Big_Integers;
begin
K := K + One;
return LFT'(Q => K,
R => Four * K + Two,
S => Zero,
T => Two * K + One);
end LFTS;
-- next z = floor (extr z 3)
function Next (T : LFT) return Big_Integer is
begin
return Big_Rationals.To_Big_Integer (Extr (T, Three));
end Next;
-- safe z n = (n == floor (extr z 4)
function Safe (T : LFT; N : Big_Integer) return Boolean is
begin
return N = Big_Rationals.To_Big_Integer (Extr (T, Four));
end Safe;
-- prod z n = comp (10, -10*n, 0, 1)
function Prod (T : LFT; N : Big_Integer) return LFT is
use Big_Integers;
begin
return Comp (LFT'(Q => Ten, R => -Ten * N, S => Zero, T => One), T);
end Prod;
procedure Print_Pi (Digit_Count : Positive) is
Z : LFT := Unit;
Y : Big_Integer;
Count : Natural := 0;
begin
loop
Y := Next (Z);
if Safe (Z, Y) then
Count := Count + 1;
Ada.Text_IO.Put (Big_Integers.Image (Y));
exit when Count >= Digit_Count;
Z := Prod (Z, Y);
else
Z := Comp (Z, LFTS);
end if;
end loop;
end Print_Pi;
N : Positive := 250;
begin
if Ada.Command_Line.Argument_Count = 1 then
N := Positive'Value (Ada.Command_Line.Argument (1));
end if;
Print_Pi (N);
end Pi_Digits;
output:
3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7
ALGOL 68
Note: This specimen retains the original Pascal coding style of code.
This codes uses 33 decimals places as a test case. Performance is O(2) based on the number of decimal places required.
#!/usr/local/bin/a68g --script #
INT base := 10;
MODE YIELDINT = PROC(INT)VOID;
PROC gen pi digits = (INT decimal places, YIELDINT yield)VOID:
BEGIN
INT nine = base - 1;
INT nines := 0, predigit := 0; # First predigit is a 0 #
[decimal places*10 OVER 3]#LONG# INT digits; # We need 3 times the digits to calculate #
FOR place FROM LWB digits TO UPB digits DO digits[place] := 2 OD; # Start with 2s #
FOR place TO decimal places + 1 DO
INT digit := 0;
FOR i FROM UPB digits BY -1 TO LWB digits DO # Work backwards #
INT x := #SHORTEN#(base*digits[i] + #LENG# digit*i);
digits[i] := x MOD (2*i-1);
digit := x OVER (2*i-1)
OD;
digits[LWB digits] := digit MOD base; digit OVERAB base;
nines :=
IF digit = nine THEN
nines + 1
ELSE
IF digit = base THEN
yield(predigit+1); predigit := 0 ;
FOR repeats TO nines DO yield(0) OD # zeros #
ELSE
IF place NE 1 THEN yield(predigit) FI; predigit := digit;
FOR repeats TO nines DO yield(nine) OD
FI;
0
FI
OD;
yield(predigit)
END;
main:(
INT feynman point = 762; # feynman point + 4 is a good test case #
# the 33rd decimal place is a shorter tricky test case #
INT test decimal places = UPB "3.1415926.......................502"-2;
INT width = ENTIER log(base*(1+small real*10));
# iterate throught the digits as they are being found #
# FOR INT digit IN # gen pi digits(test decimal places#) DO ( #,
## (INT digit)VOID: (
printf(($n(width)d$,digit))
)
# OD #);
print(new line)
)Output:
3141592653589793238462643383279502
ARM Assembly
/* ARM assembly Raspberry PI */
/* program calculPi.s Spigot algorithm */
/* conversion to Pascal */
/************************************/
/* Constantes */
/************************************/
/* for this file see task include a file in language ARM assembly*/
.include "../constantes.inc"
.equ NBVAL, 3000
.equ LEN, 10 * NBVAL / 3 @ 10000
//.include "../ficmacros32.inc" @ for debugging developper
/************************************/
/* Initialized data */
/************************************/
.data
szCarriageReturn: .asciz "\n"
szMessStart: .asciz "Program 32 bits start.\n"
szVal9: .asciz "9"
szVal0: .asciz "0"
.align 2
/************************************/
/* UnInitialized data */
/************************************/
.bss
sZoneConv: .skip 24
ibuffer: .skip 4 * (LEN+1)
/************************************/
/* code section */
/************************************/
.text
.global main
main: @ entry of program
ldr r0,iAdrszMessStart
bl affichageMess
ldr r12,iN
ldr r10,iLen
mov r1,#0
ldr r8,iAdribuffer
mov r2,#2
1: @ init start array loop
str r2,[r8,r1, lsl #2] @ store 2
add r1,#1
cmp r1,r10
blt 1b
mov r9,#0 @ nine
mov r7,#0 @ predigit
mov r6,#0 @ j
2: @ begin loop 1
mov r5,#0 @ q
sub r4,r10,#1 @ i
3: @ loop 2
ldr r0,[r8,r4, lsl #2] @ load value
mov r1,#10
mul r0,r1,r0 @ val * 10
mul r1,r5,r4 @ q *i
add r0,r1
lsl r1,r4,#1 @ divisor =i *2
sub r1,#1 @ - 1
bl division
str r3,[r8,r4, lsl #2] @ modulo 2i-1
mov r5,r2 @ q
subs r4,#1 @ decremente i
bgt 3b @ end loop 2
mov r0,r5
mov r1,#10
bl division
str r3,[r8,#4] @ poste 0 = q mod 10
mov r5,r2 @ q = q/10
cmp r5,#9
beq 7f
cmp r5,#10
beq 5f
mov r0,r7
bl afficherPredigit
mov r7,r5 @ predigit=q
cmp r9,#0 @ nine
beq 8f
mov r1,#1 @ else
4:
cmp r1,r9
bgt 41f
ldr r0,iAdrszVal9
bl affichageMess
add r1,#1
b 4b
41:
mov r9,#0 @ raz nine
b 8f
5: @ q = 10
add r0,r7,#1
bl afficherPredigit
mov r7,#0 @ predigit=0
mov r1,#1
6:
cmp r1,r9
bgt 61f
ldr r0,iAdrszVal0
bl affichageMess
add r1,#1
b 6b
61:
mov r9,#0 @ raz nine
b 8f
7:
add r9,#1
8:
add r6,#1
cmp r6,r12
ble 2b @ end loop 1 ?
mov r0,r7
bl afficherPredigit
100: @ standard end of the program
mov r0, #0 @ return code
mov r7, #EXIT @ request to exit program
svc 0 @ perform the system call
iAdrsZoneConv: .int sZoneConv
iAdrszCarriageReturn: .int szCarriageReturn
iAdrszMessStart: .int szMessStart
iAdribuffer: .int ibuffer
iN: .int NBVAL
iLen: .int LEN
iAdrszVal9: .int szVal9
iAdrszVal0: .int szVal0
/******************************************************************/
/* Display predigit */
/******************************************************************/
/* r0 contains decimale */
afficherPredigit:
push {r4,lr} @ save registers
mov r4,r0
ldr r1,iAdrsZoneConv
bl conversion10 @ decimal conversion
mov r2,#0
strb r2,[r1,r0]
ldr r0,iAdrsZoneConv
bl affichageMess
100:
pop {r4,pc} @ restaur registers
/***************************************************/
/* ROUTINES INCLUDE */
/***************************************************/
/* for this file see task include a file in language ARM assembly*/
.include "../affichage.inc"- Output:
Program 32 bits start. 03141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961
Arturo
q: 1
r: 0
t: 1
k: 1
n: 3
l: 3
d: 0
dotWritten: false
while [true][
if? (n*t) > (4*q)+r-t [
d: d+1
prints n
unless dotWritten [
prints "."
dotWritten: true
d: d+1
]
if 0 = d%80 -> prints "\n"
nr: 10*(r - n*t)
n: ((10*(r + 3*q)) / t) - 10*n
q: q*10
r: nr
]
else [
nr: (r + 2*q) * l
nn: ((q*(2 + 7*k)) + r*l) / (t*l)
q: q*k
t: t*l
l: l+2
k: k+1
n: nn
r: nr
]
]
- Output:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208 99862803482534211706798214808651328230664709384460955058223172535940812848111745 02841027019385211055596446229489549303819644288109756659334461284756482337867831 65271201909145648566923460348610454326648213393607260249141273724587006606315588 17488152092096282925409171536436789259036001133053054882046652138414695194151160 94330572703657595919530921861173819326117931051185480744623799627495673518857527 24891227938183011949129833673362440656643086021394946395224737190702179860943702 77053921717629317675238467481846766940513200056812714526356082778577134275778960 91736371787214684409012249534301465495853710507922796892589235420199561121290219 60864034418159813629774771309960518707211349999998372978049951059731732816096318 59502445945534690830264252230825334468503526193118817101000313783875288658753320 83814206171776691473035982534904287554687311595628638823537875937519577818577805 32171226806613001927876611195909216420198938095257201065485863278865936153381827 96823030195203530185296899577362259941389124972177528347913151557485724245415069 59508295331168617278558890750983817546374649393192550604009277016711390098488240 12858361603563707660104710181942955596198946767837449448255379774726847104047534 64620804668425906949129331367702898915210475216205696602405803815019351125338243 00355876402474964732639141992726042699227967823547816360093417216412199245863150 30286182974555706749838505494588586926995690927210797509302955321165344987202755 96023648066549911988183479775356636980742654252786255181841757467289097777279380 00816470600161452491921732172147723501414419735685481613611573525521334757418494 68438523323907394143334547762416862518983569485562099219222184272550254256887671 79049460165346680498862723279178608578438382796797668145410095388378636095068006 42251252051173929848960841284886269456042419652850222106611863067442786220391949 45047123713786960956364371917287467764657573962413890865832645995813390478027590 ...
AutoHotkey
Could be optimized with Ipp functions, but runs fast enough for me as-is. Does not work in AHKLx64.
#NoEnv
#SingleInstance, Force
SetBatchLines, -1
#Include mpl.ahk
dot:=".", i:=0
, MP_SET(q, "1")
, MP_SET(r, "0")
, MP_SET(t, "1")
, MP_SET(k, "1")
, MP_SET(n, "3")
, MP_SET(l, "3")
, MP_SET(ONE, "1")
, MP_SET(TWO, "2")
, MP_SET(THREE, "3")
, MP_SET(FOUR, "4")
, MP_SET(SEVEN, "7")
, MP_SET(TEN, "10")
Loop
{
MP_MUL(q4, q, FOUR)
, MP_ADD(q4r, q4, r)
, MP_SUB(q4rt, q4r, t)
, MP_MUL(tn, t, n)
If (MP_CMP(q4rt,tn) = -1)
{
s := MP_DEC(n) . dot
OutputDebug %s%
dot := ""
, i++
, MP_MUL(tn, t, n)
, MP_SUB(rtn, r, tn)
, MP_MUL(nr, rtn, TEN)
, MP_MUL(q3, q, THREE)
, MP_ADD(q3r, q3, r)
, MP_DIV(q3rt, remainder, q3r, t)
, MP_SUB(q3rtn, q3rt, n)
, MP_MUL(n, q3rtn, TEN)
, MP_MUL(tmp, q, TEN)
, MP_CPY(q, tmp)
, MP_CPY(r, nr)
}
Else
{
MP_MUL(q2, q, TWO)
, MP_ADD(q2r, q2, r)
, MP_MUL(nr, q2r, l)
, MP_MUL(k7, k, SEVEN)
, MP_ADD(k72, k7, TWO)
, MP_MUL(qk, q, k72)
, MP_MUL(rl, r, l)
, MP_ADD(qkrl, qk, rl)
, MP_MUL(tl, t, l)
, MP_DIV(nn, remainder, qkrl, tl)
, MP_MUL(tmp, q, k)
, MP_CPY(q, tmp)
, MP_MUL(tmp, t, l)
, MP_CPY(t, tmp)
, MP_ADD(tmp, l, TWO)
, MP_CPY(l, tmp)
, MP_ADD(tmp, k, ONE)
, MP_CPY(k, tmp)
, MP_CPY(n, nn)
, MP_CPY(r, nr)
}
}
bash
base=10;
gen_pi_digits(){
base_places=$1 yield=$2
((nine=base - 1));
local nines=0 predigit=0; # first predigit is_a 0 #
declare -a digits; # [base_places*10 over 3]#long# ; # we need 3 times the digits to calculate #
lwb_digits=1;
((upb_digits=base_places*10 / 3));
for((place=lwb_digits; place<=upb_digits; place++)); do digits[$place]=2; done; # start with 2s #
for((place=1; place<=base_places + 1; place++)); do
digit=0;
for((i=upb_digits; i>=lwb_digits; i--)); do # work backwards #
((x=base*digits[i] + digit*i));
((digits[i]=x%(2*i-1)));
((digit=x/(2*i-1)))
done;
((digits[lwb_digits]=digit % base)); ((digit/=base));
# nines := #
if((digit == nine)); then
((nines=nines + 1))
else
if((digit == base)); then
((out=predigit+1));
$yield $out; predigit=0;
for((repeats=1; repeats<=nines; repeats++)); do $yield 0; done
else
if((place != 1)); then $yield $predigit; fi; predigit=$digit;
for((repeats=1; repeats<=nines; repeats++)); do $yield $nine; done
fi;
nines=0
fi
done;
$yield $predigit
};
print_digit(){
echo -n $1
}
# feynman_point=762; # feynman_point + 4 is a good test case
# the 33rd decimal place is a shorter tricky test case #
target="3.1415926.......................502";
((test_decimal_places=${#target}-2)); # upb
# iterate throught the digits as they are being found #
gen_pi_digits $test_decimal_places print_digit
echo # (new_line)
BASIC
Applesoft BASIC
10 REM ADOPTED FROM COMMODORE BASIC
20 N = 100: REM N MAY BE INCREASED, BUT WILL SLOW EXECUTION
30 LN = INT(10*N/3)+16
40 ND = 1
50 DIM A(LN)
60 N9 = 0
70 PD = 0:REM FIRST PRE-DIGIT IS A 0
80 REM
90 FOR J = 1 TO LN
100 A(J-1) = 2:REM START WITH 2S
110 NEXT J
120 REM
130 FOR J = 1 TO N
140 Q = 0
150 FOR I = LN TO 1 STEP -1:REM WORK BACKWARDS
160 X = 10*A(I-1) + Q*I
170 A(I-1) = X - (2*I-1)*INT(X/(2*I-1)):REM X - INT ( X / Y) * Y
180 Q = INT(X/(2*I - 1))
190 NEXT I
200 A(0) = Q-10*INT(Q/10)
210 Q = INT(Q/10)
220 IF Q=9 THEN N9 = N9 + 1: GOTO 450
240 IF Q<>10 THEN GOTO 350
250 REM Q == 10
260 D = PD+1: GOSUB 500
270 IF N9 <= 0 THEN GOTO 320
280 FOR K = 1 TO N9
290 D = 0: GOSUB 500
300 NEXT K
310 REM END IF
320 PD = 0
330 N9 = 0
335 GOTO 450
340 REM Q <> 10
350 D = PD: GOSUB 500
360 PD = Q
370 IF N9 = 0 THEN GOTO 450
380 FOR K = 1 TO N9
390 D = 9: GOSUB 500
400 NEXT K
410 N9 = 0
450 NEXT J
460 PRINT PD
470 END
480 REM
490 REM OUTPUT DIGITS
500 IF ND=0 THEN PRINT D;: RETURN
510 IF D=0 THEN RETURN
520 PRINT D;".";
530 ND = 0
550 RETURN
Atari 8-bit
10 REM ADOPTED FROM COMMODORE BASIC
20 N = 100: REM N MAY BE INCREASED, BUT WILL SLOW EXECUTION
30 LN = INT(10*N/3)+16
40 ND = 1
50 DIM A(LN)
60 N9 = 0
70 PD = 0:REM FIRST PRE-DIGIT IS A 0
80 REM
90 FOR J = 1 TO LN
100 A(J-1) = 2:REM START WITH 2S
110 NEXT J
120 REM
130 FOR J = 1 TO N
140 Q = 0
150 FOR I = LN TO 1 STEP -1:REM WORK BACKWARDS
160 X = 10*A(I-1) + Q*I
170 A(I-1) = X - (2*I-1)*INT(X/(2*I-1)):REM X - INT ( X / Y) * Y
180 Q = INT(X/(2*I - 1))
190 NEXT I
200 A(0) = Q-10*INT(Q/10)
210 Q = INT(Q/10)
220 IF Q=9 THEN N9 = N9 + 1: GOTO 450
240 IF Q<>10 THEN GOTO 350
250 REM Q == 10
260 D = PD+1: GOSUB 500
270 IF N9 <= 0 THEN GOTO 320
280 FOR K = 1 TO N9
290 D = 0: GOSUB 500
300 NEXT K
310 REM END IF
320 PD = 0
330 N9 = 0
335 GOTO 450
340 REM Q <> 10
350 D = PD: GOSUB 500
360 PD = Q
370 IF N9 = 0 THEN GOTO 450
380 FOR K = 1 TO N9
390 D = 9: GOSUB 500
400 NEXT K
410 N9 = 0
450 NEXT J
460 PRINT PD
470 END
480 REM
490 REM OUTPUT DIGITS
500 IF ND=0 THEN PRINT D;: RETURN
510 IF D=0 THEN RETURN
520 PRINT D;".";
530 ND = 0
550 RETURN
BASIC256
below, and originally published by Stanley Rabinowitz in [1].
cls
n =1000
len = 10*n \ 4
needdecimal = true
dim a(len)
nines = 0
predigit = 0 # {First predigit is a 0}
for j = 1 to len
a[j-1] = 2 # {Start with 2s}
next j
for j = 1 to n
q = 0
for i = len to 1 step -1
# {Work backwards}
x = 10*a[i-1] + q*i
a[i-1] = x % (2*i - 1)
q = x \ (2*i - 1)
next i
a[0] = q % 10
q = q \ 10
if q = 9 then
nines = nines + 1
else
if q = 10 then
d = predigit+1: gosub outputd
if nines > 0 then
for k = 1 to nines
d = 0: gosub outputd
next k
end if
predigit = 0
nines = 0
else
d = predigit: gosub outputd
predigit = q
if nines <> 0 then
for k = 1 to nines
d = 9: gosub outputd
next k
nines = 0
end if
end if
end if
next j
print predigit
end
outputd:
if needdecimal then
if d = 0 then return
print d + ".";
needdecimal = false
else
print d;
end if
returnOutput:
3.14159265358979323846264338327950288419716939937510582097494459230781...
Chipmunk Basic
100 REM adopted from Applesoft BASIC
110 n = 100 : rem n may be increased, but will slow execution
120 ln = int(10*n/3)+16
130 nd = 1
140 dim a(ln)
150 n9 = 0
160 pd = 0 : rem First pre-digit is a 0
170 rem
180 for j = 1 to ln
190 a(j-1) = 2 : rem Start wirh 2S
200 next j
210 rem
220 for j = 1 to n
230 q = 0
240 for i = ln to 1 step -1 : rem Work backwards
250 x = 10*a(i-1)+q*i
260 a(i-1) = x-(2*i-1)*int(x/(2*i-1)) : rem X - Int ( X / Y) * Y
270 q = int(x/(2*i-1))
280 next i
290 a(0) = q-10*int(q/10)
300 q = int(q/10)
310 if q = 9 then n9 = n9+1 : goto 510
320 if q <> 10 then goto 440
330 rem Q == 10
340 d = pd+1 : gosub 560
350 if n9 <= 0 then goto 400
360 for k = 1 to n9
370 d = 0 : gosub 560
380 next k
390 rem End If
400 pd = 0
410 n9 = 0
420 goto 510
430 rem Q <> 10
440 d = pd : gosub 560
450 pd = q
460 if n9 = 0 then goto 510
470 for k = 1 to n9
480 d = 9 : gosub 560
490 next k
500 n9 = 0
510 next j
520 print str$(pd)
530 end
540 rem
550 rem Output digits
560 if nd = 0 then print str$(d); : return
570 if d = 0 then return
580 print str$(d);".";
590 nd = 0
600 return
Commodore BASIC
This works with Commodore Basic V2
10 PRINT CHR$(147)
20 N = 100: REM N MAY BE INCREASED, BUT WILL SLOW EXECUTION
30 LN = INT(10*N/3)+16
40 ND = 1
50 DIM A(LN)
60 N9 = 0
70 PD = 0:REM FIRST PRE-DIGIT IS A 0
80 REM
90 FOR J = 1 TO LN
100 A(J-1) = 2:REM START WITH 2S
110 NEXT J
120 REM
130 FOR J = 1 TO N
140 Q = 0
150 FOR I = LN TO 1 STEP -1:REM WORK BACKWARDS
160 X = 10*A(I-1) + Q*I
170 A(I-1) = X - (2*I-1)*INT(X/(2*I-1)):REM X - INT ( X / Y) * Y
180 Q = INT(X/(2*I - 1))
190 NEXT I
200 A(0) = Q-10*INT(Q/10)
210 Q = INT(Q/10)
220 IF Q=9 THEN N9 = N9 + 1: GOTO 450
240 IF Q<>10 THEN GOTO 350
250 REM Q == 10
260 D = PD+1: GOSUB 500
270 IF N9 <= 0 THEN GOTO 320
280 FOR K = 1 TO N9
290 D = 0: GOSUB 500
300 NEXT K
310 REM END IF
320 PD = 0
330 N9 = 0
335 GOTO 450
340 REM Q <> 10
350 D = PD: GOSUB 500
360 PD = Q
370 IF N9 = 0 THEN GOTO 450
380 FOR K = 1 TO N9
390 D = 9: GOSUB 500
400 NEXT K
410 N9 = 0
450 NEXT J
460 PRINT RIGHT$(STR$(PD),1)
470 END
480 REM
490 REM OUTPUT DIGITS
500 IF ND=0 THEN PRINT RIGHT$(STR$(D),1);: RETURN
510 IF D=0 THEN RETURN
520 PRINT RIGHT$(STR$(D),1);".";
530 ND = 0
550 RETURN
GW-BASIC
The Chipmunk Basic solution works without any changes.
Integer Basic
Integer version was derived from the Pascal_spigot without any optimisation. It is more than 33% faster than the Applesoft version since it runs natively with integers.
10 REM PI CALCULATION WITH SPIGOT
100 N=100: REM MAX N=260 TO AVOID OVERFLOW
110 LEN=(10*N)/3
120 J=0:K=0:Q=0:NINES=0:PREDIGIT=0
125 DIM A(LEN)
130 REM VARIABLES FOR THE SUB
140 RESULT=0:I=0:X=0
200 REM MAIN
210 FOR J=1 TO LEN:A(J)=2: NEXT J
220 REM START WITH 222...
230 NINES=0
240 PREDIGIT=0
250 REM FIRST PREDIGIT =0
260 FOR J=1 TO N
270 I=N-J: GOSUB 1000:Q=RESULT
275 A(1)=Q MOD 10
280 Q=Q/10
290 IF Q=9 THEN NINES=NINES+1
300 IF Q=10 THEN GOSUB 800
310 IF (Q<>9) AND (Q<>10) THEN GOSUB 900
320 NEXT J
330 PRINT PREDIGIT
340 END
800 PRINT PREDIGIT+1;
805 IF NINES=0 THEN GOTO 820
810 FOR K=1 TO NINES: PRINT 0;: NEXT K
820 PREDIGIT=0:NINES=0
830 RETURN
900 PRINT PREDIGIT;
910 PREDIGIT=Q
920 IF NINES<>0 THEN GOSUB 950
930 RETURN
950 FOR K=1 TO NINES: PRINT 9;: NEXT K
960 NINES=0
970 RETURN
1000 I=I*10/3+16
1010 IF I>LEN THEN I=LEN
1020 RESULT=0
1030 REM REPEAT
1040 X=10*A(I)+RESULT*I
1050 RESULT=X/(2*I-1)
1060 A(I)=X MOD (2*I-1)
1070 I=I-1
1080 IF I>0 THEN GOTO 1040
1090 RETURN
IS-BASIC
100 PROGRAM "PI.bas"
110 LET N=100 ! Nuber of digits
120 LET LN=INT(10*N/3)+16
130 DIM A(LN)
140 LET PD,N9=0:LET ND=1
150 FOR J=1 TO LN
160 LET A(J-1)=2
170 NEXT
180 FOR J=1 TO N
190 LET Q=0
200 FOR I=LN TO 1 STEP-1
210 LET X=10*A(I-1)+Q*I
220 LET A(I-1)=X-(2*I-1)*INT(X/(2*I-1))
230 LET Q=INT(X/(2*I-1))
240 NEXT
250 LET A(0)=Q-10*INT(Q/10)
260 LET Q=INT(Q/10)
270 SELECT CASE Q
280 CASE 9
290 LET N9=N9+1
300 CASE 10
310 LET D=PD+1:CALL WRITE
320 IF N9>0 THEN
330 FOR K=1 TO N9
340 LET D=0:CALL WRITE
350 NEXT
360 END IF
370 LET PD,N9=0
380 CASE ELSE
390 LET D=PD:CALL WRITE
400 LET PD=Q
410 IF N9<>0 THEN
420 FOR K=1 TO N9
430 LET D=9:CALL WRITE
440 NEXT
450 LET N9=0
460 END IF
470 END SELECT
480 NEXT
490 PRINT STR$(PD)(1)
500 END
510 DEF WRITE
520 IF ND=0 THEN
530 PRINT STR$(D)(1);
540 ELSE IF D<>0 THEN
550 PRINT STR$(D)(1);".";
560 LET ND=0
570 END IF
580 END DEFMSX Basic
The Chipmunk Basic solution works without any changes.
Osborne 1 MBASIC
Osborne 1 program is slightly different to allow it to keep the numbers all on the main screen rather than scrolling off to the right...
10 REM ADOPTED FROM COMMODORE BASIC
15 CR=0
20 N = 100: REM N MAY BE INCREASED, BUT WILL SLOW EXECUTION
30 LN = INT(10*N/3)+16
40 ND = 1
50 DIM A(LN)
60 N9 = 0
70 PD = 0:REM FIRST PRE-DIGIT IS A 0
80 REM
90 FOR J = 1 TO LN
100 A(J-1) = 2:REM START WITH 2S
110 NEXT J
120 REM
130 FOR J = 1 TO N
140 Q = 0
150 FOR I = LN TO 1 STEP -1:REM WORK BACKWARDS
160 X = 10*A(I-1) + Q*I
170 A(I-1) = X - (2*I-1)*INT(X/(2*I-1)):REM X - INT ( X / Y) * Y
180 Q = INT(X/(2*I - 1))
190 NEXT I
200 A(0) = Q-10*INT(Q/10)
210 Q = INT(Q/10)
220 IF Q=9 THEN N9 = N9 + 1: GOTO 450
240 IF Q<>10 THEN GOTO 350
250 REM Q == 10
260 D = PD+1: GOSUB 500
270 IF N9 <= 0 THEN GOTO 320
280 FOR K = 1 TO N9
290 D = 0: GOSUB 500
300 NEXT K
310 REM END IF
320 PD = 0
330 N9 = 0
335 GOTO 450
340 REM Q <> 10
350 D = PD: GOSUB 500
360 PD = Q
370 IF N9 = 0 THEN GOTO 450
380 FOR K = 1 TO N9
390 D = 9: GOSUB 500
400 NEXT K
410 N9 = 0
450 NEXT J
460 PRINT STR$(PD)
470 END
480 REM
490 REM OUTPUT DIGITS
500 IF ND=0 AND CR<22 THEN PRINT STR$(D);:CR=CR+1: RETURN
502 IF ND=0 AND CR=22 THEN PRINT STR$(D):CR=0: RETURN
510 IF D=0 THEN RETURN
520 PRINT STR$(D);".";
530 ND = 0
550 RETURN
TRS-80 Model 4 BASIC
10 REM ADOPTED FROM COMMODORE BASIC
20 N = 100: REM N MAY BE INCREASED, BUT WILL SLOW EXECUTION
30 LN = INT(10*N/3)+16
40 ND = 1
50 DIM A(LN)
60 N9 = 0
70 PD = 0:REM FIRST PRE-DIGIT IS A 0
80 REM
90 FOR J = 1 TO LN
100 A(J-1) = 2:REM START WITH 2S
110 NEXT J
120 REM
130 FOR J = 1 TO N
140 Q = 0
150 FOR I = LN TO 1 STEP -1:REM WORK BACKWARDS
160 X = 10*A(I-1) + Q*I
170 A(I-1) = X - (2*I-1)*INT(X/(2*I-1)):REM X - INT ( X / Y) * Y
180 Q = INT(X/(2*I - 1))
190 NEXT I
200 A(0) = Q-10*INT(Q/10)
210 Q = INT(Q/10)
220 IF Q=9 THEN N9 = N9 + 1: GOTO 450
240 IF Q<>10 THEN GOTO 350
250 REM Q == 10
260 D = PD+1: GOSUB 500
270 IF N9 <= 0 THEN GOTO 320
280 FOR K = 1 TO N9
290 D = 0: GOSUB 500
300 NEXT K
310 REM END IF
320 PD = 0
330 N9 = 0
335 GOTO 450
340 REM Q <> 10
350 D = PD: GOSUB 500
360 PD = Q
370 IF N9 = 0 THEN GOTO 450
380 FOR K = 1 TO N9
390 D = 9: GOSUB 500
400 NEXT K
410 N9 = 0
450 NEXT J
460 PRINT STR$(PD)
470 END
480 REM
490 REM OUTPUT DIGITS
500 IF ND=0 THEN PRINT STR$(D);: RETURN
510 IF D=0 THEN RETURN
520 PRINT STR$(D);".";
530 ND = 0
550 RETURN
BBC BASIC
BASIC version
WIDTH 80
M% = (HIMEM-END-1000) / 4
DIM B%(M%)
FOR I% = 0 TO M% : B%(I%) = 20 : NEXT
E% = 0
L% = 2
FOR C% = M% TO 14 STEP -7
D% = 0
A% = C%*2-1
FOR P% = C% TO 1 STEP -1
D% = D%*P% + B%(P%)*&64
B%(P%) = D% MOD A%
D% DIV= A%
A% -= 2
NEXT
CASE TRUE OF
WHEN D% = 99: E% = E% * 100 + D% : L% += 2
WHEN C% = M%: PRINT ;(D% DIV 100) / 10; : E% = D% MOD 100
OTHERWISE:
PRINT RIGHT$(STRING$(L%,"0") + STR$(E% + D% DIV 100),L%);
E% = D% MOD 100 : L% = 2
ENDCASE
NEXT
Assembler version
The first 250,000 digits output have been verified.
DIM P% 32
[OPT 2 :.pidig mov ebp,eax :.pi1 imul edx,ecx : mov eax,[ebx+ecx*4]
imul eax,100 : add eax,edx : cdq : div ebp : mov [ebx+ecx*4],edx
mov edx,eax : sub ebp,2 : loop pi1 : mov eax,edx : ret :]
WIDTH 80
M% = (HIMEM-END-1000) / 4
DIM B%(M%) : B% = ^B%(0)
FOR I% = 0 TO M% : B%(I%) = 20 : NEXT
E% = 0
L% = 2
FOR C% = M% TO 14 STEP -7
D% = 0
A% = C%*2-1
D% = USR(pidig)
CASE TRUE OF
WHEN D% = 99: E% = E% * 100 + D% : L% += 2
WHEN C% = M%: PRINT ;(D% DIV 100) / 10; : E% = D% MOD 100
OTHERWISE:
PRINT RIGHT$(STRING$(L%,"0") + STR$(E% + D% DIV 100),L%);
E% = D% MOD 100 : L% = 2
ENDCASE
NEXT
Output:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208 99862803482534211706798214808651328230664709384460955058223172535940812848111745 02841027019385211055596446229489549303819644288109756659334461284756482337867831 65271201909145648566923460348610454326648213393607260249141273724587006606315588 17488152092096282925409171536436789259036001133053054882046652138414695194151160 94330572703657595919530921861173819326117931051185480744623799627495673518857527 24891227938183011949129833673362440656643086021394946395224737190702179860943702 77053921717629317675238467481846766940513200056812714526356082778577134275778960 91736371787214684409012249534301465495853710507922796892589235420199561121290219 60864034418159813629774771309960518707211349999998372978049951059731732816096318 ....
bc
The digits of Pi are printed 20 per line, by successively recomputing pi with higher precision. The computation is not accurate to the entire scale (for example, scale = 4; 4*a(1) prints 3.1412 instead of the expected 3.1415), so the program includes two excess digits in the scale. Fixed number of guarding digits will eventually fail because Pi can contain arbitrarily long sequence of consecutive 9s (or consecutive 0s), though for this task it might not matter in practice. The program proceeds more and more slowly but exploits bc's unlimited precision arithmetic.
The program uses three features of GNU bc: long variable names, # comments (for the #! line), and the print command (for zero padding).
#!/usr/bin/bc -l
scaleinc= 20
define zeropad ( n ) {
auto m
for ( m= scaleinc - 1; m > 0; --m ) {
if ( n < 10^m ) {
print "0"
}
}
return ( n )
}
wantscale= scaleinc - 2
scale= wantscale + 2
oldpi= 4*a(1)
scale= wantscale
oldpi= oldpi / 1
oldpi
while( 1 ) {
wantscale= wantscale + scaleinc
scale= wantscale + 2
pi= 4*a(1)
scale= 0
digits= ((pi - oldpi) * 10^wantscale) / 1
zeropad( digits )
scale= wantscale
oldpi= pi / 1
}
Output:
3.141592653589793238 46264338327950288419 71693993751058209749 44592307816406286208 99862803482534211706 79821480865132823066 47093844609550582231 72535940812848111745 02841027019385211055 59644622948954930381 96442881097566593344 61284756482337867831 65271201909145648566 92346034861045432664 82133936072602491412 73724587006606315588 17488152092096282925 40917153643678925903 60011330530548820466 52138414695194151160 94330572703657595919 ....
Bracmat
( pi
= f,q r t k n l,first
. !arg:((=?f),?q,?r,?t,?k,?n,?l)
& yes:?first
& whl
' ( 4*!q+!r+-1*!t+-1*!n*!t:<0
& f$!n
& ( !first:yes
& f$"."
& no:?first
|
)
& "compute and update variables for next cycle"
& 10*(!r+-1*!n*!t):?nr
& div$(10*(3*!q+!r).!t)+-10*!n:?n
& !q*10:?q
& !nr:?r
| "compute and update variables for next cycle"
& (2*!q+!r)*!l:?nr
& div$(!q*(7*!k+2)+!r*!l.!t*!l):?nn
& !q*!k:?q
& !t*!l:?t
& !l+2:?l
& !k+1:?k
& !nn:?n
& !nr:?r
)
)
& pi$((=.put$!arg),1,0,1,1,3,3)Output:
3.1415926535897932384626433832795028841971693993751058209749445923078164062 862089986280348253421170679821480865132823066470938446095505822317253594081 284811174502841027019385211055596446229489549303819644288109756659334461284 756482337867831652712019091456485669234603486104543266482133936072602491412 73724587006606315588174881520...
C
There are many ways to do this, with quite different performance profiles. A simple measurement of 6 programs:
| Digits | Spigot 1 | Spigot 2 | Machin 1 | Machin 2 | AGM | Chudnovsky |
|---|---|---|---|---|---|---|
| 1,000 | 0.008 | 0.009 | 0.001 | 0.001 | 0.000 | 0.000 |
| 10,000 | 0.402 | 0.589 | 0.020 | 0.016 | 0.003 | 0.002 |
| 100,000 | 39.400 | 85.600 | 1.740 | 1.480 | 0.084 | 0.002 |
| 1,000,000 | 177.900 | 156.800 | 1.474 | 0.333 | ||
| 10,000,000 | 25.420 | 5.715 |
- Spigot 1: plain C (no GMP), modified Winter/Flammenkamp, correct to 1+M digits
- Spigot 2: C+GMP, as used in Computer Language Benchmarks Game
- Machin 1: C+GMP, shown below
- Machin 2: C+GMP, as below but using Chien-Lih 1997 formula
- AGM: C+GMP, essentially from the Arithmetic-geometric mean/Calculate Pi task. This has performance only slightly slower than MPFR.
- Chudnovsky: Hanhong Xue's code from GMP web site.
Using Machin's formula. The "continuous printing" part is silly: the algorithm really calls for a preset number of digits, so the program repeatedly calculates Pi digits with increasing length and chop off leading digits already displayed. But it's still faster than the unbounded Spigot method by an order of magnitude, at least for the first 100k digits.
#include <stdio.h>
#include <stdlib.h>
#include <gmp.h>
mpz_t tmp1, tmp2, t5, t239, pows;
void actan(mpz_t res, unsigned long base, mpz_t pows)
{
int i, neg = 1;
mpz_tdiv_q_ui(res, pows, base);
mpz_set(tmp1, res);
for (i = 3; ; i += 2) {
mpz_tdiv_q_ui(tmp1, tmp1, base * base);
mpz_tdiv_q_ui(tmp2, tmp1, i);
if (mpz_cmp_ui(tmp2, 0) == 0) break;
if (neg) mpz_sub(res, res, tmp2);
else mpz_add(res, res, tmp2);
neg = !neg;
}
}
char * get_digits(int n, size_t* len)
{
mpz_ui_pow_ui(pows, 10, n + 20);
actan(t5, 5, pows);
mpz_mul_ui(t5, t5, 16);
actan(t239, 239, pows);
mpz_mul_ui(t239, t239, 4);
mpz_sub(t5, t5, t239);
mpz_ui_pow_ui(pows, 10, 20);
mpz_tdiv_q(t5, t5, pows);
*len = mpz_sizeinbase(t5, 10);
return mpz_get_str(0, 0, t5);
}
int main(int c, char **v)
{
unsigned long accu = 16384, done = 0;
size_t got;
char *s;
mpz_init(tmp1);
mpz_init(tmp2);
mpz_init(t5);
mpz_init(t239);
mpz_init(pows);
while (1) {
s = get_digits(accu, &got);
/* write out digits up to the last one not preceding a 0 or 9*/
got -= 2; /* -2: length estimate may be longer than actual */
while (s[got] == '0' || s[got] == '9') got--;
printf("%.*s", (int)(got - done), s + done);
free(s);
done = got;
/* double the desired digits; slows down at least cubically */
accu *= 2;
}
return 0;
}
C#
using System;
using System.Numerics;
namespace PiCalc {
internal class Program {
private readonly BigInteger FOUR = new BigInteger(4);
private readonly BigInteger SEVEN = new BigInteger(7);
private readonly BigInteger TEN = new BigInteger(10);
private readonly BigInteger THREE = new BigInteger(3);
private readonly BigInteger TWO = new BigInteger(2);
private BigInteger k = BigInteger.One;
private BigInteger l = new BigInteger(3);
private BigInteger n = new BigInteger(3);
private BigInteger q = BigInteger.One;
private BigInteger r = BigInteger.Zero;
private BigInteger t = BigInteger.One;
public void CalcPiDigits() {
BigInteger nn, nr;
bool first = true;
while (true) {
if ((FOUR*q + r - t).CompareTo(n*t) == -1) {
Console.Write(n);
if (first) {
Console.Write(".");
first = false;
}
nr = TEN*(r - (n*t));
n = TEN*(THREE*q + r)/t - (TEN*n);
q *= TEN;
r = nr;
} else {
nr = (TWO*q + r)*l;
nn = (q*(SEVEN*k) + TWO + r*l)/(t*l);
q *= k;
t *= l;
l += TWO;
k += BigInteger.One;
n = nn;
r = nr;
}
}
}
private static void Main(string[] args) {
new Program().CalcPiDigits();
}
}
}
Adopted Version:
using System;
using System.Collections.Generic;
using System.Linq;
using System.Numerics;
namespace EnumeratePi {
class Program {
private const int N = 60;
private const string ZS = " +-";
static void Main() {
Console.WriteLine("Digits of PI");
Console.WriteLine(new string('=', N + 13));
Console.WriteLine("Decimal : {0}", string.Concat(PiDigits(10).Take(N).Select(_ => _.ToString("d"))));
Console.WriteLine("Binary : {0}", string.Concat(PiDigits(2).Take(N).Select(_ => _.ToString("d"))));
Console.WriteLine("Quaternary : {0}", string.Concat(PiDigits(4).Take(N).Select(_ => _.ToString("d"))));
Console.WriteLine("Octal : {0}", string.Concat(PiDigits(8).Take(N).Select(_ => _.ToString("d"))));
Console.WriteLine("Hexadecimal: {0}", string.Concat(PiDigits(16).Take(N).Select(_ => _.ToString("x"))));
Console.WriteLine("Alphabetic : {0}", string.Concat(PiDigits(26).Take(N).Select(_ => (char) ('A' + _))));
Console.WriteLine("Fun : {0}", string.Concat(PiDigits(ZS.Length).Take(N).Select(_ => ZS[(int)_])));
Console.WriteLine("Nibbles : {0}", string.Concat(PiDigits(0x10).Take(N/2).Select(_ => string.Format("{0:x1} ", _))));
Console.WriteLine("Bytes : {0}", string.Concat(PiDigits(0x100).Take(N/3).Select(_ => string.Format("{0:x2} ", _))));
Console.WriteLine("Words : {0}", string.Concat(PiDigits(0x10000).Take(N/5).Select(_ => string.Format("{0:x4} ", _))));
Console.WriteLine("Dwords : {0}", string.Concat(PiDigits(0x100000000).Take(N/9).Select(_ => string.Format("{0:x8} ", _))));
Console.WriteLine(new string('=', N + 13));
Console.WriteLine("* press any key to exit *");
Console.ReadKey();
}
/// <summary>Enumerates the digits of PI.</summary>
/// <param name="b">Base of the Numeral System to use for the resulting digits (default = Base.Decimal (10)).</param>
/// <returns>The digits of PI.</returns>
static IEnumerable<long> PiDigits(long b = 10) {
BigInteger
k = 1,
l = 3,
n = 3,
q = 1,
r = 0,
t = 1
;
// skip integer part
var nr = b * (r - t * n);
n = b * (3 * q + r) / t - b * n;
q *= b;
r = nr;
for (; ; ) {
var tn = t * n;
if (4 * q + r - t < tn) {
yield return (long)n;
nr = b * (r - tn);
n = b * (3 * q + r) / t - b * n;
q *= b;
} else {
t *= l;
nr = (2 * q + r) * l;
var nn = (q * (7 * k) + 2 + r * l) / t;
q *= k;
l += 2;
++k;
n = nn;
}
r = nr;
}
}
}
}
- Output:
Digits of PI ========================================================================= Decimal : 141592653589793238462643383279502884197169399375105820974944 Binary : 001001000011111101101010100010001000010110100011000010001101 Quaternary : 021003331222202020112203002031030103012120220232000313001303 Octal : 110375524210264302151423063050560067016321122011160210514763 Hexadecimal: 243f6a8885a308d313198a2e03707344a4093822299f31d0082efa98ec4e Alphabetic : DRSQLOLYRTRODNLHNQTGKUDQGTUIRXNEQBCKBSZIVQQVGDMELMUEXROIQIYA Fun : + -++ +---- + -++ -+++++ --+----- +++- +-+-+-+- +-++ + Nibbles : 2 4 3 f 6 a 8 8 8 5 a 3 0 8 d 3 1 3 1 9 8 a 2 e 0 3 7 0 7 3 Bytes : 24 3f 6a 88 85 a3 08 d3 13 19 8a 2e 03 70 73 44 a4 09 38 22 Words : 243f 6a88 85a3 08d3 1319 8a2e 0370 7344 a409 3822 299f 31d0 Dwords : 243f6a88 85a308d3 13198a2e 03707344 a4093822 299f31d0 ========================================================================= * press any key to exit *
C++
#include <iostream>
#include <boost/multiprecision/cpp_int.hpp>
using namespace boost::multiprecision;
class Gospers
{
cpp_int q, r, t, i, n;
public:
// use Gibbons spigot algorith based on the Gospers series
Gospers() : q{1}, r{0}, t{1}, i{1}
{
++*this; // move to the first digit
}
// the ++ prefix operator will move to the next digit
Gospers& operator++()
{
n = (q*(27*i-12)+5*r) / (5*t);
while(n != (q*(675*i-216)+125*r)/(125*t))
{
r = 3*(3*i+1)*(3*i+2)*((5*i-2)*q+r);
q = i*(2*i-1)*q;
t = 3*(3*i+1)*(3*i+2)*t;
i++;
n = (q*(27*i-12)+5*r) / (5*t);
}
q = 10*q;
r = 10*r-10*n*t;
return *this;
}
// the dereference operator will give the current digit
int operator*()
{
return (int)n;
}
};
int main()
{
Gospers g;
std::cout << *g << "."; // print the first digit and the decimal point
for(;;) // run forever
{
std::cout << *++g; // increment to the next digit and print
}
}
- Output:
3.1415926535897932384626433832795028841971693993751058209749445923078164062862 . . .
Clojure
(ns pidigits
(:gen-class))
(def calc-pi
; integer division rounding downwards to -infinity
(let [div (fn [x y] (long (Math/floor (/ x y))))
; Computations performed after yield clause in Python code
update-after-yield (fn [[q r t k n l]]
(let [nr (* 10 (- r (* n t)))
nn (- (div (* 10 (+ (* 3 q) r)) t) (* 10 n))
nq (* 10 q)]
[nq nr t k nn l]))
; Update of else clause in Python code: if (< (- (+ (* 4 q) r) t) (* n t))
update-else (fn [[q r t k n l]]
(let [nr (* (+ (* 2 q) r) l)
nn (div (+ (* q 7 k) 2 (* r l)) (* t l))
nq (* k q)
nt (* l t)
nl (+ 2 l)
nk (+ 1 k)]
[nq nr nt nk nn nl]))
; Compute the lazy sequence of pi digits translating the Python code
pi-from (fn pi-from [[q r t k n l]]
(if (< (- (+ (* 4 q) r) t) (* n t))
(lazy-seq (cons n (pi-from (update-after-yield [q r t k n l]))))
(recur (update-else [q r t k n l]))))]
; Use Clojure big numbers to perform the math (avoid integer overflow)
(pi-from [1N 0N 1N 1N 3N 3N])))
;; Indefinitely Output digits of pi, with 40 characters per line
(doseq [[i q] (map-indexed vector calc-pi)]
(when (= (mod i 40) 0)
(println))
(print q))
- Output:
3141592653589793238462643383279502884197 1693993751058209749445923078164062862089 9862803482534211706798214808651328230664 7093844609550582231725359408128481117450 ...
Common Lisp
(defun pi-spigot ()
(labels
((g (q r t1 k n l)
(cond
((< (- (+ (* 4 q) r) t1)
(* n t1))
(princ n)
(g (* 10 q)
(* 10 (- r (* n t1)))
t1
k
(- (floor (/ (* 10 (+ (* 3 q) r))
t1))
(* 10 n))
l))
(t
(g (* q k)
(* (+ (* 2 q) r) l)
(* t1 l)
(+ k 1)
(floor (/ (+ (* q (+ (* 7 k) 2))
(* r l))
(* t1 l)))
(+ l 2))))))
(g 1 0 1 1 3 3)))
- Output:
CL-USER> (pi-spigot) 3141592653589793238462643383279502884197169399375105820974944592307816406286 ...
Crystal
require "big"
def pi
q, r, t, k, n, l = [1, 0, 1, 1, 3, 3].map { |n| BigInt.new(n) }
dot_written = false
loop do
if 4*q + r - t < n*t
yield n
unless dot_written
yield '.'
dot_written = true
end
nr = 10*(r - n*t)
n = ((10*(3*q + r)) / t) - 10*n
q *= 10
r = nr
else
nr = (2*q + r) * l
nn = (q*(7*k + 2) + r*l) / (t*l)
q *= k
t *= l
l += 2
k += 1
n = nn
r = nr
end
end
end
pi { |digit_or_dot| print digit_or_dot; STDOUT.flush }
- Output:
3.141592653589793238462643383279502884197169399375105820974944592307816406286 ...
D
This modified Spigot algorithm does not continue infinitely, because its required memory grow as the number of digits need to print.
import std.stdio, std.conv, std.string;
struct PiDigits {
immutable uint nDigits;
int opApply(int delegate(ref string /*chunk of pi digit*/) dg){
// Maximum width for correct output, for type ulong.
enum size_t width = 9;
enum ulong scale = 10UL ^^ width;
enum ulong initDigit = 2UL * 10UL ^^ (width - 1);
enum string formatString = "%0" ~ text(width) ~ "d";
immutable size_t len = 10 * nDigits / 3;
auto arr = new ulong[len];
arr[] = initDigit;
ulong carry;
foreach (i; 0 .. nDigits / width) {
ulong sum;
foreach_reverse (j; 0 .. len) {
auto quo = sum * (j + 1) + scale * arr[j];
arr[j] = quo % (j*2 + 1);
sum = quo / (j*2 + 1);
}
auto yield = format(formatString, carry + sum/scale);
if (dg(yield))
break;
carry = sum % scale;
}
return 0;
}
}
void main() {
foreach (d; PiDigits(100))
writeln(d);
}
- Output:
314159265 358979323 846264338 327950288 419716939 937510582 097494459 230781640 628620899 862803482 534211706
Alternative version
import std.stdio, std.bigint;
void main() {
int ndigits = 0;
auto q = BigInt(1);
auto r = BigInt(0);
auto t = q;
auto k = q;
auto n = BigInt(3);
auto l = n;
bool first = true;
while (ndigits < 1_000) {
if (4 * q + r - t < n * t) {
write(n); ndigits++;
if (ndigits % 70 == 0) writeln();
if (first) { first = false; write('.'); }
auto nr = 10 * (r - n * t);
n = ((10 * (3 * q + r)) / t) - 10 * n;
q *= 10;
r = nr;
} else {
auto nr = (2 * q + r) * l;
auto nn = (q * (7 * k + 2) + r * l) / (t * l);
q *= k;
t *= l;
l += 2;
k++;
n = nn;
r = nr;
}
}
}
- Output:
3.141592653589793238462643383279502884197169399375105820974944592307816 4062862089986280348253421170679821480865132823066470938446095505822317 2535940812848111745028410270193852110555964462294895493038196442881097 5665933446128475648233786783165271201909145648566923460348610454326648 2133936072602491412737245870066063155881748815209209628292540917153643 6789259036001133053054882046652138414695194151160943305727036575959195 3092186117381932611793105118548074462379962749567351885752724891227938 1830119491298336733624406566430860213949463952247371907021798609437027 7053921717629317675238467481846766940513200056812714526356082778577134 2757789609173637178721468440901224953430146549585371050792279689258923 5420199561121290219608640344181598136297747713099605187072113499999983 7297804995105973173281609631859502445945534690830264252230825334468503 5261931188171010003137838752886587533208381420617177669147303598253490 4287554687311595628638823537875937519577818577805321712268066130019278 76611195909216420198
Delphi
250,000 digits of pi on a BBC micro is pretty impressive, so a translation of the BBC code has been used for this Delphi 7 solution. A Delphi TMemo replaces the BBC screen, and the output is copied from the TMemo to a disk file at the end of the program.
The number of digits written depends on the variable M, and is found by experiment to be very close to 2*M/7. The value of M is limited by the danger of overflow in D, which can approach (but doesn't exceed) 400*M.
With M = floor( (2^31 - 1)/400 ), the Delphi version writes 1,533,913 decimal places, which have been verified against [sorry, defeated by the captcha as usual]
introcs dot cs dot princeton dot edu slash java slash data slash pi-10million.txt
unit Pi_BBC_Main;
interface
uses
Classes, Controls, Forms, Dialogs, StdCtrls;
type
TForm1 = class(TForm)
btnRunSpigotAlgo: TButton;
memScreen: TMemo;
procedure btnRunSpigotAlgoClick(Sender: TObject);
procedure FormCreate(Sender: TObject);
private
fScreenWidth : integer;
fLineBuffer : string;
procedure ClearText();
procedure AddText( const s : string);
procedure FlushText();
end;
var
Form1: TForm1;
implementation
{$R *.dfm}
uses SysUtils;
// Button clicked to run algorithm
procedure TForm1.btnRunSpigotAlgoClick(Sender: TObject);
var
// BBC Basic variables. Delphi longint is 32 bits.
B : array of longint;
A, C, D, E, I, L, M, P : longint;
// Added for Delphi version
temp : string;
h, j, t : integer;
begin
fScreenWidth := 80;
ClearText();
M := 5368709; // floor( (2^31 - 1)/400 )
// DIM B%(M%) in BBC Basic declares an array [0..M%], i.e. M% + 1 elements
SetLength( B, M + 1);
for I := 0 to M do B[I] := 20;
E := 0;
L := 2;
// FOR C% = M% TO 14 STEP -7
// In Delphi (or at least Delphi 7) the step size in a for loop has to be 1.
// So the BBC Basic FOR loop has been replaced by a repeat loop.
C := M;
repeat
D := 0;
A := C*2 - 1;
for P := C downto 1 do begin
D := D*P + B[P]*$64; // hex notation copied from BBC version
B[P] := D mod A;
D := D div A;
dec( A, 2);
end;
// The BBC CASE statement here amounts to a series of if ... else
if (D = 99) then begin
E := E*100 + D;
inc( L, 2);
end
else if (C = M) then begin
AddText( SysUtils.Format( '%2.1f', [1.0*(D div 100) / 10.0] ));
E := D mod 100;
end
else begin
// PRINT RIGHT$(STRING$(L%,"0") + STR$(E% + D% DIV 100),L%);
// This can't be done so concisely in Delphi 7
SetLength( temp, L);
for j := 1 to L do temp[j] := '0';
temp := temp + SysUtils.IntToStr( E + D div 100);
t := Length( temp);
AddText( Copy( temp, t - L + 1, L));
E := D mod 100;
L := 2;
end;
dec( C, 7);
until (C < 14);
FlushText();
// Delphi addition: Write screen output to a file for checking
h := SysUtils.FileCreate( 'C:\Delphi\PiDigits.txt'); // h = file handle
for j := 0 to memScreen.Lines.Count - 1 do
SysUtils.FileWrite( h, memScreen.Lines[j][1], Length( memScreen.Lines[j]));
SysUtils.FileClose( h);
end;
{=========================== Auxiliary routines ===========================}
// Form created
procedure TForm1.FormCreate(Sender: TObject);
begin
fScreenWidth := 80; // in case not set by the algotithm
ClearText();
end;
// This Delphi version builds each screen line in a buffer and puts
// the line into the TMemo when the buffer is full.
// This is faster than writing to the TMemo a few characters at a time,
// but note that the buffer must be flushed at the end of the program.
procedure TForm1.ClearText();
begin
memScreen.Lines.Clear();
fLineBuffer := '';
end;
procedure TForm1.AddText( const s : string);
var
nrChars, nrLeft : integer;
begin
nrChars := Length( s);
nrLeft := fScreenWidth - Length( fLineBuffer); // nr chars left in line
if (nrChars <= nrLeft) then fLineBuffer := fLineBuffer + s
else begin
fLineBuffer := fLineBuffer + Copy( s, 1, nrLeft);
memScreen.Lines.Add( fLineBuffer);
fLineBuffer := Copy( s, nrLeft + 1, nrChars - nrLeft);
end;
end;
procedure TForm1.FlushText();
begin
if (Length(fLineBuffer) > 0) then begin
memScreen.Lines.Add( fLineBuffer);
fLineBuffer := '';
end;
end;
end.
- Output:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208 99862803482534211706798214808651328230664709384460955058223172535940812848111745 02841027019385211055596446229489549303819644288109756659334461284756482337867831 ...
EDSAC order code
The task is impossible if taken literally, because a program with finite memory cannot generate a non-repeating infinite sequence.
Like many of the solutions posted here, this program implements the spigot algorithm of Rabinowitz and Wagon. The code takes up 192 EDSAC locations, leaving 832 for storage, which is enough for 252 correct digits of pi.
With only a few changes (noted in the comments), the same code can be used to output the first 2070 digits of e.
[EDSAC program, Initial Orders 2.
Calculates digits of pi by spigot algorithm.
Easily edited to calculate digits of e (see "if pi" and "if e" in comments).
Based on http://pi314.net/eng/goutte.php
See also https://www.cut-the-knot.org/Curriculum/Algorithms/SpigotForPi.shtml
Uses 17-bit values throughout.
Array index and counters are stored in the address field,
i.e. with the least significant bit in bit 1.
For integer arithmetic, the least significant bit is bit 0.
Variables don't need to be initialized at load time, so they overwrite
locations 6..12 of initial orders to save space.]
[6] P F [index into remainder array]
[7] P F [carry in the spigot algorithm]
[8] P F [negative count of digits]
[9] P F [pending digit, always < 9]
[10] P F [negative count of pending 9's]
[11] P F [9 or 0 in top 5 bits, for printing]
[12] P F [negative count of characters in current line]
[Array corresponding to Remainder row on http://pi314.net/eng/goutte.php]
T 53 K [refer to array via B parameter]
P 13 F [start array immediately after variables]
[Subroutine for short (17-bit) integer division.
Input: dividend at 4F, divisor at 5F.
Output: remainder at 4F, quotient at 5F.
Working locations 0F, 1F. 37 locations.]
T 987 K
GKA3FT34@A5FUFT35@A4FRDS35@G13@T1FA35@LDE4@T1FT5FA4FS35@G22@
T4FA5FA36@T5FT1FAFS35@E34@T1FA35@RDT35@A5FLDT5FE15@EFPFPD
T 845 K
G K
[Constants]
[Enter the editable numbers as addresses, e.g. P 100 F for 100.
Reducing the maximum array index will make the program take less time.
A maximum index of 831 (i.e. using all available memory) will give
252 correct digits of pi, or 2070 correct digits of e.]
[0] P 831 F [maximum array index <-- EDIT HERE, don't exceed 831]
[1] P 252 F [number of digits <-- EDIT HERE]
[2] P 72 F [digits per line <-- EDIT HERE]
[3] P D [short-value 1]
[4] P 5 F [short-value 10]
[5] J F [10*(2^12)]
[6] X F [add to T order to make V order]
[7] M F [used to convert T order to A order]
[8] # F [figures shift (for printer)]
[9] @ F [carriage return]
[10] & F [line feed]
[Main routine. Enter with acc = 0.]
[11] O 8 @ [set printer to figures; also used to print '9']
S 2 @ [load negative characters per line]
T 12 F [initialize character count]
S 1 @ [load negated number of digits]
T 8 F [initialize digit count]
T 10 F [clear negative count of 9's]
S 2 F [load -2 (any value < -1 would do)]
T 9 F [initialize digit buffer]
[Start algorithm: fill the remainder array with 2's (or 1's for e)
The code is a bit neater if we work backwards.]
A @ [maximum index]
[20] A 81 @ [make T order for array entry]
T 23 @ [plant in code]
[if pi] A 2 F [acc := 2]
[if e] [A 3 @] [acc := 1]
[23] T F [store in array entry]
A 23 @ [dec address in array]
S 2 F
S 81 @ [finished array?]
E 20 @ [loop back if not
Outer loop. Here for next digit.]
[28] T F [clear acc]
[Multiply remainder array by 10.
NB To preserve integer scaling, we need product times 2^16.]
H 5 @ [mult reg := 10*(2^12)]
A @ [acc := maximum index]
[31] A 81 @ [make T order for array entry]
U 37 @ [plant in code]
A 6 @ [convert to V order, same address]
T 35 @ [plant in code]
[35] V F [acc := array entry * 10*(2^12)]
L 4 F [shift to complete mult by 10*(2^16)]
[37] T F [store result in array]
A 37 @ [load T order]
S 2 F [dec address]
S 81 @ [test for done]
E 31 @ [loop back if not]
T F [clear acc]
T 7 F [clear carry]
A @ [acc := maximum index]
T 6 F [initialize array index]
[Inner loop to get next digit.
Work backwards through remainder array.]
[46] T F [clear acc]
A 6 F [load index]
A 81 @ [make T order for array entry]
U 61 @ [plant in code]
A 7 @ [convert to A order]
T 52 @ [plant in code]
[52] A F [load array element]
A 7 F [add carry from last time round loop]
T 4 F [sum to 4F for division routine]
A 6 F [acc := index as address = 2*(index as integer)]
[if pi] A 3 @ [plus 1]
[if e] [R D] [shift right, address --> integer]
T 5 F [to 5F for division routine (for e, 5F = index/2)]
[58] A 58 @ [call routine to divide 4F by 5F]
G 987 F
A 4 F [load remainder]
[61] T F [update element of remainder array]
[if pi: 4 orders]
H 5 F [mult reg := quotient]
V 6 F [multiply by index
NB need to shift 15 left to preserve integer scaling]
L F [shift 13 left]
L 1 F [shift 2 more left (for e, just use quotient)
[if e: 1 order, plus 3 no-ops to keep addresses the same]
[A 5 F] [load quotient]
[XF XF XF]
T 7 F [update carry for next time round loop]
A 6 F [load index]
S 2 F [decrement]
U 6 F [store back]
[We want to terminate after doing index = 1]
S 2 F [dec again]
E 46 @ [jump back if index >= 1]
[Treatment of index = 0 is different]
T F [clear acc]
A B [load rem{0)]
A 7 F [add carry]
T 4 F [sum to 4F for division routine]
A 4 @ [load 10]
T 5 F [to 5F for division routine]
[78] A 78 @ [call division routine]
G 987 F
A 4 F [load remainder]
[81] T B [store in rem{0}; also used to manufacture orders]
[82] A 82 @ [call subroutine to deal with quotient (clears acc)]
G 93 @
A 8 F [load negative digit count]
A 2 F [increment]
U 8 F [store back]
G 28 @ [if not yet 0, loop for next digit]
[Fake a zero digit to flush the last genuine digit(s)]
T 5 F [store fake digit 0 in 5F]
[89] A 89 @ [call subroutine to deal with digit]
G 93 @
O 8 @ [set figures: dummy character to flush print buffer]
Z F [stop]
[Subroutine to handle buffering and printing of digits.
Here with quotient from spigot algorithm still in 5F.
The quotient at 5F is usually the new decimal digit.
But the quotient can be 10, in which case we must treat it as 0
and ripple a carry through the previously-computed digits.
Hence the need for buffering.]
[93] A 3 F [make and plant return link]
T 130 @
A 5 F [load quotient]
S 4 @ [subtract 10]
E 105 @ [jump if quotient >= 10]
A 3 @ [add 1]
G 109 @ [jump if quotient < 9]
[Here if quotient = 9. Update count of 9's,
don't do anything with the buffer.]
T F [clear acc]
A 10 F [load negative count of 9's]
S 2 F [subtract 1]
T 10 F [update count]
E 130 @ [exit with acc = 0]
[Here if quotient >= 10. Take digit = quotient - 10,
and ripple a carry through the buffered digits.]
[105] T 5 F [store (quotient - 10) formed above]
T 11 F [store 0 to print '0' not '9']
A 3 @ [add 1 to buffered digit]
E 112 @ [join common code]
[Here if quotient < 9. Flush the stored digits.]
[109] T F [clear acc]
A 11 @ [load any O order (code for O is 9)]
T 11 F [store to print '9']
[112] A 9 F [load buffered digit, plus 1 if quotient >= 10]
G 118 @ [skip printing if buffer is empty]
L 1024 F [shift digits to top 5 bits]
T 1 F [store in 1F for printing]
[116] A 116 @ [call print routine]
G 131 @
[118] T F [clear acc]
A 5 F [load quotient (possibly modified as above)]
T 9 F [store in buffer]
[121] A 10 F [load negative count of 9's]
E 130 @ [if none, exit with acc = 0]
A 2 F [inc count]
T 10 F
A 11 F [load 9 (or 0 if there's a carry)]
T 1 F [to 1F for printing]
[127] A 127 @ [call print routine (clears acc)]
G 131 @
E 121 @ [jump back (always)]
[130] E F [return to caller]
[Subroutine to print character at 1F.
Also prints CR LF if necessary.]
[131] A 3 F [make and plant link for return]
T 141 @
A 12 F [load negative character count]
G 138 @ [jump if not end of line]
S 2 @ [reset character count]
O 9 @ [print CR LF]
O 10 @
[138] O 1 F [print character]
A 2 F [add 1]
T 12 F
[141] E F
E 11 Z [define entry point]
P F [acc = 0 on entry]- Output:
314159265358979323846264338327950288419716939937510582097494459230781640 628620899862803482534211706798214808651328230664709384460955058223172535 940812848111745028410270193852110555964462294895493038196442881097566593 344612847564823378678316527120190914
Elixir
defmodule Pi do
def calc, do: calc(1,0,1,1,3,3,0)
defp calc(q,r,t,k,n,l,c) when c==50 do
IO.write "\n"
calc(q,r,t,k,n,l,0)
end
defp calc(q,r,t,k,n,l,c) when (4*q + r - t) < n*t do
IO.write n
calc(q*10, 10*(r-n*t), t, k, div(10*(3*q+r), t) - 10*n, l, c+1)
end
defp calc(q,r,t,k,_n,l,c) do
calc(q*k, (2*q+r)*l, t*l, k+1, div(q*7*k+2+r*l, t*l), l+2, c)
end
end
Pi.calc
- Output:
Hit Ctrl-C to stop it.
C:\Elixir>elixir pi.exs 31415926535897932384626433832795028841971693993751 05820974944592307816406286208998628034825342117067 98214808651328230664709384460955058223172535940812 84811174502841027019385211055596446229489549303819 64428810975665933446128475648233786783165271201909 14564856692346034861045432664821339360726024914127 37245870066063155881748815209209628292540917153643 67892590360011330530548820466521384146951941511609 43305727036575959195309218611738193261179310511854 80744623799627495673518857527248912279381830119491 29833673362440656643086021394946395224737190702179 86094370277053921717629317675238467481846766940513 20005681271452635608277857713427577896091736371787 214684409012249534301
Erlang
% Implemented by Arjun Sunel
-module(pi_calculation).
-export([main/0]).
main() ->
pi(1,0,1,1,3,3,0).
pi(Q,R,T,K,N,L,C) ->
if C=:=50 ->
io:format("\n"),
pi(Q,R,T,K,N,L,0) ;
true ->
if
(4*Q + R-T) < (N*T) ->
io:format("~p",[N]),
P = 10*(R-N*T),
pi(Q*10 , P, T , K , ((10*(3*Q+R)) div T)-10*N , L,C+1);
true ->
P = (2*Q+R)*L,
M = (Q*(7*K)+2+(R*L)) div (T*L),
H = L+2,
J =K+ 1,
pi(Q*K, P , T*L ,J,M,H,C)
end
end.
- Output:
31415926535897932384626433832795028841971693993751 05820974944592307816406286208998628034825342117067 98214808651328230664709384460955058223172535940812 84811174502841027019385211055596446229489549303819 64428810975665933446128475648233786783165271201909 14564856692346034861045432664821339360726024914127 37245870066063155881748815209209628292540917153643 67892590360011330530548820466521384146951941511609 43305727036575959195309218611738193261179310511854 80744623799627495673518857527248912279381830119491 29833673362440656643086021394946395224737190702179 86094370277053921717629317675238467481846766940513 20005681271452635608277857713427577896091736371787 21468440901224953430146549585371050792279689258923 54201995611212902196086403441815981362977477130996 05187072113499999983729780499510597317328160963185 95024459455346908302642522308253344685035261931188 17101000313783875288658753320838142061717766914730 35982534904287554687311595628638823537875937519577 81857780532171226806613001927876611195909216420198 93809525720106548586327886593615338182796823030195 20353018529689957736225994138912497217752834791315 15574857242454150695950829533116861727855889075098 38175463746493931925506040092770167113900984882401 28583616035637076601047101819429555961989467678374 4944825537977472684710404753464620
F#
Translation of Haskell
let rec g q r t k n l = seq {
if 4I*q+r-t < n*t
then
yield n
yield! (g (10I*q) (10I*(r-n*t)) t k ((10I*(3I*q+r))/t - 10I*n) l)
else
yield! (g (q*k) ((2I*q+r)*l) (t*l) (k+1I) ((q*(7I*k+2I)+r*l)/(t*l)) (l+2I))
}
let π = (g 1I 0I 1I 1I 3I 3I)
Seq.take 1 π |> Seq.iter (printf "%A.")
// 6 digits beginning at position 762 of π are '9'
Seq.take 767 (Seq.skip 1 π) |> Seq.iter (printf "%A")
- Output:
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999
As an Unfold
Haskell can probably do this as an unfold, it has not so I shall in F#
// Generate Pi as above using unfold. Nigel Galloway: March 15th., 2022
let π()=Seq.unfold(fun(q,r,t,k,n,l)->Some(if 4I*q+r-t < n*t then(Some(int n),((10I*q),(10I*(r-n*t)),t,k,((10I*(3I*q+r))/t-10I*n),l)) else (None,((q*k),((2I*q+r)*l),(t*l),(k+1I),((q*(7I*k+2I)+r*l)/(t*l)),(l+2I)))))(1I,0I,1I,1I,3I,3I)|>Seq.choose id
π()|>Seq.take 767|>Seq.iter(printf "%d")
Factor
USING: combinators.extras io kernel locals math prettyprint ;
IN: rosetta-code.pi
:: calc-pi-digits ( -- )
1 0 1 1 3 3 :> ( q! r! t! k! n! l! ) [
4 q * r + t - n t * < [
n pprint flush
r n t * - 10 *
3 q * r + 10 * t /i n 10 * - n! r!
q 10 * q!
] [
2 q * r + l *
7 k * q * 2 + r l * + t l * /i n! r!
k q * q!
t l * t!
l 2 + l!
k 1 + k!
] if
] forever ;
MAIN: calc-pi-digits
Fortran
This is a modernized version of the example Fortran programme written by S. Rabinowitz in 1991. It works in base 100000 and the key step is the initialisation of all elements of VECT to 2. The format code of I5.5 means I5 output but with all leading spaces made zero so that 66 comes out as "00066", not " 66".
program pi
implicit none
integer,dimension(3350) :: vect
integer,dimension(201) :: buffer
integer :: more,karray,num,k,l,n
more = 0
vect = 2
do n = 1,201
karray = 0
do l = 3350,1,-1
num = 100000*vect(l) + karray*l
karray = num/(2*l - 1)
vect(l) = num - karray*(2*l - 1)
end do
k = karray/100000
buffer(n) = more + k
more = karray - k*100000
end do
write (*,'(i2,"."/(1x,10i5.5))') buffer
end program pi
The output is accumulated in BUFFER then written in one go at the end, but it could be written as successive values as each is calculated without much extra nitpickery: instead of BUFFER(N) = MORE + K for example just WRITE (*,"(I5.5)") MORE + K and no need for array BUFFER.
3. 14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679 82148086513282306647093844609550582231725359408128 48111745028410270193852110555964462294895493038196 44288109756659334461284756482337867831652712019091 45648566923460348610454326648213393607260249141273 72458700660631558817488152092096282925409171536436 78925903600113305305488204665213841469519415116094 33057270365759591953092186117381932611793105118548 07446237996274956735188575272489122793818301194912 98336733624406566430860213949463952247371907021798 60943702770539217176293176752384674818467669405132 00056812714526356082778577134275778960917363717872 14684409012249534301465495853710507922796892589235 42019956112129021960864034418159813629774771309960 51870721134999999837297804995105973173281609631859 50244594553469083026425223082533446850352619311881 71010003137838752886587533208381420617177669147303 59825349042875546873115956286388235378759375195778 18577805321712268066130019278766111959092164201989
This is an alternate version using an unbounded spigot. Higher precision is accomplished by using the Fortran Multiple Precision Library, FMLIB (http://myweb.lmu.edu/dmsmith/fmlib.html), provided by Dr. David M. Smith (dsmith@lmu.edu), Mathematics Professor (Emeritus) at Loyola Marymount University. We use the default precision which is about 50 significant digits.
!================================================
program pi_spigot_unbounded
!================================================
do
call print_next_pi_digit()
end do
contains
!------------------------------------------------
subroutine print_next_pi_digit()
!------------------------------------------------
use fmzm
type (im) :: q, r, t, k, n, l, nr
logical :: dot=.false., init=.false.
save :: q, r, t, k, n, l
if (.not.init) then
q=to_im(1)
r=to_im(0)
t=to_im(1)
k=to_im(1)
n=to_im(3)
l=to_im(3)
init=.true.
end if
if (4*q+r-t < n*t) then
write(6,fmt='(i1)',advance='no') to_int(n)
if (.not.dot) then
write(6,fmt='(a1)',advance='no') '.'
dot=.true.
end if
flush(6)
nr = 10 * ( r - n*t )
n = 10 * ( (3*q + r) / t - n )
q = 10 * q
r = nr
else
nr = (2*q + r) * l
n = ( (q * (7*k + 2) + r*l) / (t*l) )
q = q * k
t = t * l
l = l + 2
k = k + 1
r = nr
end if
end subroutine
end program
FreeBASIC
' version 05-07-2018
' compile with: fbc -s console
' unbounded spigot
' Ctrl-c to end program or close console window
#Include "gmp.bi"
Dim As UInteger num, ndigit, fp = Not 0
Dim As mpz_ptr q,r,t,k,n,l,tmp1,tmp2
q = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(q,1)
r = Allocate(Len(__Mpz_struct)) : Mpz_init(r)
t = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(t,1)
k = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(k,1)
n = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(n,3)
l = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(l,3)
tmp1 = Allocate(Len(__Mpz_struct)) : Mpz_init(tmp1)
tmp2 = Allocate(Len(__Mpz_struct)) : Mpz_init(tmp2)
Do
mpz_mul_2exp(tmp1, q, 2)
mpz_add(tmp1,tmp1,r)
mpz_sub(tmp1,tmp1,t)
mpz_mul(tmp2, n, t)
If mpz_cmp(tmp1, tmp2) < 0 Then
Print mpz_get_ui(n); : ndigit += 1 : If ndigit Mod 50 = 0 Then Print " :";ndigit
If fp Then Print "."; : fp = Not fp : Print :ndigit = 0
mpz_sub(tmp1, r, tmp2)
mpz_mul_ui(tmp1, tmp1, 10)
mpz_mul_ui(tmp2, q, 3)
mpz_add(tmp2, tmp2, r)
mpz_mul_ui(tmp2, tmp2, 10)
mpz_set(r, tmp1)
mpz_mul_ui(tmp1, n, 10)
mpz_tdiv_q(tmp2, tmp2, t)
mpz_sub(n, tmp2, tmp1)
mpz_mul_ui(q, q, 10)
Else
mpz_mul(tmp2, r, l)
mpz_mul(tmp1, q, k)
mpz_mul_ui(tmp1, tmp1, 7)
mpz_add(tmp1, tmp1, tmp2)
mpz_mul_2exp(tmp2, q, 1)
mpz_add(tmp2, tmp2, r)
mpz_mul(tmp2, tmp2, l)
mpz_mul(t, t, l)
mpz_tdiv_q(tmp1, tmp1, t)
mpz_mul(q, q, k)
mpz_add_ui(k, k, 1)
mpz_add_ui(l, l, 2)
mpz_set(n, tmp1)
mpz_set(r, tmp2)
End If
Loop
- Output:
3. 14159265358979323846264338327950288419716939937510 :50 58209749445923078164062862089986280348253421170679 :100 82148086513282306647093844609550582231725359408128 :150 48111745028410270193852110555964462294895493038196 :200 44288109756659334461284756482337867831652712019091 :250 ...... 59284936959414340814685298150539471789004518357551 :20300 54125223590590687264878635752541911288877371766374 :20350 86027660634960353679470269232297186832771739323619 :20400 20077745221262475186983349515101986426988784717193 :20450 96649769070825217423365662725928440620430214113719 :20500
FunL
The code for compute_pi() is from [2]. The number of digits may be given on the command line as an argument. If there's no argument, the program will run until interrupted.
def compute_pi =
def g( q, r, t, k, n, l ) =
if 4*q + r - t < n*t
n # g( 10*q, 10*(r - n*t), t, k, (10*(3*q + r))\t - 10*n, l )
else
g( q*k, (2*q + r)*l, t*l, k + 1, (q*(7*k + 2) + r*l)\(t*l), l + 2 )
g( 1, 0, 1, 1, 3, 3 )
if _name_ == '-main-'
print( compute_pi().head() + '.' )
if args.isEmpty()
for d <- compute_pi().tail()
print( d )
else
for d <- compute_pi().tail().take( int(args(0)) )
print( d )
println()FutureBasic
This old-school code still works on Mac OS Monterey and is expected to work on Ventura, but it needs a modern refactoring.
_maxlong = 0x7fffffff
begin globals
long kf, ks
xref mf(_maxLong - 1) as long
xref ms(_maxLong - 1) as long
long cnt, n, temp, nd
long col, col1
long lloc, stor(50)
end globals
local mode
local fn FmtStr( nn as long, s as Str255 ) as Str255
long l
Str255 f
l = s[0]
select case
case ( nn => l ) : f = string$( nn-l, 32 ) + s
case ( -nn > l ) : f = s + string$( -nn-l, 32 )
case else : f = s
end select
end fn = f
local mode
local fn FmtInt( nn as long, s as Str255 ) as Str255
if ( left$( s, 1 ) = " " ) then s = mid$( s, 2 )
end fn = fn FmtStr( nn, s )
local fn yprint( m as long )
if ( cnt < n )
col++
if ( col == 11 )
col = 1
col1++
if ( col1 == 6 )
col1 = 0
print
print fn FmtInt( 4, str$( m mod 10) );
else
print fn FmtInt( 3, str$ (m mod 10) );
end if
else
print mid$( str$( m ), 2 ) ;
end if
end if
cnt++
end fn
local fn xprint( m as long )
long ii, wk, wk1
if ( m < 8 )
ii = 1
while ( ii <= lloc )
fn yprint( stor(ii) )
ii++
wend
lloc = 0
else
if ( m > 9 )
wk = m / 10
m = m mod 10
wk1 = lloc
while ( wk1 >= 1 )
wk += stor(wk1)
stor(wk1) = wk mod 10
wk = wk/10
wk1--
wend
end if
end if
lloc++
stor(lloc) = m
end fn
local mode
local fn shift( l1 as ^long, l2 as ^long, lp as long, lmod as long )
long k
if ( l2.nil& > 0 )
k = ( l2.nil& ) / lmod
else
k = -( -l2.nil& / lmod ) - 1
end if
l2.nil& = l2.nil& - k*lmod
l1.nil& = l1.nil& + k*lp
end fn
local fn Main( nDig as long )
long i
n = nDig
stor(0) = 0
mf = fn malloc( ( n + 10 ) * sizeof(long) )
if ( 0 == mf ) then stop "Out of memory"
ms = fn malloc( ( n + 10 ) * sizeof(long) )
if ( 0 == ms ) then stop "Out of memory"
print : printf @"Approximation of π to %ld digits", n
cnt = 0
kf = 25
ks = 57121
mf(1) = 1
i = 2
while ( i <= n )
mf(i) = -16
mf(i + 1) = 16
i += 2
wend
i = 1
while ( i <= n )
ms(i) = -4
ms(i + 1) = 4
i += 2
wend
print : print " 3.";
while ( cnt < n )
i = 0
i++
while ( i <= n - cnt )
mf(i) = mf(i) * 10
ms(i) = ms(i) * 10
i++
wend
i = ( n - cnt + 1 )
i--
while ( i >= 2 )
temp = 2 * i - 1
fn shift( @mf(i - 1), @mf(i), temp - 2, temp * kf )
fn shift( @ms(i - 1), @ms(i), temp - 2, temp * ks )
i--
wend
nd = 0
fn shift( @nd, @mf(1), 1, 5 )
fn shift( @nd, @ms(1), 1, 239 )
fn xprint( nd )
wend
print : print "Done"
fn free( ms )
fn free( mf )
end fn
window 1
CFTimeInterval t
t = fn CACurrentMediaTime
// Here we specify the number of decimal places
fn Main( 4000 )
print : printf @"Compute time: %.3f ms",(fn CACurrentMediaTime-t)*1000
HandleEventsOutput:
Approximation of π to 4000 digits 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959 5082953311 6861727855 8890750983 8175463746 4939319255 0604009277 0167113900 9848824012 8583616035 6370766010 4710181942 9555961989 4676783744 9448255379 7747268471 0404753464 6208046684 2590694912 9331367702 8989152104 7521620569 6602405803 8150193511 2533824300 3558764024 7496473263 9141992726 0426992279 6782354781 6360093417 2164121992 4586315030 2861829745 5570674983 8505494588 5869269956 9092721079 7509302955 3211653449 8720275596 0236480665 4991198818 3479775356 6369807426 5425278625 5181841757 4672890977 7727938000 8164706001 6145249192 1732172147 7235014144 1973568548 1613611573 5255213347 5741849468 4385233239 0739414333 4547762416 8625189835 6948556209 9219222184 2725502542 5688767179 0494601653 4668049886 2723279178 6085784383 8279679766 8145410095 3883786360 9506800642 2512520511 7392984896 0841284886 2694560424 1965285022 2106611863 0674427862 2039194945 0471237137 8696095636 4371917287 4677646575 7396241389 0865832645 9958133904 7802759009 9465764078 9512694683 9835259570 9825822620 5224894077 2671947826 8482601476 9909026401 3639443745 5305068203 4962524517 4939965143 1429809190 6592509372 2169646151 5709858387 4105978859 5977297549 8930161753 9284681382 6868386894 2774155991 8559252459 5395943104 9972524680 8459872736 4469584865 3836736222 6260991246 0805124388 4390451244 1365497627 8079771569 1435997700 1296160894 4169486855 5848406353 4220722258 2848864815 8456028506 0168427394 5226746767 8895252138 5225499546 6672782398 6456596116 3548862305 7745649803 5593634568 1743241125 1507606947 9451096596 0940252288 7971089314 5669136867 2287489405 6010150330 8617928680 9208747609 1782493858 9009714909 6759852613 6554978189 3129784821 6829989487 2265880485 7564014270 4775551323 7964145152 3746234364 5428584447 9526586782 1051141354 7357395231 1342716610 2135969536 2314429524 8493718711 0145765403 5902799344 0374200731 0578539062 1983874478 0847848968 3321445713 8687519435 0643021845 3191048481 0053706146 8067491927 8191197939 9520614196 6342875444 0643745123 7181921799 9839101591 9561814675 1426912397 4894090718 6494231961 5679452080 9514655022 5231603881 9301420937 6213785595 6638937787 0830390697 9207734672 2182562599 6615014215 0306803844 7734549202 6054146659 2520149744 2850732518 6660021324 3408819071 0486331734 6496514539 0579626856 1005508106 6587969981 6357473638 4052571459 1028970641 4011097120 6280439039 7595156771 5770042033 7869936007 2305587631 7635942187 3125147120 5329281918 2618612586 7321579198 4148488291 6447060957 5270695722 0917567116 7229109816 9091528017 3506712748 5832228718 3520935396 5725121083 5791513698 8209144421 0067510334 6711031412 6711136990 8658516398 3150197016 5151168517 1437657618 3515565088 4909989859 9823873455 2833163550 7647918535 8932261854 8963213293 3089857064 2046752590 7091548141 6549859461 6371802709 8199430992 4488957571 2828905923 2332609729 9712084433 5732654893 8239119325 9746366730 5836041428 1388303203 8249037589 8524374417 0291327656 1809377344 4030707469 2112019130 2033038019 7621101100 4492932151 6084244485 9637669838 9522868478 3123552658 2131449576 8572624334 4189303968 6426243410 7732269780 2807318915 4411010446 8232527162 0105265227 2111660396 Done Elapsed time: 729.405 ms
Go
Code below is a simplistic translation of Haskell code in Unbounded Spigot Algorithms for the Digits of Pi. This is the algorithm specified for the pidigits benchmark of the Computer Language Benchmarks Game. (The standard Go distribution includes source submitted to the benchmark site, and that code runs stunning faster than the code below.)
package main
import (
"fmt"
"math/big"
)
type lft struct {
q,r,s,t big.Int
}
func (t *lft) extr(x *big.Int) *big.Rat {
var n, d big.Int
var r big.Rat
return r.SetFrac(
n.Add(n.Mul(&t.q, x), &t.r),
d.Add(d.Mul(&t.s, x), &t.t))
}
var three = big.NewInt(3)
var four = big.NewInt(4)
func (t *lft) next() *big.Int {
r := t.extr(three)
var f big.Int
return f.Div(r.Num(), r.Denom())
}
func (t *lft) safe(n *big.Int) bool {
r := t.extr(four)
var f big.Int
if n.Cmp(f.Div(r.Num(), r.Denom())) == 0 {
return true
}
return false
}
func (t *lft) comp(u *lft) *lft {
var r lft
var a, b big.Int
r.q.Add(a.Mul(&t.q, &u.q), b.Mul(&t.r, &u.s))
r.r.Add(a.Mul(&t.q, &u.r), b.Mul(&t.r, &u.t))
r.s.Add(a.Mul(&t.s, &u.q), b.Mul(&t.t, &u.s))
r.t.Add(a.Mul(&t.s, &u.r), b.Mul(&t.t, &u.t))
return &r
}
func (t *lft) prod(n *big.Int) *lft {
var r lft
r.q.SetInt64(10)
r.r.Mul(r.r.SetInt64(-10), n)
r.t.SetInt64(1)
return r.comp(t)
}
func main() {
// init z to unit
z := new(lft)
z.q.SetInt64(1)
z.t.SetInt64(1)
// lfts generator
var k int64
lfts := func() *lft {
k++
r := new(lft)
r.q.SetInt64(k)
r.r.SetInt64(4*k+2)
r.t.SetInt64(2*k+1)
return r
}
// stream
for {
y := z.next()
if z.safe(y) {
fmt.Print(y)
z = z.prod(y)
} else {
z = z.comp(lfts())
}
}
}
Groovy
Solution:
BigInteger q = 1, r = 0, t = 1, k = 1, n = 3, l = 3
String nn
boolean first = true
while (true) {
(nn, first, q, r, t, k, n, l) = (4*q + r - t < n*t) \
? ["${n}${first?'.':''}", false, 10*q, 10*(r - n*t), t , k , 10*(3*q + r)/t - 10*n , l ] \
: ['' , first, q*k , (2*q + r)*l , t*l, k + 1, (q*(7*k + 2) + r*l)/(t*l), l + 2]
print nn
}
Output (thru first 1000 iterations):
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337
Haskell
The code from [3]:
pi_ = g (1, 0, 1, 1, 3, 3)
where
g (q, r, t, k, n, l) =
if 4 * q + r - t < n * t
then n :
g
( 10 * q
, 10 * (r - n * t)
, t
, k
, div (10 * (3 * q + r)) t - 10 * n
, l)
else g
( q * k
, (2 * q + r) * l
, t * l
, k + 1
, div (q * (7 * k + 2) + r * l) (t * l)
, l + 2)
Complete command-line program
#!/usr/bin/runhaskell
import Control.Monad
import System.IO
pi_ = g(1,0,1,1,3,3) where
g (q,r,t,k,n,l) =
if 4*q+r-t < n*t
then n : g (10*q, 10*(r-n*t), t, k, div (10*(3*q+r)) t - 10*n, l)
else g (q*k, (2*q+r)*l, t*l, k+1, div (q*(7*k+2)+r*l) (t*l), l+2)
digs = insertPoint digs'
where insertPoint (x:xs) = x:'.':xs
digs' = map (head . show) pi_
main = do
hSetBuffering stdout $ BlockBuffering $ Just 80
forM_ digs putChar
- Output:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198
Quicker, Unverified Algorithm
Snippet verbatim from source .pdf:
piG3 = g(1,180,60,2) where
g(q,r,t,i) = let (u,y)=(3*(3*i+1)*(3*i+2),div(q*(27*i-12)+5*r)(5*t))
in y : g(10*q*i*(2*i-1),10*u*(q*(5*i-2)+r-y*t),t*u,i+1)
This is more efficient because each term converges in less than one step, so no checking needs to be done partway through the iteration. Only caveat is that the convergence is on average slightly over one digit, so there is a chance that, if one checked enough digits, one may find a gap where a digit would be incorrect. Though it seems to be OK for the first 100k digits, or so.
Output is the same. link to complete program at tio.run
Icon and Unicon
based on Jeremy Gibbons' Haskell solution.
J
pi=: 3 :0
echo"0 '3.1'
i=. 0
while. i=. i + 1 do.
echo -/ 1 10 * <.@o. 10x ^ 1 0 + i
end.
)
Example use:
pi''
3
.
1
4
1
5
9
2
6
5
3
...
Java
import java.math.BigInteger ;
public class Pi {
final BigInteger TWO = BigInteger.valueOf(2) ;
final BigInteger THREE = BigInteger.valueOf(3) ;
final BigInteger FOUR = BigInteger.valueOf(4) ;
final BigInteger SEVEN = BigInteger.valueOf(7) ;
BigInteger q = BigInteger.ONE ;
BigInteger r = BigInteger.ZERO ;
BigInteger t = BigInteger.ONE ;
BigInteger k = BigInteger.ONE ;
BigInteger n = BigInteger.valueOf(3) ;
BigInteger l = BigInteger.valueOf(3) ;
public void calcPiDigits(){
BigInteger nn, nr ;
boolean first = true ;
while(true){
if(FOUR.multiply(q).add(r).subtract(t).compareTo(n.multiply(t)) == -1){
System.out.print(n) ;
if(first){System.out.print(".") ; first = false ;}
nr = BigInteger.TEN.multiply(r.subtract(n.multiply(t))) ;
n = BigInteger.TEN.multiply(THREE.multiply(q).add(r)).divide(t).subtract(BigInteger.TEN.multiply(n)) ;
q = q.multiply(BigInteger.TEN) ;
r = nr ;
System.out.flush() ;
}else{
nr = TWO.multiply(q).add(r).multiply(l) ;
nn = q.multiply((SEVEN.multiply(k))).add(TWO).add(r.multiply(l)).divide(t.multiply(l)) ;
q = q.multiply(k) ;
t = t.multiply(l) ;
l = l.add(TWO) ;
k = k.add(BigInteger.ONE) ;
n = nn ;
r = nr ;
}
}
}
public static void main(String[] args) {
Pi p = new Pi() ;
p.calcPiDigits() ;
}
}
Output :
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480 ...
JavaScript
Spigot Algorithm
Calculate one digit of π at a time and write it out. process.stdout.write will work in Node.js; to make this work in a browser, change it to document.body.textContent += .
let q = 1n, r = 180n, t = 60n, i = 2n;
for (;;) {
let y = (q*(27n*i-12n)+5n*r)/(5n*t);
let u = 3n*(3n*i+1n)*(3n*i+2n);
r = 10n*u*(q*(5n*i-2n)+r-y*t);
q = 10n*q*i*(2n*i-1n);
t = t*u;
i = i+1n;
process.stdout.write(y.toString());
if (i === 3n) { process.stdout.write('.'); }
}
Web Page version
This shows how to load the previous code into a webpage that writes digits out without freezing the browser
<html><head><script src='https://rawgit.com/andyperlitch/jsbn/v1.1.0/index.js'></script></head>
<body style="width: 100%"><tt id="pi"></tt><tt>...</tt>
<script async defer>
function bi(n, b) { return new jsbn.BigInteger(n.toString(), b ? b : 10); };
var one=bi(1), two=bi(2), three=bi(3), four=bi(4), seven=bi(7), ten=bi(10);
function calcPi() {
var q=bi(1), r=bi(0), t=bi(1), k=bi(1), n=bi(3), l=bi(3);
var digit=0, firstrun=1;
var p=document.getElementById('pi');
function w(s) { p.appendChild(document.createTextNode(s));}
function continueCalcPi(q, r, t, k, n, l) {
while (true) {
if (q.multiply(four).add(r).subtract(t).compareTo(n.multiply(t)) < 0) {
w(n.toString());
if (digit==0 && firstrun==1) { w('.'); firstrun=0; };
digit = (digit+1) % 256;
var nr = (r.subtract(n.multiply(t))).multiply(ten);
n = (q.multiply(three).add(r)).multiply(ten).divide(t).subtract(n.multiply(ten));
q = q.multiply(ten);
r = nr;
if (digit%8==0) {
if (digit%64==0) {
p.appendChild(document.createElement('br'));
}
w(' ');
return setTimeout(function() { continueCalcPi(q, r, t, k, n, l); }, 50);
};
} else {
var nr = q.shiftLeft(1).add(r).multiply(l);
var nn = q.multiply(k).multiply(seven).add(two).add(r.multiply(l)).divide(t.multiply(l));
q = q.multiply(k);
t = t.multiply(l);
l = l.add(two);
k = k.add(one);
n = nn;
r = nr;
}
}
}
continueCalcPi(q, r, t, k, n, l);
}
calcPi();
</script>
</body></html>
Web Page using BigInt
Above converted to use BigInt
<html>
<head>
</head>
<body style="width: 100%">
<tt id="pi"></tt>
<tt>...</tt>
<script async defer>
function calcPi() {
let q=1n, r=0n, t=1n, k=1n, n=3n, l=3n, nr, nn, digit=0, firstrun=1;
const p=document.getElementById('pi');
function w(s) { p.appendChild(document.createTextNode(s));}
// function continueCalcPi(q, r, t, k, n, l) { // (see note)
function continueCalcPi() {
while (true) {
if (q*4n+r-t < n*t) {
w(n.toString());
if (digit==0 && firstrun==1) { w('.'); firstrun=0; };
digit = (digit+1) % 256;
nr = (r-n*t)*10n;
n = (q*3n+r)*10n/t-n*10n;
q *= 10n;
r = nr;
if (digit%8==0) {
if (digit%64==0) {
p.appendChild(document.createElement('br'));
}
w('\xA0');
// return setTimeout(function() { continueCalcPi(q, r, t, k, n, l); }, 50);
return setTimeout(continueCalcPi, 50);
};
} else {
nr = (q*2n+r)*l;
nn = (q*k*7n+2n+r*l)/(t*l);
q *= k;
t *= l;
l += 2n;
k += 1n;
n = nn;
r = nr;
}
}
}
continueCalcPi(q, r, t, k, n, l);
}
calcPi();
</script>
</body>
</html>
Note: removing the parameters to continueCalcPi() as shown may eat (even) more memory, not entirely sure about that.
Simple Approximation
Returns an approximation of Pi.
var calcPi = function() {
var n = 20000;
var pi = 0;
for (var i = 0; i < n; i++) {
var temp = 4 / (i*2+1);
if (i % 2 == 0) {
pi += temp;
}
else {
pi -= temp;
}
}
return pi;
}
jq
The focus in this section is on the Gibbons spigot algorithm as it is relatively simple and therefore provides a gentle introduction to how such algorithms can be implemented in jq.
Since the Gibbons algorithm quickly fails in the absence of support for large integers, we shall assume BigInt support, such as provided by BigInt.jq.
The jq program presented here closely follows the Groovy and Python examples on this page. The spigot generator is named "next", and is driven by an annotation function, "decorate"; thus the main program is just "S0 | decorate(next)" where S0 is the initial state. One advantage of this approach is that the generator's state is exposed, thus making it easy to restart the stream at any point.
The annotation defined here results in a triple for each digit of pi: [index, digit, space], where "space" is the sum of the lengths of the strings in the six-dimensional state vector, [q, r, t, k, n, l]. The output shows that the space requirements of the Gibbons spigot grow very slightly more than linearly.
# The Gibbons spigot, in the mold of the [[#Groovy]] and [[#Python]] programs shown on this page.
# The "bigint" functions needed are:
# long_minus long_add long_multiply long_div
def pi_spigot:
# S is the sixtuple:
# q r t k n l
# 0 1 2 3 4 5
def long_lt(x;y): if x == y then false else lessOrEqual(x;y) end;
def check:
long_lt(long_minus(long_add(long_multiply("4"; .[0]); .[1]) ; .[2]);
long_multiply(.[4]; .[2]));
# state: [d, S] where digit is null or a digit ready to be printed
def next:
.[1] as $S
| $S[0] as $q | $S[1] as $r | $S[2] as $t | $S[3] as $k | $S[4] as $n | $S[5] as $l
| if $S|check
then [$n,
[long_multiply("10"; $q),
long_multiply("10"; long_minus($r; long_multiply($n;$t))),
$t,
$k,
long_minus( long_div(long_multiply("10";long_add(long_multiply("3"; $q); $r)); $t );
long_multiply("10";$n)),
$l ]]
else [null,
[long_multiply($q;$k),
long_multiply( long_add(long_multiply("2";$q); $r); $l),
long_multiply($t;$l),
long_add($k; "1"),
long_div( long_add(long_multiply($q; long_add(long_multiply("7";$k); "2")) ; long_multiply($r;$l));
long_multiply($t;$l) ),
long_add($l; "2") ]]
end;
# Input: input to the filter "nextstate"
# Output: [count, space, digit] for successive digits produced by "nextstate"
def decorate( nextstate ):
# For efficiency it is important that the recursive
# function have arity 0 and be tail-recursive:
def count:
.[0] as $count
| .[1] as $state
| $state[0] as $value
| ($state[1] | map(length) | add) as $space
| (if $value then [$count, $space, $value] else empty end),
( [if $value then $count+1 else $count end, ($state | nextstate)] | count);
[0, .] | count;
# q=1, r=0, t=1, k=1, n=3, l=3
[null, ["1", "0", "1", "1", "3", "3"]] | decorate(next)
;
pi_spigot- Output:
$ jq -M -n -c -f pi.bigint.jq
[0,9,"3"]
[1,14,"1"]
[2,29,"4"]
[3,36,"1"]
[4,51,"5"]
[5,69,"9"]
[6,80,"2"]
[7,95,"6"]
[8,115,"5"]
[9,125,"3"]
[10,142,"5"]
[11,167,"8"]
[12,181,"9"]
[13,197,"7"]
[14,226,"9"]
[15,245,"3"]
[16,263,"2"]
[17,276,"3"]
[18,300,"8"]
[19,320,"4"]
[20,350,"6"]
[21,363,"2"]
[22,383,"6"]
[23,408,"4"]
[24,429,"3"]
[25,442,"3"]
[26,475,"8"]
[27,502,"3"]
[28,510,"2"]
[29,531,"7"]
[30,563,"9"]
[31,611,"5"]
[32,613,"0"]
[33,628,"2"]
[34,649,"8"]
[35,676,"8"]
[36,711,"4"]
[37,720,"1"]
[38,748,"9"]
[39,783,"7"]
[40,792,"1"]
[41,814,"6"]
[42,849,"9"]
[43,870,"3"]
[44,886,"9"]
[45,923,"9"]
[46,939,"3"]
[47,967,"7"]
[48,1004,"5"]
[49,1041,"1"]
[50,1043,"0"]
[51,1059,"5"]
[52,1103,"8"]
[53,1133,"2"]
[54,1135,"0"]
[55,1165,"9"]
[56,1195,"7"]
[57,1212,"4"]
[58,1242,"9"]
[59,1273,"4"]
[60,1297,"4"]
[61,1313,"5"]
[62,1358,"9"]
[63,1375,"2"]
[64,1421,"3"]
[65,1423,"0"]
[66,1447,"7"]
[67,1493,"8"]
[68,1501,"1"]
[69,1533,"6"]
[70,1579,"4"]
[71,1581,"0"]
[72,1613,"6"]
[73,1630,"2"]
[74,1662,"8"]
[75,1701,"6"]
[76,1733,"2"]
[77,1735,"0"]
[78,1781,"8"]
[79,1792,"9"]
[80,1816,"9"]
[81,1849,"8"]
[82,1889,"6"]
[83,1898,"2"]
[84,1961,"8"]
[85,1963,"0"]
[86,1988,"3"]
[87,2013,"4"]
[88,2054,"8"]
[89,2071,"2"]
[90,2104,"5"]
[91,2129,"3"]
[92,2162,"4"]
[93,2195,"2"]
[94,2220,"1"]
[95,2230,"1"]
[96,2287,"7"]
[97,2289,"0"]
[98,2314,"6"]
[99,2340,"7"]
[100,2373,"9"]
[101,2414,"8"]
[102,2448,"2"]
[103,2458,"1"]
[104,2484,"4"]
[105,2534,"8"]
[106,2536,"0"]
[107,2569,"8"]
[108,2602,"6"]
[109,2645,"5"]
[110,2662,"1"]
[111,2696,"3"]
[112,2707,"2"]
[113,2756,"8"]
[114,2775,"2"]
[115,2825,"3"]
[116,2827,"0"]
[117,2853,"6"]
[118,2887,"6"]
[119,2914,"4"]
[120,2964,"7"]
[121,2966,"0"]
[122,3008,"9"]
[123,3027,"3"]
[124,3061,"8"]
[125,3088,"4"]
[126,3114,"4"]
[127,3165,"6"]
[128,3167,"0"]
[129,3202,"9"]
[130,3237,"5"]
[131,3287,"5"]
[132,3289,"0"]
[133,3316,"5"]
[134,3360,"8"]
[135,3387,"2"]
[136,3414,"2"]
[137,3456,"3"]
[138,3466,"1"]
[139,3510,"7"]
[140,3529,"2"]
[141,3564,"5"]
[142,3583,"3"]
[143,3610,"5"]
[144,3653,"9"]
[145,3697,"4"]
[146,3699,"0"]
[147,3752,"8"]
[148,3770,"1"]
[149,3789,"2"]
[150,3825,"8"]
[151,3852,"4"]
[152,3905,"8"]
[153,3933,"1"]
[154,3960,"1"]
[155,3970,"1"]
[156,4006,"7"]
[157,4033,"4"]
[158,4102,"5"]
[159,4104,"0"]
[160,4124,"2"]
[161,4159,"8"]
[162,4203,"4"]
[163,4248,"1"]
[164,4250,"0"]
[165,4269,"2"]
[166,4348,"7"]
[167,4350,"0"]
[168,4361,"1"]
[169,4405,"9"]
[170,4424,"3"]
[171,4460,"8"]
[172,4497,"5"]
[173,4542,"2"]
[174,4569,"1"]
[175,4605,"1"]
[176,4607,"0"]
[177,4644,"5"]
[178,4672,"5"]
[179,4691,"5"]
[180,4727,"9"]
[181,4764,"6"]
[182,4792,"4"]
[183,4820,"4"]
[184,4865,"6"]
[185,4893,"2"]
[186,4913,"2"]
[187,4949,"9"]
[188,4968,"4"]
[189,5005,"8"]
[190,5042,"9"]
[191,5070,"5"]
[192,5098,"4"]
[193,5144,"9"]
[194,5198,"3"]
[195,5200,"0"]
[196,5219,"3"]
[197,5266,"8"]
[198,5276,"1"]
[199,5313,"9"]
[200,5350,"6"]
[201,5387,"4"]
[202,5416,"4"]
[203,5435,"2"]
[204,5471,"8"]
[205,5526,"8"]
[206,5556,"1"]
[207,5558,"0"]
[208,5594,"9"]
[209,5632,"7"]
[210,5660,"5"]
[211,5689,"6"]
[212,5726,"6"]
[213,5746,"5"]
[214,5792,"9"]
[215,5821,"3"]
[216,5849,"3"]
[217,5887,"4"]
[218,5906,"4"]
[219,5961,"6"]
[220,5981,"1"]
[221,6002,"2"]
[222,6038,"8"]
[223,6068,"4"]
[224,6096,"7"]
[225,6134,"5"]
[226,6163,"6"]
[227,6191,"4"]
[228,6238,"8"]
[229,6267,"2"]
[230,6296,"3"]
[231,6316,"3"]
[232,6344,"7"]
[233,6383,"8"]
[234,6411,"6"]
[235,6440,"7"]
[236,6487,"8"]
[237,6525,"3"]
[238,6545,"1"]
[239,6574,"6"]
[240,6621,"5"]
[241,6641,"2"]
[242,6688,"7"]
[243,6717,"1"]
[244,6782,"2"]
[245,6784,"0"]
[246,6795,"1"]
[247,6852,"9"]
[248,6854,"0"]
[249,6910,"9"]
[250,6929,"1"]
[251,6959,"4"]
[252,6988,"5"]
[253,7027,"6"]
[254,7046,"4"]
[255,7085,"8"]
[256,7115,"5"]
[257,7153,"6"]
[258,7181,"6"]
[259,7229,"9"]
[260,7258,"2"]
[261,7288,"3"]
[262,7317,"4"]
[263,7383,"6"]
[264,7385,"0"]
[265,7415,"3"]
[266,7435,"4"]
[267,7474,"8"]
[268,7530,"6"]
[269,7569,"1"]
[270,7571,"0"]
[271,7609,"4"]
[272,7639,"5"]
[273,7678,"4"]
[274,7716,"3"]
[275,7736,"2"]
[276,7766,"6"]
[277,7805,"6"]
[278,7826,"4"]
[279,7873,"8"]
[280,7912,"2"]
[281,7933,"1"]
[282,7971,"3"]
[283,7991,"3"]
[284,8030,"9"]
[285,8060,"3"]
[286,8118,"6"]
[287,8120,"0"]
[288,8168,"7"]
[289,8189,"2"]
[290,8264,"6"]
[291,8266,"0"]
[292,8287,"2"]
[293,8317,"4"]
[294,8374,"9"]
[295,8395,"1"]
[296,8443,"4"]
[297,8464,"1"]
[298,8485,"2"]
[299,8524,"7"]
[300,8544,"3"]
[301,8593,"7"]
[302,8623,"2"]
...
Julia
Julia comes with built-in support for computing π in arbitrary precision (using the GNU MPFR library). This implementation computes π at precisions that are repeatedly doubled as more digits are needed, printing one digit at a time and never terminating (until it runs out of memory) as specified:
let prec = precision(BigFloat), spi = "", digit = 1
while true
if digit > lastindex(spi)
prec *= 2
setprecision(prec)
spi = string(big(π))
end
print(spi[digit])
digit += 1
end
end
Output:
3.141592653589793238462643383279502884195e69399375105820974944592307816406286198e9862803482534211706798214808651328230664709384460955058223172535940812848115e450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724586997e0631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526357e8277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201...
Kotlin
// version 1.1.2
import java.math.BigInteger
val ZERO = BigInteger.ZERO
val ONE = BigInteger.ONE
val TWO = BigInteger.valueOf(2L)
val THREE = BigInteger.valueOf(3L)
val FOUR = BigInteger.valueOf(4L)
val SEVEN = BigInteger.valueOf(7L)
val TEN = BigInteger.TEN
fun calcPi() {
var nn: BigInteger
var nr: BigInteger
var q = ONE
var r = ZERO
var t = ONE
var k = ONE
var n = THREE
var l = THREE
var first = true
while (true) {
if (FOUR * q + r - t < n * t) {
print(n)
if (first) { print ("."); first = false }
nr = TEN * (r - n * t)
n = TEN * (THREE * q + r) / t - TEN * n
q *= TEN
r = nr
}
else {
nr = (TWO * q + r) * l
nn = (q * SEVEN * k + TWO + r * l) / (t * l)
q *= k
t *= l
l += TWO
k += ONE
n = nn
r = nr
}
}
}
fun main(args: Array<String>) = calcPi()
- Output:
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745...
Lambdatalk
1) We can build a lambdatalk function using the lib_BN javascript library.
{require lib_BN}
{def genpi
{def genpi.loop
{lambda {:n :pi :q :r :t :i :z}
{if {> :z :n}
then :pi
else {let { {:n :n} {:pi :pi} {:q :q} {:r :r} {:t :t} {:i :i} {:z :z}
{:digit {BN./ {BN.+ {BN.* {BN.- {BN.* :i 27} 12} :q}
{BN.* :r 5} }
{BN.* :t 5} } }
{:u {BN.* {BN.+ {BN.* :i 3} 1}
{BN.* 3 {BN.+ {BN.* :i 3} 2} } } }
} {genpi.loop :n
{BN.+ :pi :digit}
{BN.* {BN.* :q 1}
{BN.* :i {BN.- {BN.* :i 2} 1} }}
{BN.* {BN.* :u 1}
{BN.+ {BN.* :q {BN.- {BN.* :i 5} 2} }
{BN.- :r {BN.* :t :digit} }}}
{BN.* :t :u}
{BN.+ :i 1}
{+ :z 1}} }}}}
{lambda {:n}
{genpi.loop :n # 1 180 60 2 0} }}
-> genpi
We can generate π with 72 digits in about 500ms.
{BN.DEC 72}
-> 72 digits
{genpi 60}
-> 3.141592653589793238462643383279502884197169399375105820974944592307816406
To go further the best is to build a javascript primitive using the script special form.
{script
LAMBDATALK.DICT["spigot"] = function() {
function generateDigitsOfPi(max) {
var pi = "";
var z = 0;
var q = 1n;
var r = 180n;
var t = 60n;
var i = 2n;
while (z < max) {
var digit = ((i * 27n - 12n) * q + r * 5n) / (t * 5n);
pi += digit;
var u = (i * 3n + 1n) * 3n * (i * 3n + 2n);
r = u * 10n * (q * (i * 5n - 2n) + (r - t * digit));
q = q * 10n * i * (i * 2n - 1n);
i = i + 1n;
t = t * u;
z++;
}
return pi
}
var args = arguments[0].trim();
return generateDigitsOfPi( args );
};
}
We can generate 1000 digits of π in about 70ms
3.{W.rest {spigot 100}}
-> 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198
Lasso
Based off Dik T. Winter's C implementation of Beeler et al. 1972, Item 120.
#!/usr/bin/lasso9
define generatePi => {
yield currentCapture
local(r = array(), i, k, b, d, c = 0, x)
with i in generateSeries(1, 2800)
do #r->insert(2000)
with k in generateSeries(2800, 1, -14)
do {
#d = 0
#i = #k
while(true) => {
#d += #r->get(#i) * 10000
#b = 2 * #i - 1
#r->get(#i) = #d % #b
#d /= #b
#i--
!#i ? loop_abort
#d *= #i
}
#x = (#c + #d / 10000)
yield (#k == 2800 ? ((#x * 0.001)->asstring(-precision = 3)) | #x->asstring(-padding=4, -padChar='0'))
#c = #d % 10000
}
}
local(pi_digits) = generatePi
loop(200) => {
stdout(#pi_digits())
}
Output (first 100 places):
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067
Liberty BASIC
Pretty slow if you run for over 100 digits...
ndigits = 0
q = 1
r = 0
t = q
k = q
n = 3
L = n
first = 666 ' ANY non-zero =='true' in LB.
while ndigits <100
if ( 4 *q +r -t) <( n *t) then
print n;
ndigits =ndigits +1
if not( ndigits mod 40) then print: print " ";
if first =666 then first = 0: print ".";
nr =10 *( r -n *t)
n =int( ( (10 *( 3 *q +r)) /t) -10 *n)
q =q *10
r =nr
else
nr =( 2 *q +r) *L
nn =(q *( 7 *k +2) +r *L) /( t *L)
q =q *k
t =t *L
L =L +2
k =k +1
n =int( nn)
r =nr
end if
scan
wend
end3.141592653589793238462643383279502884197 1693993751058209749445923078164062862089 98628034825342117067
Lua
a = {}
n = 1000
len = math.modf( 10 * n / 3 )
for j = 1, len do
a[j] = 2
end
nines = 0
predigit = 0
for j = 1, n do
q = 0
for i = len, 1, -1 do
x = 10 * a[i] + q * i
a[i] = math.fmod( x, 2 * i - 1 )
q = math.modf( x / ( 2 * i - 1 ) )
end
a[1] = math.fmod( q, 10 )
q = math.modf( q / 10 )
if q == 9 then
nines = nines + 1
else
if q == 10 then
io.write( predigit + 1 )
for k = 1, nines do
io.write(0)
end
predigit = 0
nines = 0
else
io.write( predigit )
predigit = q
if nines ~= 0 then
for k = 1, nines do
io.write( 9 )
end
nines = 0
end
end
end
end
print( predigit )
03141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086 ...
M2000 Interpreter
We can ask for 200 digits, but we can remove Digits-- in While Digits {} to print endless number of digits (without good precision). Algorithm developed after reading [4] and [5]
A Faster version handling console refresh time (and os shared time). M2000 run on an environment, which is loop event, and console is actual a form, a window. We can stop execution using Esc, Ctrl+C and Break keys, without stopping the interpreter (which is an application for Windows Os, written in Visual Basic 6,a s an ActiveX dll with a window manager on top of Vb forms).
Module Checkpi {
Module FindPi(Digits){
Digits++
n=Int(3.32*Digits)
PlusOne=Lambda N=0% -> {
=N
N++
}
PlusTwo=Lambda N=1% -> {
=N
N+=2
}
Dim A(n)<<PlusOne(), B(n)<<PlusTwo()
Dim Ten(n), CarrierOver(n), Sum(n),Remainder(n)=2
OutPutDigits=Digits
Predigits=Stack
CallBack=lambda fl=true, Chars=0 (x)->{
Print x;
Chars++
If fl then Print "." : Print " "; : fl=false : Chars=0 : exit
If Chars=50 then {
Print
Print " ";
Chars=0
Refresh
} else.if (Chars mod 5)=0 then {
Print " ";
Refresh
}
\\ explicitly refresh output layer, using Fast ! mode of speed
}
Print "Pi=";
While Digits {
NextDigit(&CallBack, &Digits)
}
print
Refresh
Sub NextDigit(&f, &D)
CarrierOver=0
For k=n-1 to 1 {
Ten(k)=Remainder(k)*10%
CarrierOver(k)=CarrierOver
Sum(k)=Ten(k)+CarrierOver(k)
q=Sum(k) div B(k)
Remainder(k)=Sum(k)-B(k)*q
CarrierOver=A(k)*q
}
Ten(0)=Remainder(0)*10%
CarrierOver(0)=CarrierOver
Sum(0)=Ten(0)+CarrierOver(0)
q=Sum(0) div 10%
Remainder(0)=Sum(0)-10%*q
if q<>9 and q<>10 then {
Stack Predigits {
While not empty {
Call f(Number)
if D>0 then D--
If D=0 then flush ' empty stack
}
Push q
}
} else.if q=9 Then {
Stack Predigits { Data q }
} else {
Stack Predigits {
While not empty {
Call f((Number+1) mod 10)
if D>0 then D--
If D=0 then flush ' empty stack
}
Push 0
}
}
End Sub
}
\\ reduce time to share with OS
\\ Need explicitly use of refresh output layer (M2000 console)
\\ Slow for a screen refresh per statement and give more time to OS
Rem Set Slow
\\ Fast is normal screen refresh, per Refresh time, and give standard time to OS
Rem Set Fast
\\ Fast ! use Refresh for screen refresh, and give less time o OS than standard
\\ Esc key work when Refresh executed (and OS get little time)
Set Fast !
FindPi 4
FindPi 28
Print Pi ' pi in M2000 is Decimal type with 29 digits (1 plus 28 after dot, is same as FindPi 28)
Refresh
FindPi 50
}
Flush ' empty stack of values
CheckPi
List ' no variables exist
Modules ? ' current module exist
Stack ' Stack of values ' has to be empty, we didn't use current stack for values.Mathematica / Wolfram Language
User can interrupt computation using "Alt+." or "Cmd+." on a Mac.
WriteString[$Output, "3."];
For[i = -1, True, i--,
WriteString[$Output, RealDigits[Pi, 10, 1, i][[1, 1]]]; Pause[.05]];
MATLAB
Requires the Variable Precision Integer (vpi) Toolbox
function pi_str = piSpigot(N)
% Return N digits of pi using Gibbons's first spigot algorithm.
% If N is omitted, the digits are printed ad infinitum.
% Uses the expansion
% pi = sum_{i=0} (i!)^2 2^{i+1} /(2i+1)!
% = 2 + 1/3 * ( 2 + 2/5 * (2 + 3/7 * ( 2 + 4/9 * ( ..... )))))
% = (2 + 1/3 *)(2 + 2/5 *)(2 + 3/7 *)...
% where the terms in the last expression represent Linear Fractional
% Transforms (LFTs).
%
% Requires the Variable Precision Integer (vpi) Toolbox
%
% Reference:
% "Unbounded Spigot Algorithms for the Digits of Pi" by J. Gibbons, 2004
% American Mathematical Monthly, vol. 113.
if nargin < 1
N = Inf;
lineLength = 50;
else
pi_str = repmat(' ',1,N);
end
q = vpi(1);
r = vpi(0);
t = vpi(1);
k = 1; % If printing more than 3E15 digits, use k = vpi(1);
i = 1;
first_digit = true;
while i <= N
threeQplusR = 3*q + r;
n = double(threeQplusR / t);
if q+threeQplusR < (n+1)*t
d = num2str(n);
if isinf(N)
fprintf(1,'%s', d);
if first_digit
fprintf(1,'.');
first_digit = false;
i = i+1;
end
if i == lineLength
fprintf(1,'\n');
i = 0;
end
else
pi_str(i) = d;
end
q = 10*q;
r = 10*(r-n*t);
i = i + 1;
else
t = (2*k+1)*t;
r = (4*k+2)*q + (2*k+1)*r;
q = k*q;
k = k + 1;
end
end
end
>> piSpigot 3.141592653589793238462643383279502884197169399375 10582097494459230781640628620899862803482534211706 79821480865132823066470938446095505822317253594081 28481117450284102701938521105559644622948954930381 96442881097566593344612847564823378678316527120190 91456485669234603486104543266482133936072602491412
MiniScript
Calculate pi using the Rabinowitz-Wagon algorithm
digits = input("Enter number of digits to calculate after decimal point: ").val
// I've seen variations of this "precision" calculation from
// 10 * digits
// to
// floor(10 * digits / 3) + 16
// A larger value provides a more precise calculation but also
// takes longer to run. Based on my testing, this calculation
// below for precision produces accurate output for inputs
// from 1 to 4000 - haven't tried larger than this.
precision = floor(10 * digits / 3) + 4
A = [2] * precision
nines = 0
predigit = 0
cnt = 0
while cnt <= digits
carry = 0
for i in range(precision - 1, 1, -1)
temp = 10 * A[i] + carry * i
A[i] = temp % (2 * i - 1)
carry = floor(temp/(2 * i - 1))
end for
A[1] = carry % 10
carry = floor(carry / 10)
current = carry
if current == 9 then
nines += 1
else if current == 10 then
print (predigit+1), ""
cnt += 1
if nines > 0 then
print "9" * nines, ""
cnt += nines
end if
predigit = 0
nines = 0
else
// the first digit produced is always a zero
// don't need to see that
if cnt != 0 then print predigit, ""
cnt += 1
predigit = current
if nines > 0 then
print "9" * nines, ""
cnt += nines
end if
nines = 0
end if
if cnt == 2 then print ".", ""
end while
print str(predigit) * (cnt < digits + 2)
- Output:
Enter number of digits to calculate after decimal point: 1000 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903690113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051329005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710199031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066139019278766111959092164201989
Nanoquery
q = 1; r = 0; t = 1
k = 1; n = 3; l = 3
nn = null; nr = null
first = true
while true
if (((4 * q) + r) - t) < (n * t)
print n
if first
print "."
first = false
end
nr = int(10 * (r - (n * t)))
n = int((10 * ((3 * q) + r)) / t) - (10 * n)
q *= 10
r = nr
else
nr = int(((2 * q) + r) * l)
nn = int((((q * (7 * k)) + 2) + (r * l)) / (t * l))
q *= k
t *= l
l += 2
k += 1
n = nn
r = nr
end if
end while- Output:
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...
NetRexx
/* NetRexx */
options replace format comments java crossref symbols binary
import java.math.BigInteger
runSample(arg)
return
-- 07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)07:11, 27 August 2022 (UTC)~~
method runSample(arg) private static
parse arg places .
if places = '' then places = -1
TWO = BigInteger.valueOf(2)
THREE = BigInteger.valueOf(3)
FOUR = BigInteger.valueOf(4)
SEVEN = BigInteger.valueOf(7)
q_ = BigInteger.ONE
r_ = BigInteger.ZERO
t_ = BigInteger.ONE
k_ = BigInteger.ONE
n_ = BigInteger.valueOf(3)
l_ = BigInteger.valueOf(3)
nn = BigInteger
nr = BigInteger
first = isTrue()
digitCt = 0
loop forever
if FOUR.multiply(q_).add(r_).subtract(t_).compareTo(n_.multiply(t_)) == -1 then do
digitCt = digitCt + 1
if places > 0 & digitCt - 1 > places then leave
say n_'\-'
if first then do
say '.\-'
first = isFalse()
end
nr = BigInteger.TEN.multiply(r_.subtract(n_.multiply(t_)))
n_ = BigInteger.TEN.multiply(THREE.multiply(q_).add(r_)).divide(t_).subtract(BigInteger.TEN.multiply(n_))
q_ = q_.multiply(BigInteger.TEN)
r_ = nr
end
else do
nr = TWO.multiply(q_).add(r_).multiply(l_)
nn = q_.multiply((SEVEN.multiply(k_))).add(TWO).add(r_.multiply(l_)).divide(t_.multiply(l_))
q_ = q_.multiply(k_)
t_ = t_.multiply(l_)
l_ = l_.add(TWO)
k_ = k_.add(BigInteger.ONE)
n_ = nn
r_ = nr
end
end
say
return
method isTrue() private static returns boolean
return (1 == 1)
method isFalse() private static returns boolean
return \isTrue()
- Output:
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...
Nim
import bigints
var
tmp1, tmp2, tmp3, acc, k = initBigInt(0)
den, num, k2 = initBigInt(1)
proc extractDigit(): int32 =
if num > acc:
return -1
tmp3 = num shl 1 + num + acc
tmp1 = tmp3 div den
tmp2 = tmp3 mod den + num
if tmp2 >= den:
return -1
result = int32(tmp1.limbs[0])
proc eliminateDigit(d: int32) =
acc -= den * d
acc *= 10
num *= 10
proc nextTerm() =
k += 1
k2 += 2
acc += num shl 1
acc *= k2
den *= k2
num *= k
var i = 0
while true:
var d: int32 = -1
while d < 0:
nextTerm()
d = extractDigit()
stdout.write chr(ord('0') + d)
inc i
if i == 40:
echo ""
i = 0
eliminateDigit d
- Output:
3141592653589793238462643383279502884197 1693993751058209749445923078164062862089 9862803482534211706798214808651328230664 7093844609550582231725359408128481117450 ...
Another version which avoids to access the internals of big integers:
import bignum
proc calcPi() =
var
q = newInt(1)
r = newInt(0)
t = newInt(1)
k = newInt(1)
n = newInt(3)
l = newInt(3)
var count = 0
while true:
if 4 * q + r - t < n * t:
stdout.write n
inc count
if count == 40: (echo ""; count = 0)
let nr = 10 * (r - n * t)
n = 10 * (3 * q + r) div t - 10 * n
q *= 10
r = nr
else:
let nr = (2 * q + r) * l
let nn = (7 * q * k + 2 + r * l) div (t * l)
q *= k
t *= l
l += 2
k += 1
n = nn
r = nr
calcPi()
- Output:
3141592653589793238462643383279502884197 1693993751058209749445923078164062862089 9862803482534211706798214808651328230664 7093844609550582231725359408128481117450 ...
OCaml
The Constructive Real library Creal contains an infinite-precision Pi, so we can just print out its digits.
open Creal;;
let block = 100 in
let segment n =
let s = to_string pi (n*block) in
String.sub s ((n-1)*block) block in
let counter = ref 1 in
while true do
print_string (segment !counter);
flush stdout;
incr counter
done
However that is cheating if you want to see an algorithm to generate Pi. Since the Spigot algorithm is already used in the pidigits program, this implements Machin's formula.
open Num
(* series for: c*atan(1/k) *)
class atan_sum c k = object
val kk = k*/k
val mutable n = 0
val mutable kpow = k
val mutable pterm = c*/k
val mutable psum = Int 0
val mutable sum = c*/k
method next =
n <- n+1; kpow <- kpow*/kk;
let t = c*/kpow//(Int (2*n+1)) in
pterm <- if n mod 2 = 0 then t else minus_num t;
psum <- sum;
sum <- sum +/ pterm
method error = abs_num pterm
method bounds = if pterm </ Int 0 then (sum, psum) else (psum, sum)
end;;
let inv i = (Int 1)//(Int i) in
let t1 = new atan_sum (Int 16) (inv 5) in
let t2 = new atan_sum (Int (-4)) (inv 239) in
let base = Int 10 in
let npr = ref 0 in
let shift = ref (Int 1) in
let d_acc = inv 10000 in
let acc = ref d_acc in
let shown = ref (Int 0) in
while true do
while t1#error >/ !acc do t1#next done;
while t2#error >/ !acc do t2#next done;
let (lo1, hi1), (lo2, hi2) = t1#bounds, t2#bounds in
let digit x = int_of_num (floor_num ((x -/ !shown) */ !shift)) in
let d, d' = digit (lo1+/lo2), digit (hi1+/hi2) in
if d = d' then (
print_int d;
if !npr = 0 then print_char '.';
flush stdout;
shown := !shown +/ ((Int d) // !shift);
incr npr; shift := !shift */ base;
) else (acc := !acc */ d_acc);
done
Oforth
: calcPiDigits
| q r t k n l |
1 ->q 0 ->r 1 ->t 1 ->k 3 ->n 3 -> l
while( true ) [
4 q * r + t - n t * < ifTrue: [
n print
r n t * - 10 *
3 q * r + 10 * t / n 10 * - ->n ->r
q 10 * ->q
]
else: [
2 q * r + l *
7 k * q * 2 + r l * + t l * / ->n ->r
k q * ->q
t l * ->t
l 2 + ->l
k 1+ ->k
]
] ;Ol
; 'numbers' is count of numbers or #false for eternal pleasure.
(define (pi numbers)
(let loop ((q 1) (r 0) (t 1) (k 1) (n 3) (l 3) (numbers numbers))
(unless (eq? numbers 0)
(if (< (- (+ (* 4 q) r) t) (* n t))
(begin
(display n)
(loop (* q 10)
(* 10 (- r (* n t)))
t
k
(- (div (* 10 (+ (* 3 q) r)) t) (* 10 n))
l
(if numbers (- numbers 1))))
(begin
(loop (* q k)
(* (+ (* 2 q) r) l)
(* t l)
(+ k 1)
(div (+ (* q (* 7 k)) 2 (* r l)) (* t l))
(+ l 2)
(if numbers (- numbers 1))))))))
(pi #false)
- Output:
31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132 82306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475 64823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925 40917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480 74462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539 21717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958 53710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328 16096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598 25349042875546873115956286388235378759375195778185778053217122680661300192787661119590921642019893809525720106548 58632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331 1686172785588907509838175463
PARI/GP
Uses the built-in Brent-Salamin arithmetic-geometric mean iteration.
pi()={
my(x=Pi,n=0,t);
print1("3.");
while(1,
if(n>=default(realprecision),
default(realprecision,default(realprecision)*2);
x=Pi
);
print1(floor(x*10^n++)%10)
)
};Swift
//
// main.swift
// pi digits
//
// Created by max goren on 11/11/21.
// Copyright © 2021 maxcodes. All rights reserved.
//
import Foundation
var r = [Int]()
var i = 0
var k = 2800
var b = 0
var c = 0
var d = 0
for _ in 0...2800 {
r.append(2000);
}
while k > 0 {
d = 0;
i = k;
while (true) {
d = d + r[i] * 10000
b = 2 * i - 1
r[i] = d % b
d = d / b
i = i - 1
if i == 0 {
break;
}
d = d * i;
}
print(c + d / 10000, "")
c = d % 10000
k = k - 14
}
Pascal
With minor editing changes as published by Stanley Rabinowitz in [6]. Minor improvement of <user>Mischi</user> { speedup ~2 ( n=10000 , rumtime 4s-> 1,44s fpc 2.6.4 -O3 }, by calculating only necessary digits up to n.
Program Pi_Spigot;
const
n = 1000;
len = 10*n div 3;
var
j, k, q, nines, predigit: integer;
a: array[0..len] of longint;
function OneLoop(i:integer):integer;
var
x: integer;
begin
{Only calculate as far as needed }
{+16 for security digits ~5 decimals}
i := i*10 div 3+16;
IF i > len then
i := len;
result := 0;
repeat {Work backwards}
x := 10*a[i] + result*i;
result := x div (2*i - 1);
a[i] := x - result*(2*i - 1);//x mod (2*i - 1)
dec(i);
until i<= 0 ;
end;
begin
for j := 1 to len do
a[j] := 2; {Start with 2s}
nines := 0;
predigit := 0; {First predigit is a 0}
for j := 1 to n do
begin
q := OneLoop(n-j);
a[1] := q mod 10;
q := q div 10;
if q = 9 then
nines := nines + 1
else
if q = 10 then
begin
write(predigit+1);
for k := 1 to nines do
write(0); {zeros}
predigit := 0;
nines := 0
end
else
begin
write(predigit);
predigit := q;
if nines <> 0 then
begin
for k := 1 to nines do
write(9);
nines := 0
end
end
end;
writeln(predigit);
end.
Output:
% ./Pi_Spigot 03141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198
Perl
Perl being what it is, there are many ways to do this with many variations. With a fixed number of digits and the Math::BigInt::GMP library installed, the [[Arithmetic-geometric mean/Calculate Pi code will be much faster than any of these methods other than some of the modules. If Math::GMP is installed, then replacing "use bigint" with "use Math::GMP qw/:constant/" in either the Raku spigot or Machin methods below will be pretty fast. They are not too bad if the Math::BigInt::GMP library is installed. With the default Math::BigInt backend, the AGM code isn't very fast and the Raku spigot and Machin methods are very slow.
Simple Spigot
This takes a numer-of-digits argument, but we can make it large (albeit using memory and some startup time). Unlike the other two, this uses no modules and does not require bigints so is worth showing.
sub pistream {
my $digits = shift;
my(@out, @a);
my($b, $c, $d, $e, $f, $g, $i, $d4, $d3, $d2, $d1);
my $outi = 0;
$digits++;
$b = $d = $e = $g = $i = 0;
$f = 10000;
$c = 14 * (int($digits/4)+2);
@a = (20000000) x $c;
print "3.";
while (($b = $c -= 14) > 0 && $i < $digits) {
$d = $e = $d % $f;
while (--$b > 0) {
$d = $d * $b + $a[$b];
$g = ($b << 1) - 1;
$a[$b] = ($d % $g) * $f;
$d = int($d / $g);
}
$d4 = $e + int($d/$f);
if ($d4 > 9999) {
$d4 -= 10000;
$out[$i-1]++;
for ($b = $i-1; $out[$b] == 1; $b--) {
$out[$b] = 0;
$out[$b-1]++;
}
}
$d3 = int($d4/10);
$d2 = int($d3/10);
$d1 = int($d2/10);
$out[$i++] = $d1;
$out[$i++] = $d2-$d1*10;
$out[$i++] = $d3-$d2*10;
$out[$i++] = $d4-$d3*10;
print join "", @out[$i-15 .. $i-15+3] if $i >= 16;
}
# We've closed the spigot. Print the remainder without rounding.
print join "", @out[$i-15+4 .. $digits-2], "\n";
}
Raku spigot
As mentioned earlier, replacing "use bigint" with "use Math::GMP qw/:constant/" will result in many orders of magnitude faster performance.
use bigint try=>"GMP";
sub stream {
my ($next, $safe, $prod, $cons, $z, $x) = @_;
$x = $x->();
sub {
while (1) {
my $y = $next->($z);
if ($safe->($z, $y)) {
$z = $prod->($z, $y);
return $y;
} else {
$z = $cons->($z, $x->());
}
}
}
}
sub extr {
use integer;
my ($q, $r, $s, $t) = @{shift()};
my $x = shift;
($q * $x + $r) / ($s * $x + $t);
}
sub comp {
my ($q, $r, $s, $t) = @{shift()};
my ($u, $v, $w, $x) = @{shift()};
[$q * $u + $r * $w,
$q * $v + $r * $x,
$s * $u + $t * $w,
$s * $v + $t * $x];
}
my $pi_stream = stream
sub { extr shift, 3 },
sub { my ($z, $n) = @_; $n == extr $z, 4 },
sub { my ($z, $n) = @_; comp([10, -10*$n, 0, 1], $z) },
\&comp,
[1, 0, 0, 1],
sub { my $n = 0; sub { $n++; [$n, 4 * $n + 2, 0, 2 * $n + 1] } },
;
$|++;
print $pi_stream->(), '.';
print $pi_stream->() while 1;
Machin's Formula
Here is an original Perl 5 code, using Machin's formula. Not the fastest program in the world. As with the previous code, using either Math::GMP or Math::BigInt::GMP instead of the default bigint Calc backend will make it run thousands of times faster.
use bigint try=>"GMP";
# Pi/4 = 4 arctan 1/5 - arctan 1/239
# expanding it with Taylor series with what's probably the dumbest method
my ($ds, $ns) = (1, 0);
my ($n5, $d5) = (16 * (25 * 3 - 1), 3 * 5**3);
my ($n2, $d2) = (4 * (239 * 239 * 3 - 1), 3 * 239**3);
sub next_term {
my ($coef, $p) = @_[1, 2];
$_[0] /= ($p - 4) * ($p - 2);
$_[0] *= $p * ($p + 2) * $coef**4;
}
my $p2 = 5;
my $pow = 1;
$| = 1;
for (my $x = 5; ; $x += 4) {
($ns, $ds) = ($ns * $d5 + $n5 * $pow * $ds, $ds * $d5);
next_term($d5, 5, $x);
$n5 = 16 * (5 * 5 * ($x + 2) - $x);
while ($d5 > $d2) {
($ns, $ds) = ($ns * $d2 - $n2 * $pow * $ds, $ds * $d2);
$n2 = 4 * (239 * 239 * ($p2 + 2) - $p2);
next_term($d2, 239, $p2);
$p2 += 4;
}
my $ppow = 1;
while ($pow * $n5 * 5**4 < $d5 && $pow * $n2 * $n2 * 239**4 < $d2) {
$pow *= 10;
$ppow *= 10;
}
if ($ppow > 1) {
$ns *= $ppow;
#FIX? my $out = $ns->bdiv($ds); # bugged?
my $out = $ns / $ds;
$ns %= $ds;
$out = ("0" x (length($ppow) - length($out) - 1)) . $out;
print $out;
}
if ( $p2 % 20 == 1) {
my $g = Math::BigInt::bgcd($ds, $ns);
$ds /= $g;
$ns /= $g;
}
}
Modules
While no current CPAN module does continuous printing, there are (usually fast) ways to get digits of Pi. Examples include:
use ntheory qw/Pi/;
say Pi(10000);
use Math::Pari qw/setprecision Pi/;
setprecision(10000);
say Pi;
use Math::MPFR;
my $pi = Math::MPFR->new();
Math::MPFR::Rmpfr_set_prec($pi, int(10000 * 3.322)+40);
Math::MPFR::Rmpfr_const_pi($pi, 0);
say Math::MPFR::Rmpfr_get_str($pi, 10, 10000, 0);
use Math::BigFloat try=>"GMP"; # Slow without Math::BigInt::GMP installed
say Math::BigFloat::bpi(10000); # For over ~2k digits, slower than AGM
use Math::Big qw/pi/; # Very slow
say pi(10000);
Phix
I already had this golf entry to hand. Prints 2400 places, change the 8400 (derived from 2400*14/4) as needed, but I've not tested > that.
with javascript_semantics integer a=10000,b,c=8400,d,e=0,g sequence f=repeat(floor(a/5),c+1) while c>0 do g=2*c d=0 b=c while b>0 do d+=f[b]*a g-=1 f[b]=remainder(d, g) d=floor(d/g) g-=1 b-=1 if b!=0 then d*=b end if end while printf(1,"%04d",e+floor(d/a)) c-=14 e = remainder(d,a) end while
Someone was benchmarking the above against Lua, so I translated the Lua entry, and upped it to 2400 places, for a fairer test.
with javascript_semantics integer n = 2400, len = floor(10*n/3) sequence a = repeat(2,len) integer nines = 0, predigit = 0 string res = "" for j=1 to n do integer q = 0 for i=len to 1 by -1 do integer x = 10*a[i]+q*i, d = 2*i-1 a[i] = remainder(x,d) q = floor(x/d) end for a[1] = remainder(q,10) q = floor(q/10) if q==9 then nines = nines+1 else integer nine = '9' if q==10 then predigit += 1 q = 0 nine = '0' end if res &= predigit+'0'&repeat(nine,nines) predigit = q nines = 0 end if end for res &= predigit+'0' puts(1,res)
Picat
go =>
pi2(1,0,1,1,3,3,0),
nl.
pi2(Q,R,T,K,N,L,C) =>
if C == 50 then
nl,
pi2(Q,R,T,K,N,L,0)
else
if (4*Q + R-T) < (N*T) then
print(N),
P := 10*(R-N*T),
pi2(Q*10, P, T, K, ((10*(3*Q+R)) div T)-10*N, L,C+1)
else
P := (2*Q+R)*L,
M := (Q*(7*K)+2+(R*L)) div (T*L),
H := L+2,
J := K+ 1,
pi2(Q*K, P, T*L, J, M, H, C)
end
end,
nl.- Output:
31415926535897932384626433832795028841971693993751 05820974944592307816406286208998628034825342117067 98214808651328230664709384460955058223172535940812 84811174502841027019385211055596446229489549303819 64428810975665933446128475648233786783165271201909 14564856692346034861045432664821339360726024914127 37245870066063155881748815209209628292540917153643 67892590360011330530548820466521384146951941511609 43305727036575959195309218611738193261179310511854 80744623799627495673518857527248912279381830119491 29833673362440656643086021394946395224737190702179 86094370277053921717629317675238467481846766940513 [Ctrl-C]
PicoLisp
The following script uses the spigot algorithm published by Jeremy Gibbons. Hit Ctrl-C to stop it.
#!/usr/bin/picolisp /usr/lib/picolisp/lib.l
(de piDigit ()
(job '((Q . 1) (R . 0) (S . 1) (K . 1) (N . 3) (L . 3))
(while (>= (- (+ R (* 4 Q)) S) (* N S))
(mapc set '(Q R S K N L)
(list
(* Q K)
(* L (+ R (* 2 Q)))
(* S L)
(inc K)
(/ (+ (* Q (+ 2 (* 7 K))) (* R L)) (* S L))
(+ 2 L) ) ) )
(prog1 N
(let M (- (/ (* 10 (+ R (* 3 Q))) S) (* 10 N))
(setq Q (* 10 Q) R (* 10 (- R (* N S))) N M) ) ) ) )
(prin (piDigit) ".")
(loop
(prin (piDigit))
(flush) )Output:
3.14159265358979323846264338327950288419716939937510582097494459 ...
PL/I
/* Uses the algorithm of S. Rabinowicz and S. Wagon, "A Spigot Algorithm */
/* for the Digits of Pi". */
(subrg, fofl, size):
Pi_Spigot: procedure options (main); /* 21 January 2012. */
declare (n, len) fixed binary;
n = 1000;
len = 10*n / 3;
begin;
declare ( i, j, k, q, nines, predigit ) fixed binary;
declare x fixed binary (31);
declare a(len) fixed binary (31);
a = 2; /* Start with 2s */
nines, predigit = 0; /* First predigit is a 0 */
do j = 1 to n;
q = 0;
do i = len to 1 by -1; /* Work backwards */
x = 10*a(i) + q*i;
a(i) = mod (x, (2*i-1));
q = x / (2*i-1);
end;
a(1) = mod(q, 10); q = q / 10;
if q = 9 then nines = nines + 1;
else if q = 10 then
do;
put edit(predigit+1) (f(1));
do k = 1 to nines;
put edit ('0')(a(1)); /* zeros */
end;
predigit, nines = 0;
end;
else
do;
put edit(predigit) (f(1)); predigit = q;
do k = 1 to nines; put edit ('9')(a(1)); end;
nines = 0;
end;
end;
put edit(predigit) (f(1));
end; /* of begin block */
end Pi_Spigot;output:
03141592653589793238462643383279502884197169399375105820974944592307816406286208 99862803482534211706798214808651328230664709384460955058223172535940812848111745 02841027019385211055596446229489549303819644288109756659334461284756482337867831 65271201909145648566923460348610454326648213393607260249141273724587006606315588 17488152092096282925409171536436789259036001133053054882046652138414695194151160 94330572703657595919530921861173819326117931051185480744623799627495673518857527 24891227938183011949129833673362440656643086021394946395224737190702179860943702 77053921717629317675238467481846766940513200056812714526356082778577134275778960 91736371787214684409012249534301465495853710507922796892589235420199561121290219 60864034418159813629774771309960518707211349999998372978049951059731732816096318 59502445945534690830264252230825334468503526193118817101000313783875288658753320 83814206171776691473035982534904287554687311595628638823537875937519577818577805 32171226806613001927876611195909216420198
Powershell
With some tweaking. Prints 100 digits a time. Total possible output limited by available memory.
Function Get-Pi ( $Digits )
{
$Big = [bigint[]](0..10)
$ndigits = 0
$Output = ""
$q = $t = $k = $Big[1]
$r = $Big[0]
$l = $n = $Big[3]
# Calculate first digit
$nr = ( $Big[2] * $q + $r ) * $l
$nn = ( $q * ( $Big[7] * $k + $Big[2] ) + $r * $l ) / ( $t * $l )
$q *= $k
$t *= $l
$l += $Big[2]
$k = $k + $Big[1]
$n = $nn
$r = $nr
$Output += [string]$n + '.'
$ndigits++
$nr = $Big[10] * ( $r - $n * $t )
$n = ( ( $Big[10] * ( 3 * $q + $r ) ) / $t ) - 10 * $n
$q *= $Big[10]
$r = $nr
While ( $ndigits -lt $Digits )
{
While ( $ndigits % 100 -ne 0 -or -not $Output )
{
If ( $Big[4] * $q + $r - $t -lt $n * $t )
{
$Output += [string]$n
$ndigits++
$nr = $Big[10] * ( $r - $n * $t )
$n = ( ( $Big[10] * ( 3 * $q + $r ) ) / $t ) - 10 * $n
$q *= $Big[10]
$r = $nr
}
Else
{
$nr = ( $Big[2] * $q + $r ) * $l
$nn = ( $q * ( $Big[7] * $k + $Big[2] ) + $r * $l ) / ( $t * $l )
$q *= $k
$t *= $l
$l += $Big[2]
$k = $k + $Big[1]
$n = $nn
$r = $nr
}
}
$Output
$Output = ""
}
}
Alternate version using .Net classes
[math]::pi
Outputs:
.Net digits of pi
3.14159265358979
Prolog
Using coroutine with freeze/2 predicate:
pi_spigot :-
pi(X),
forall(member(Y, X), write(Y)).
pi(OUT) :-
pi(1, 180, 60, 2, OUT).
pi(Q, R, T, I, OUT) :-
freeze(OUT,
( OUT = [Digit | OUT_]
-> U is 3 * (3 * I + 1) * (3 * I + 2),
Y is (Q * (27 * I - 12) + 5 * R) // (5 * T),
Digit is Y,
Q2 is 10 * Q * I * (2 * I - 1),
R2 is 10 * U * (Q * (5 * I - 2) + R - Y * T),
T2 is T * U,
I2 is I + 1,
pi(Q2, R2, T2, I2, OUT_)
; true)).
PureBasic
Calculate Pi, limited to ~24 M-digits for memory and speed reasons.
#SCALE = 10000
#ARRINT= 2000
Procedure Pi(Digits)
Protected First=#True, Text$
Protected Carry, i, j, sum
Dim Arr(Digits)
For i=0 To Digits
Arr(i)=#ARRINT
Next
i=Digits
While i>0
sum=0
j=i
While j>0
sum*j+#SCALE*arr(j)
Arr(j)=sum%(j*2-1)
sum/(j*2-1)
j-1
Wend
Text$ = RSet(Str(Carry+sum/#SCALE),4,"0")
If First
Text$ = ReplaceString(Text$,"3","3.")
First = #False
EndIf
Print(Text$)
Carry=sum%#SCALE
i-14
Wend
EndProcedure
If OpenConsole()
SetConsoleCtrlHandler_(?Ctrl,#True)
Pi(24*1024*1024)
EndIf
End
Ctrl:
PrintN(#CRLF$+"Ctrl-C was pressed")
End
Python
def calcPi():
q, r, t, k, n, l = 1, 0, 1, 1, 3, 3
while True:
if 4*q+r-t < n*t:
yield n
nr = 10*(r-n*t)
n = ((10*(3*q+r))//t)-10*n
q *= 10
r = nr
else:
nr = (2*q+r)*l
nn = (q*(7*k)+2+(r*l))//(t*l)
q *= k
t *= l
l += 2
k += 1
n = nn
r = nr
import sys
pi_digits = calcPi()
i = 0
for d in pi_digits:
sys.stdout.write(str(d))
i += 1
if i == 40: print(""); i = 0
output
3141592653589793238462643383279502884197 1693993751058209749445923078164062862089 9862803482534211706798214808651328230664 7093844609550582231725359408128481117450 2841027019385211055596446229489549303819 6442881097566593344612847564823378678316 5271201909145648566923460348610454326648 2133936072602491412737245870066063155881 7488152092096282925409171536436789259036 0011330530548820466521384146951941511609 4330572703657595919530921861173819326117 ...
Quackery
Quackery does not have variables, it has ancillary stacks. To expedite translation from Oforth, the first two definitions implement words equivalent to the Forth words VALUE and TO.
[ immovable
]this[ share ]done[ ] is value ( --> x )
[ ]'[ replace ] is to ( x --> )
[ value 1 ] is Q ( --> x )
[ value 0 ] is R ( --> x )
[ value 1 ] is T ( --> x )
[ value 1 ] is K ( --> x )
[ value 3 ] is N ( --> x )
[ value 3 ] is L ( --> x )
[ value 0 ] is chcount ( --> x )
[ echo
chcount dup 79 =
if cr
1+ 80 mod to chcount ] is printch
[ 4 Q * R + T - N T * < iff
[ N printch
R N T * - 10 *
3 Q * R + 10 * T / N 10 * - to N to R
Q 10 * to Q ]
else
[ 2 Q * R + L *
7 K * Q * 2 + R L * + T L * / to N to R
K Q * to Q
T L * to T
L 2 + to L
K 1+ to K ]
chcount again ]- Output:
31415926535897932384626433832795028841971693993751058209749445923078164062862089 98628034825342117067982148086513282306647093844609550582231725359408128481117450 28410270193852110555964462294895493038196442881097566593344612847564823378678316 52712019091456485669234603486104543266482133936072602491412737245870066063155881 74881520920962829254091715364367892590360011330530548820466521384146951941511609 43305727036575959195309218611738193261179310511854807446237996274956735188575272 48912279381830119491298336733624406566430860213949463952247371907021798609437027 70539217176293176752384674818467669405132000568127145263560827785771342757789609 17363717872146844090122495343014654958537105079227968925892354201995611212902196 08640344181598136297747713099605187072113499999983729780499510597317328160963185 95024459455346908302642522308253344685035261931188171010003137838752886587533208 38142061717766914730359825349042875546873115956286388235378759375195778185778053 21712268066130019278766111959092164201989380952572010654858632788659361533818279 68230301952035301852968995773622599413891249721775283479131515574857242454150695 95082953311686172785588907509838175463746493931925506040092770167113900984882401 28583616035637076601047101819429555961989467678374494482553797747268471040475346
… and so on.
R
suppressMessages(library(gmp))
ONE <- as.bigz("1")
TWO <- as.bigz("2")
THREE <- as.bigz("3")
FOUR <- as.bigz("4")
SEVEN <- as.bigz("7")
TEN <- as.bigz("10")
q <- as.bigz("1")
r <- as.bigz("0")
t <- as.bigz("1")
k <- as.bigz("1")
n <- as.bigz("3")
l <- as.bigz("3")
char_printed <- 0
how_many <- 1000
first <- TRUE
while (how_many > 0) {
if ((FOUR * q + r - t) < (n * t)) {
if (char_printed == 80) {
cat("\n")
char_printed <- 0
}
how_many <- how_many - 1
char_printed <- char_printed + 1
cat(as.integer(n))
if (first) {
cat(".")
first <- FALSE
char_printed <- char_printed + 1
}
nr <- as.bigz(TEN * (r - n * t))
n <- as.bigz(((TEN * (THREE * q + r)) %/% t) - (TEN * n))
q <- as.bigz(q * TEN)
r <- as.bigz(nr)
} else {
nr <- as.bigz((TWO * q + r) * l)
nn <- as.bigz((q * (SEVEN * k + TWO) + r * l) %/% (t * l))
q <- as.bigz(q * k)
t <- as.bigz(t * l)
l <- as.bigz(l + TWO)
k <- as.bigz(k + ONE)
n <- as.bigz(nn)
r <- as.bigz(nr)
}
}
cat("\n")
Output:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208 99862803482534211706798214808651328230664709384460955058223172535940812848111745 02841027019385211055596446229489549303819644288109756659334461284756482337867831 65271201909145648566923460348610454326648213393607260249141273724587006606315588 17488152092096282925409171536436789259036001133053054882046652138414695194151160 94330572703657595919530921861173819326117931051185480744623799627495673518857527 24891227938183011949129833673362440656643086021394946395224737190702179860943702 77053921717629317675238467481846766940513200056812714526356082778577134275778960 91736371787214684409012249534301465495853710507922796892589235420199561121290219 60864034418159813629774771309960518707211349999998372978049951059731732816096318 59502445945534690830264252230825334468503526193118817101000313783875288658753320 83814206171776691473035982534904287554687311595628638823537875937519577818577805 32171226806613001927876611195909216420198
Racket
Utilizing Jeremy Gibbons spigot algorithm and racket generator:
#lang racket
(require racket/generator)
(define pidig
(generator ()
(let loop ([q 1] [r 0] [t 1] [k 1] [n 3] [l 3])
(if (< (- (+ r (* 4 q)) t) (* n t))
(begin (yield n)
(loop (* q 10) (* 10 (- r (* n t))) t k
(- (quotient (* 10 (+ (* 3 q) r)) t) (* 10 n))
l))
(loop (* q k) (* (+ (* 2 q) r) l) (* t l) (+ 1 k)
(quotient (+ (* (+ 2 (* 7 k)) q) (* r l)) (* t l))
(+ l 2))))))
(for ([i (in-naturals)])
(display (pidig))
(when (zero? i) (display "." ))
(when (zero? (modulo i 80)) (newline)))
Output:
3.14159265358979323846264338327950288419716939937510...
Raku
(formerly Perl 6)
# based on http://www.mathpropress.com/stan/bibliography/spigot.pdf
sub stream(&next, &safe, &prod, &cons, $z is copy, @x) {
gather loop {
$z = safe($z, my $y = next($z)) ??
prod($z, take $y) !!
cons($z, @x[$++])
}
}
sub extr([$q, $r, $s, $t], $x) {
($q * $x + $r) div ($s * $x + $t)
}
sub comp([$q,$r,$s,$t], [$u,$v,$w,$x]) {
[$q * $u + $r * $w,
$q * $v + $r * $x,
$s * $u + $t * $w,
$s * $v + $t * $x]
}
my $pi :=
stream -> $z { extr($z, 3) },
-> $z, $n { $n == extr($z, 4) },
-> $z, $n { comp([10, -10*$n, 0, 1], $z) },
&comp,
<1 0 0 1>,
(1..*).map: { [$_, 4 * $_ + 2, 0, 2 * $_ + 1] }
for ^Inf -> $i {
print $pi[$i];
once print '.'
}
REXX
version 1
This REXX program calculates decimal digits of using John Machin's formula.
It should be noted that the program's mechanism spits out the next (new) decimal digit(s) of .
The REXX program uses the following formula to calculate :
┌─ ─┐ ┌─ ─┐
π │ 1 │ │ 1 │ John
─── = 4 ∙ arctan│ ─── │ - arctan│ ───── │ Machin's
4 │ 5 │ │ 239 │ formula
└─ ─┘ └─ ─┘
which expands into:
┌─ ─┐
│ 1 1 1 1 1 1 │
4 ∙ │ ─── - ────── + ────── - ────── + ────── - ──────── + ... │
│ 1 3 5 7 9 11 │
│ 1∙5 3∙5 5∙5 7∙5 9∙5 11∙5 │
└─ ─┘
┌─ ─┐
│ 1 1 1 1 1 1 │
- │ ─── - ────── + ────── - ────── + ────── - ──────── + ... │
│ 1 3 5 7 9 11 │
│ 1∙239 3∙239 5∙239 7∙239 9∙239 11∙239 │
└─ ─┘
/*REXX program spits out decimal digits of pi (one digit at a time) until Ctrl─Break.*/
parse arg digs oFID . /*obtain optional argument from the CL.*/
if digs=='' | digs=="," then digs= 1e6 /*Not specified? Then use the default.*/
if oFID=='' | oFID=="," then oFID='PI_SPIT.OUT' /* " " " " " " */
write= digs<0 /*if ODIGS is <0, also spit pi to file.*/
numeric digits abs(digs) + 4 /*with bigger digs, spitting is slower.*/
call time 'Reset' /*reset the wall─clock (elapsed) timer.*/
signal on halt /*───► HALT when Ctrl─Break is pressed.*/
spit= 0 /*the index of the spitted pi dec. digs*/
pi=0; v=5; vv=v*v; g=239; gg=g*g; s= 16 /*assign some values to some variables.*/
r= 4 /*calculate π with increasing accuracy */
do n=1 by 2 until old=pi; old= pi /*just calculate pi with odd integers*/
pi= pi + s / (n*v) - r / (n*g) /* ··· using John Machin's formula.*/
s= -s; r= -r; v= v * vv; g= g * gg /*compute some variables for shortcuts.*/
if n>3 then spit= spit + 1 /*maintain a lag for pi digits rounding*/
if spit<4 then iterate /*Not enough digs yet? Then don't show*/
$= substr(pi, spit-3, 1) /*lag behind the true pi calculation. */
call charout , $ /*write the spitted digits to the term.*/
if write then call charout oFID, $ /* " " " " " a file?*/
end /*n*/
$= substr(pi, spit - 2); L= length($) - 4 /*handle any residual decimal digits. */
if L>0 then do /*if any residual digits, then show 'em*/
call charout , substr($, 1, L) /*write to term. */
if write then call charout oFID, substr($, 1, L) /* " " file? */
end
say /*stick a fork in it, we're all done. */
exit: say; say n%2+1 'iterations took' format(time("Elapsed"),,2) 'seconds.'; exit 0
halt: say; say 'PI_SPIT halted via use of Ctrl─Break.'; signal exit /*show iterations.*/
- output [until the Ctrl─Break key (or equivalent) was pressed]:
(Shown at four-fifth size.)
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081 284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185 480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051 320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099 605187072113499999983729780499 PI_SPIT halted via use of Ctrl─Break. 447 iterations took 4.30 seconds.
version 2
This REXX version is a translation of Icon with some speed optimizations.
This algorithm is limited to the number of decimal digits as specified with the numeric digits ddd (line or statement six).
/*REXX program spits out decimal digits of pi (one digit at a time) until Ctrl-Break.*/
signal on halt /*───► HALT when Ctrl─Break is pressed.*/
parse arg digs oFID . /*obtain optional argument from the CL.*/
if digs=='' | digs=="," then digs= 300 /*Not specified? Then use the default.*/
if oFID=='' | oFID=="," then oFID='PI_SPIT2.OUT' /* " " " " " " */
numeric digits digs /*with bigger digs, spitting is slower.*/
q=1; r=0; t=1; k=1; n=3; L=3; z=0 /*define some REXX variables. */
dot=1 /*DOT≡a flag when a dot in pi is shown.*/
do until z==digs; qq= q+q /* qq is a fast version of: q*2 */
tn= t*n /* t*n is used twice (below). */
if qq+qq+r-t < tn then do; z= z+1 /* qq+qq is faster than qq*2 */
call charout , n
call charout oFID, n
if dot then do; dot=0; call charout , .
call charout oFID, .
end
nr= (r - tn) * 10
n = ((( (qq+q+r) * 10) / t) - n*10) %1
q = q*10
end
else do; nr= (qq+r) * L
tL= t*L
n = (q * (k*7 + 2) + r*L) / tL %1
q = q*k
t = tL
L = L+2
k = k+1
end /* %1≡fast way doing TRUNC of a number.*/
r=nr
end /*forever*/
exit /*stick a fork in it, we're all done. */
halt: say; say 'PI_SPIT2 halted via use of Ctrl-Break.'; exit
RPL
It is not easy to print character by character with RPL. Something could be done with the DISP instruction, but it would require to manage the cursor position - and anyway the emulator does not emulate DISP !
Rabinowitz & Wagon algorithm
There is probably a way to translate the BBC BASIC approach into something that uses only the stack, but it has been preferred here to use 'global' variables with the names used by the BBC BASIC program - except for e, which represents the Euler constant in RPL.
≪ IF 1 FS?C THEN "π = " 'output' STO END
SWAP →STR WHILE DUP2 SIZE > REPEAT "0" SWAP + END
output SWAP + 'output' STO DROP
≫
'PRINT' STO
≪ → m
≪ 1 SF {} 1 m START #20d + NEXT
#0d 'ee' STO #2d 'l' STO
m 14 FOR c
#0 'd' STO c 2 * 1 - R→B 'a' STO
c 1 FOR p
DUP p GET 100 * d p * + 'd' STO
p d a / LAST ROT * - PUT
'd' a STO/ 'a' 2 STO-
-1 STEP
IF d #99d ==
THEN ee 100 * d + 'ee' STO #2h 'l' STO+
ELSE
IF c m ==
THEN
d 100 / B→R 10 / 0 PRINT
d 100 / LAST ROT * - 'ee' STO
ELSE
ee d 100 / + B→R l B→R PRINT
d 100 / LAST ROT * - 'ee' STO #2h 'l' STO
END
END
-7 STEP
DROP output
≫ ≫
'M→π' STO
200 M→π
- Output:
1: "π = 3.14159265358979323846264338327950288419716939937510582"
Faster Rabinowitz & Wagon implementation
This much faster version favors the stack to local variables.
≪ SWAP →STR
IF 1 FS?C THEN "." + ELSE WHILE DUP2 SIZE > REPEAT "0" SWAP + END output SWAP + END
'output' STO DROP
≫
'WRITE' STO
≪ DUP 50 * 3 / IP → n m
≪ 1 SF 0 {} m + 2 CON 0
1 n START
DROP 0
m 1 FOR p
p * OVER p GET 100000 * +
p DUP + 1 - MOD LAST / IP
ROT p 4 ROLL PUT SWAP
-1 STEP
DUP 100000 MOD LAST / IP
5 ROLL + 5 WRITE
ROT ROT
NEXT
3 DROPN output
≫ ≫
'N→π' STO
100 N→π
- Output:
"3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011"
Ruby
pi_digits = Enumerator.new do |y|
q, r, t, k, n, l = 1, 0, 1, 1, 3, 3
loop do
if 4*q+r-t < n*t
y << n
nr = 10*(r-n*t)
n = ((10*(3*q+r)) / t) - 10*n
q *= 10
r = nr
else
nr = (2*q+r) * l
nn = (q*(7*k+2)+r*l) / (t*l)
q *= k
t *= l
l += 2
k += 1
n = nn
r = nr
end
end
end
print pi_digits.next, "."
loop { print pi_digits.next }
Rust
use num_bigint::BigInt;
fn main() {
calc_pi();
}
fn calc_pi() {
let mut q = BigInt::from(1);
let mut r = BigInt::from(0);
let mut t = BigInt::from(1);
let mut k = BigInt::from(1);
let mut n = BigInt::from(3);
let mut l = BigInt::from(3);
let mut first = true;
loop {
if &q * 4 + &r - &t < &n * &t {
print!("{}", n);
if first {
print!(".");
first = false;
}
let nr = (&r - &n * &t) * 10;
n = (&q * 3 + &r) * 10 / &t - &n * 10;
q *= 10;
r = nr;
} else {
let nr = (&q * 2 + &r) * &l;
let nn = (&q * &k * 7 + 2 + &r * &l) / (&t * &l);
q *= &k;
t *= &l;
l += 2;
k += 1;
n = nn;
r = nr;
}
}
}
Scala
object Pi {
class PiIterator extends Iterable[BigInt] {
var r: BigInt = 0
var q, t, k: BigInt = 1
var n, l: BigInt = 3
def iterator: Iterator[BigInt] = new Iterator[BigInt] {
def hasNext = true
def next(): BigInt = {
while ((4 * q + r - t) >= (n * t)) {
val nr = (2 * q + r) * l
val nn = (q * (7 * k) + 2 + (r * l)) / (t * l)
q = q * k
t = t * l
l = l + 2
k = k + 1
n = nn
r = nr
}
val ret = n
val nr = 10 * (r - n * t)
n = ((10 * (3 * q + r)) / t) - (10 * n)
q = q * 10
r = nr
ret
}
}
}
def main(args: Array[String]): Unit = {
val it = new PiIterator
println("" + (it.head) + "." + (it.take(300).mkString))
}
}
Output:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998 62803482534211706798214808651328230664709384460955058223172535940812848111745028410 27019385211055596446229489549303819644288109756659334461284756482337867831652712019 09145648566923460348610454326648213393607260249141273
Scheme
(import (rnrs))
(define (calc-pi yield)
(let loop ((q 1) (r 0) (t 1) (k 1) (n 3) (l 3))
(if (< (- (+ (* 4 q) r) t) (* n t))
(begin
(yield n)
(loop (* q 10)
(* 10 (- r (* n t)))
t
k
(- (div (* 10 (+ (* 3 q) r)) t) (* 10 n))
l))
(begin
(loop (* q k)
(* (+ (* 2 q) r) l)
(* t l)
(+ k 1)
(div (+ (* q (* 7 k)) 2 (* r l)) (* t l))
(+ l 2))))))
(let ((i 0))
(calc-pi
(lambda (d)
(display d)
(set! i (+ i 1))
(if (= 40 i)
(begin
(newline)
(set! i 0))))))
Output:
3141592653589793238462643383279502884197 1693993751058209749445923078164062862089 9862803482534211706798214808651328230664 7093844609550582231725359408128481117450 2841027019385211055596446229489549303819 6442881097566593344612847564823378678316 5271201909145648566923460348610454326648 2133936072602491412737245870066063155881 7488152092096282925409171536436789259036 0011330530548820466521384146951941511609 4330572703657595919530921861173819326117 9310511854807446237996274956735188575272 4891227938183011949129833673362440656643 0860213949463952247371907021798609437027 7053921717629317675238467481846766940513 2000568127145263560827785771342757789609 ...
Seed7
$ include "seed7_05.s7i";
include "bigint.s7i";
const proc: main is func
local
var bigInteger: q is 1_;
var bigInteger: r is 0_;
var bigInteger: t is 1_;
var bigInteger: k is 1_;
var bigInteger: n is 3_;
var bigInteger: l is 3_;
var bigInteger: nn is 0_;
var bigInteger: nr is 0_;
var boolean: first is TRUE;
begin
while TRUE do
if 4_ * q + r - t < n * t then
write(n);
if first then
write(".");
first := FALSE;
end if;
nr := 10_ * (r - n * t);
n := 10_ * (3_ * q + r) div t - 10_ * n;
q *:= 10_;
r := nr;
flush(OUT);
else
nr := (2_ * q + r) * l;
nn := (q * (7_ * k + 2_) + r * l) div (t * l);
q *:= k;
t *:= l;
l +:= 2_;
incr(k);
n := nn;
r := nr;
end if;
end while;
end func;Original source: [7]
Sidef
Classical Algorithm
func pi(callback) {
var (q, r, t, k, n, l) = (1, 0, 1, 1, 3, 3)
loop {
if ((4*q + r - t) < n*t) {
callback(n)
static _dot = callback('.')
var nr = 10*(r - n*t)
n = ((10*(3*q + r)) // t - 10*n)
q *= 10
r = nr
}
else {
var nr = ((2*q + r) * l)
var nn = ((q*(7*k + 2) + r*l) // (t*l))
q *= k
t *= l
l += 2
k += 1
n = nn
r = nr
}
}
}
STDOUT.autoflush(true)
pi(func(digit){ print digit })
Quicker, Unverified Algorithm
From the same .pdf mentioned throughout this task, from the last page. The original algorithm was written in Haskell, this is a translation which has also been optimized to avoid redundant multiplications. Same output, but the algorithm is based on one of Gosper’s series that yields more than one digit per term on average, so no test is made partway through the iteration. This is capable of producing approximately 100,000 digits at tio.run in the maximum 60 seconds allowed.
func p(c) { var(q,r,t,g,j,h,k,a,b,y) = (1,180,60,60,54,10,10,15,3)
loop { c(y=(q*(a+=27) +5*r)//5*t); static _ = c('.')
r=10*(g+=j+=54)*(q*(b+=5) +r -y*t); q*=h+=k+=40; t*=g } }
STDOUT.autoflush(1):p(func(d){print d})
Simula
CLASS BIGNUM;
BEGIN
BOOLEAN PROCEDURE TISZERO(T); TEXT T;
TISZERO := T = "0";
TEXT PROCEDURE TSHL(T); TEXT T;
TSHL :- IF TISZERO(T) THEN T ELSE T & "0";
TEXT PROCEDURE TSHR(T); TEXT T;
TSHR :- IF T.LENGTH = 1 THEN "0" ELSE T.SUB(1, T.LENGTH - 1);
INTEGER PROCEDURE TSIGN(T); TEXT T;
TSIGN := IF TISZERO(T) THEN 0
ELSE IF T.SUB(1, 1) = "-" THEN -1
ELSE 1;
TEXT PROCEDURE TABS(T); TEXT T;
TABS :- IF TSIGN(T) < 0 THEN T.SUB(2, T.LENGTH - 1) ELSE T;
TEXT PROCEDURE TNEGATE(T); TEXT T;
TNEGATE :- IF TSIGN(T) <= 0 THEN TABS(T) ELSE ("-" & T);
TEXT PROCEDURE TREVERSE(T); TEXT T;
BEGIN
INTEGER I, J;
I := 1; J := T.LENGTH;
WHILE I < J DO
BEGIN CHARACTER C1, C2;
T.SETPOS(I); C1 := T.GETCHAR;
T.SETPOS(J); C2 := T.GETCHAR;
T.SETPOS(I); T.PUTCHAR(C2);
T.SETPOS(J); T.PUTCHAR(C1);
I := I + 1;
J := J - 1;
END;
TREVERSE :- T;
END TREVERSE;
INTEGER PROCEDURE TCMPUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER ALEN, BLEN, RESULT;
ALEN := A.LENGTH; BLEN := B.LENGTH;
IF ALEN < BLEN THEN
RESULT := -1
ELSE IF ALEN > BLEN THEN
RESULT := 1
ELSE BEGIN
INTEGER CMP, I; BOOLEAN DONE;
A.SETPOS(1);
B.SETPOS(1);
I := 1;
WHILE I <= ALEN AND NOT DONE DO
BEGIN
I := I + 1;
CMP := RANK(A.GETCHAR) - RANK(B.GETCHAR);
IF NOT (CMP = 0) THEN
DONE := TRUE;
END;
RESULT := CMP;
END;
TCMPUNSIGNED := RESULT;
END TCMPUNSIGNED;
INTEGER PROCEDURE TCMP(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF ANEG AND BNEG THEN
TCMP := -TCMPUNSIGNED(TABS(A), TABS(B))
ELSE IF NOT ANEG AND BNEG THEN
TCMP := 1
ELSE IF ANEG AND NOT BNEG THEN
TCMP := -1
ELSE
TCMP := TCMPUNSIGNED(A, B);
END TCMP;
TEXT PROCEDURE TADDUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER CARRY, I, J;
TEXT BF;
I := A.LENGTH;
J := B.LENGTH;
BF :- BLANKS(MAX(I, J) + 1);
WHILE I >= 1 OR J >= 1 DO BEGIN
INTEGER X, Y, Z;
IF I >= 1 THEN BEGIN
A.SETPOS(I); I := I - 1; X := RANK(A.GETCHAR) - RANK('0');
END;
IF J >= 1 THEN BEGIN
B.SETPOS(J); J := J - 1; Y := RANK(B.GETCHAR) - RANK('0');
END;
Z := X + Y + CARRY;
IF Z < 10 THEN
BEGIN BF.PUTCHAR(CHAR(Z + RANK('0'))); CARRY := 0;
END ELSE
BEGIN BF.PUTCHAR(CHAR(MOD(Z, 10) + RANK('0'))); CARRY := 1;
END;
END;
IF CARRY > 0 THEN
BF.PUTCHAR(CHAR(CARRY + RANK('0')));
BF :- TREVERSE(BF.STRIP);
TADDUNSIGNED :- BF;
END TADDUNSIGNED;
TEXT PROCEDURE TADD(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF NOT ANEG AND BNEG THEN ! (+7)+(-5) = (7-5) = 2 ;
TADD :- TSUBUNSIGNED(A, TABS(B))
ELSE IF ANEG AND NOT BNEG THEN ! (-7)+(+5) = (5-7) = -2 ;
TADD :- TSUBUNSIGNED(B, TABS(A))
ELSE IF ANEG AND BNEG THEN ! (-7)+(-5) = -(7+5) = -12 ;
TADD :- TNEGATE(TADDUNSIGNED(TABS(A), TABS(B)))
ELSE ! (+7)+(+5) = (7+5) = 12 ;
TADD :- TADDUNSIGNED(A, B);
END TADD;
TEXT PROCEDURE TSUBUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER I, J, CARRY;
I := A.LENGTH; J := B.LENGTH;
IF I < J OR I = J AND A < B THEN
TSUBUNSIGNED :- TNEGATE(TSUBUNSIGNED(B, A)) ELSE
BEGIN
TEXT BF;
BF :- BLANKS(MAX(I, J) + 1);
WHILE I >= 1 OR J >= 1 DO
BEGIN
INTEGER X, Y, Z;
IF I >= 1 THEN
BEGIN A.SETPOS(I); I := I - 1;
X := RANK(A.GETCHAR) - RANK('0');
END;
IF J >= 1 THEN
BEGIN B.SETPOS(J); J := J - 1;
Y := RANK(B.GETCHAR) - RANK('0');
END;
Z := X - Y - CARRY;
IF Z >= 0 THEN
BEGIN
BF.PUTCHAR(CHAR(RANK('0') + Z));
CARRY := 0;
END ELSE
BEGIN
BF.PUTCHAR(CHAR(RANK('0') + MOD(10 + Z, 10)));
CARRY := 1; ! (Z / 10);
END;
END;
BF :- BF.STRIP;
BF :- TREVERSE(BF);
BF.SETPOS(1);
WHILE BF.LENGTH > 1 AND THEN BF.GETCHAR = '0' DO
BEGIN
BF :- BF.SUB(2, BF.LENGTH - 1);
BF.SETPOS(1);
END;
TSUBUNSIGNED :- BF;
END;
END TSUBUNSIGNED;
TEXT PROCEDURE TSUB(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF ANEG AND BNEG THEN ! (-7)-(-5) = -(7-5) = -2 ;
TSUB :- TNEGATE(TSUBUNSIGNED(TABS(A), TABS(B)))
ELSE IF NOT ANEG AND BNEG THEN ! (+7)-(-5) = (7+5) = 12 ;
TSUB :- TADDUNSIGNED(A, TABS(B))
ELSE IF ANEG AND NOT BNEG THEN ! (-7)-(+5) = -(7+5) = -12 ;
TSUB :- TNEGATE(TADDUNSIGNED(TABS(A), B))
ELSE ! (+7)-(+5) = (7-5) = 2 ;
TSUB :- TSUBUNSIGNED(A, B);
END TSUB;
TEXT PROCEDURE TMULUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER ALEN, BLEN;
ALEN := A.LENGTH; BLEN := B.LENGTH;
IF ALEN < BLEN THEN
TMULUNSIGNED :- TMULUNSIGNED(B, A)
ELSE BEGIN
TEXT PRODUCT; INTEGER J;
PRODUCT :- "0";
FOR J := 1 STEP 1 UNTIL BLEN DO BEGIN
TEXT PART; INTEGER I, Y, CARRY;
B.SETPOS(J); Y := RANK(B.GETCHAR) - RANK('0');
PART :- BLANKS(ALEN + BLEN + 1); PART.SETPOS(1);
FOR I := ALEN STEP -1 UNTIL 1 DO BEGIN
INTEGER X, Z;
A.SETPOS(I); X := RANK(A.GETCHAR) - RANK('0');
Z := X * Y + CARRY;
IF Z < 10 THEN BEGIN
PART.PUTCHAR(CHAR(RANK('0') + Z));
CARRY := 0;
END ELSE BEGIN
PART.PUTCHAR(CHAR(RANK('0') + MOD(Z, 10)));
CARRY := Z // 10;
END;
END;
IF CARRY > 0 THEN
PART.PUTCHAR(CHAR(RANK('0') + CARRY));
PART :- PART.SUB(1, PART.POS - 1);
PART :- TREVERSE(PART);
PART.SETPOS(1);
WHILE PART.LENGTH > 1 AND THEN PART.GETCHAR = '0' DO
BEGIN
PART :- PART.SUB(2, PART.LENGTH - 1);
PART.SETPOS(1);
END;
PRODUCT :- TADDUNSIGNED(TSHL(PRODUCT), PART);
END;
TMULUNSIGNED :- PRODUCT;
END;
END TMULUNSIGNED;
TEXT PROCEDURE TMUL(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF ANEG AND BNEG THEN ! (-7)*(-5) = (7*5) => 35 ;
TMUL :- TMULUNSIGNED(TABS(A), TABS(B))
ELSE IF NOT ANEG AND BNEG THEN ! (+7)*(-5) = -(7*5) => -35 ;
TMUL :- TNEGATE(TMULUNSIGNED(A, TABS(B)))
ELSE IF ANEG AND NOT BNEG THEN ! (-7)*(+5) = -(7*5) => -35 ;
TMUL :- TNEGATE(TMULUNSIGNED(TABS(A), B))
ELSE ! (+7)*(+5) = (7*5) => 35 ;
TMUL :- TMULUNSIGNED(A, B);
END TMUL;
CLASS DIVMOD(DIV,MOD); TEXT DIV,MOD;;
REF(DIVMOD) PROCEDURE TDIVMODUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER CC;
REF(DIVMOD) RESULT;
IF TISZERO(B) THEN
ERROR("DIVISION BY ZERO");
CC := TCMPUNSIGNED(A, B);
IF CC < 0 THEN
RESULT :- NEW DIVMOD("0", A)
ELSE IF CC = 0 THEN
RESULT :- NEW DIVMOD("1", "0")
ELSE BEGIN
INTEGER ALEN, BLEN, AIDX;
TEXT Q, R;
ALEN := A.LENGTH; BLEN := B.LENGTH;
Q :- BLANKS(ALEN); Q.SETPOS(1);
R :- BLANKS(ALEN); R.SETPOS(1);
R := A.SUB(1, BLEN - 1); R.SETPOS(BLEN);
FOR AIDX := BLEN STEP 1 UNTIL ALEN DO
BEGIN
INTEGER COUNT; BOOLEAN DONE;
IF TISZERO(R.STRIP) THEN
R.SETPOS(1);
A.SETPOS(AIDX); R.PUTCHAR(A.GETCHAR);
WHILE NOT DONE DO
BEGIN
TEXT DIFF;
DIFF :- TSUBUNSIGNED(R.STRIP, B);
IF TSIGN(DIFF) < 0 THEN
DONE := TRUE
ELSE BEGIN
R := DIFF; R.SETPOS(DIFF.LENGTH + 1);
COUNT := COUNT + 1;
END;
END;
IF (NOT (COUNT = 0)) OR (NOT (Q.POS = 1)) THEN
Q.PUTCHAR(CHAR(COUNT + RANK('0')));
END;
RESULT :- NEW DIVMOD(Q.STRIP, R.STRIP);
END;
TDIVMODUNSIGNED :- RESULT;
END TDIVMODUNSIGNED;
REF(DIVMOD) PROCEDURE TDIVMOD(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG; REF(DIVMOD) RESULT;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF ANEG AND BNEG THEN
BEGIN
RESULT :- TDIVMOD(TABS(A), TABS(B));
RESULT.MOD :- TNEGATE(RESULT.MOD);
END
ELSE IF NOT ANEG AND BNEG THEN
BEGIN
RESULT :- TDIVMOD(A, TABS(B));
RESULT.DIV :- TNEGATE(RESULT.DIV);
END
ELSE IF ANEG AND NOT BNEG THEN
BEGIN
RESULT :- TDIVMOD(TABS(A), B);
RESULT.DIV :- TNEGATE(RESULT.DIV);
RESULT.MOD :- TNEGATE(RESULT.MOD);
END
ELSE
RESULT :- TDIVMODUNSIGNED(A, B);
TDIVMOD :- RESULT;
END TDIVMOD;
TEXT PROCEDURE TDIV(A, B); TEXT A, B;
TDIV :- TDIVMOD(A, B).DIV;
TEXT PROCEDURE TMOD(A, B); TEXT A, B;
TMOD :- TDIVMOD(A, B).MOD;
END BIGNUM;EXTERNAL CLASS BIGNUM;
BIGNUM
BEGIN
PROCEDURE CALCPI;
BEGIN
INTEGER I;
TEXT Q, R, T, K, N, L;
COMMENT
! q, r, t, k, n, l = 1, 0, 1, 1, 3, 3
;
Q :- COPY("1");
R :- COPY("0");
T :- COPY("1");
K :- COPY("1");
N :- COPY("3");
L :- COPY("3");
WHILE TRUE DO
BEGIN
COMMENT
! if 4*q+r-t < n*t
;
IF TCMP(TSUB(TADD(TMUL("4",Q),R),T),TMUL(N,T)) < 0 THEN
BEGIN
TEXT NR;
OUTTEXT(N);
I := I + 1;
IF I = 40 THEN
BEGIN
OUTIMAGE;
I := 0;
END;
COMMENT
! nr = 10*(r-n*t)
! n = ((10*(3*q+r))//t)-10*n
! q *= 10
! r = nr
;
NR :- TMUL("10",TSUB(R,TMUL(N,T)));
N :- TSUB(TDIV(TMUL("10",TADD(TMUL("3",Q),R)),T),TMUL("10",N));
Q :- TMUL("10",Q);
R :- NR;
END
ELSE
BEGIN
TEXT NR, NN;
COMMENT
! nr = (2*q+r)*l
! nn = (q*(7*k)+2+(r*l))//(t*l)
! q *= k
! t *= l
! l += 2
! k += 1
! n = nn
! r = nr
;
NR :- TMUL(TADD(TMUL("2",Q),R),L);
NN :- TDIV(TADD(TADD(TMUL(Q,TMUL("7",K)),"2"),TMUL(R,L)),TMUL(T,L));
Q :- TMUL(Q,K);
T :- TMUL(T,L);
L :- TADD(L,"2");
K :- TADD(K,"1");
N :- NN;
R :- NR;
END;
END;
END CALCPI;
CALCPI;
END.Output:
3141592653589793238462643383279502884197 1693993751058209749445923078164062862089 9862803482534211706798214808651328230664 7093844609550582231725359408128481117450 2841027019385211055596446229489549303819 6442881097566593344612847564823378678316 5271201909145648566923460348610454326648 2133936072602491412737245870066063155881 7488152092096282925409171536436789259036 0011330530548820466521384146951941511609 4330572703657595919530921861173819326117 9310511854807446237996274956735188575272 4891227938183011949129833673362440656643 0860213949463952247371907021798609437027 7053921717629317675238467481846766940513 2000568127145263560827785771342757789609 ...
Standard ML
(* https://www.cs.ox.ac.uk/people/jeremy.gibbons/publications/spigot.pdf *)
fun gibbons _ _ _ _ _ _ 0 = ()
| gibbons q r t k n l count =
let
val (q',r',t',k',n',l',count') =
if 4*q+r-t < n*t
then (10*q,10*(r-n*t),t,k,(10*(3*q+r)) div t-10*n,l,count-1) before print (IntInf.toString n)
else (q*k,(2*q+r)*l,t*l,k+1,(q*(7*k+2)+r*l) div (t*l),l+2,count)
in
gibbons q' r' t' k' n' l' count'
end
fun doGibbons n = gibbons 1 0 1 1 3 3 n
fun timeGibbons n =
let
val timer1 = Timer.startCPUTimer ()
val () = doGibbons n
val {usr=usr, sys=sys} = Timer.checkCPUTimer timer1
in
print "\n----------------------\n";
print ("usr: " ^ Time.toString usr ^ "\n");
print ("sys: " ^ Time.toString sys ^ "\n")
end
fun main () = timeGibbons 5000
Tailspin
Used the compact algorithm from Gibbons paper. Tailspin will at some point have arbitrary precision integers, currently we have to link into java and use BigInteger. Using java code can be slightly awkward as the argument in Tailspin comes before the method, so divide and subtract read backwards.
use 'java:java.math' stand-alone
def zero: 0 -> math/BigInteger::valueOf;
def one: 1 -> math/BigInteger::valueOf;
def two: 2 -> math/BigInteger::valueOf;
def three: 3 -> math/BigInteger::valueOf;
def four: 4 -> math/BigInteger::valueOf;
def seven: 7 -> math/BigInteger::valueOf;
def ten: 10 -> math/BigInteger::valueOf;
templates g&{q:, r:, t:, k:, n:, l:}
def u: $four -> q::multiply -> r::add;
def nt: $n -> t::multiply;
$ -> #
when <?($t -> u::subtract <..~$nt>)> do
$n -> !OUT::write
$ -> \(<=1> $!\) -> '.' -> !OUT::write
def v: $three -> q::multiply -> r::add -> ten::multiply;
def quot: $t -> v::divide;
0 -> g&{q: $ten -> q::multiply,
r: $nt -> r::subtract -> ten::multiply,
t: $t,
k: $k,
n: $ten -> n::multiply -> quot::subtract,
l: $l } !
otherwise
def tl: $t -> l::multiply;
def rl: $r -> l::multiply;
def term: $q -> seven::multiply -> k::multiply -> two::add -> rl::add;
$ -> g&{q: $q -> k::multiply,
r: $two -> q::multiply -> r::add -> l::multiply,
t: $tl,
k: $k -> one::add,
n: $tl -> term::divide,
l: $l -> two::add} !
end g
1 -> g&{q:$one, r:$zero, t:$one, k:$one, n:$three, l:$three} -> !VOID- Output:
3.141592653589793238462643383279502884197169399375105820974...keeps going until stopped.
Tcl
Based on the reference in the D code.
package require Tcl 8.6
# http://www.cut-the-knot.org/Curriculum/Algorithms/SpigotForPi.shtml
# http://www.mathpropress.com/stan/bibliography/spigot.pdf
proc piDigitsBySpigot n {
yield [info coroutine]
set A [lrepeat [expr {int(floor(10*$n/3.)+1)}] 2]
set Alen [llength $A]
set predigits {}
while 1 {
set carry 0
for {set i $Alen} {[incr i -1] > 0} {} {
lset A $i [expr {
[set val [expr {[lindex $A $i] * 10 + $carry}]]
% [set modulo [expr {2*$i + 1}]]
}]
set carry [expr {$val / $modulo * $i}]
}
lset A 0 [expr {[set val [expr {[lindex $A 0]*10 + $carry}]] % 10}]
set predigit [expr {$val / 10}]
if {$predigit < 9} {
foreach p $predigits {yield $p}
set predigits [list $predigit]
} elseif {$predigit == 9} {
lappend predigits $predigit
} else {
foreach p $predigits {yield [incr p]}
set predigits [list 0]
}
}
}
The pi digit generation requires picking a limit to the number of digits; the bigger the limit, the more digits can be safely computed. A value of 10k yields values relatively rapidly.
coroutine piDigit piDigitsBySpigot 10000
fconfigure stdout -buffering none
while 1 {
puts -nonewline [piDigit]
}
TypeScript
type AnyWriteableObject={write:((textToOutput:string)=>any)};
function calcPi(pipe:AnyWriteableObject) {
let q = 1n, r=0n, t=1n, k=1n, n=3n, l=3n;
while (true) {
if (q * 4n + r - t < n* t) {
pipe.write(n.toString());
let nr = (r - n * t) * 10n;
n = (q * 3n + r) * 10n / t - n * 10n ;
q = q * 10n;
r = nr;
} else {
let nr = (q * 2n + r) * l;
let nn = (q * k * 7n + 2n + r * l) / (t * l);
q = q * k;
t = t * l;
l = l + 2n;
k = k + 1n;
n = nn;
r = nr;
}
}
}
calcPi(process.stdout);
Notes:
1. Typescript has bigint support https://www.typescriptlang.org/docs/handbook/release-notes/typescript-3-2.html#bigint Literals are write with a n sufix: 10n
2. Pi function receives any object that has a write function. Using node.js we can pass to it process.stdout
Async version
type AnyWriteableObject = {write:((textToOutput:string)=>Promise<any>)};
async function calcPi<T extends AnyWriteableObject>(pipe:T) {
let q = 1n, r=0n, t=1n, k=1n, n=3n, l=3n;
while (true) {
if (q * 4n + r - t < n* t) {
await pipe.write(n.toString());
let nr = (r - n * t) * 10n;
n = (q * 3n + r) * 10n / t - n * 10n ;
q = q * 10n;
r = nr;
} else {
let nr = (q * 2n + r) * l;
let nn = (q * k * 7n + 2n + r * l) / (t * l);
q = q * k;
t = t * l;
l = l + 2n;
k = k + 1n;
n = nn;
r = nr;
}
}
}
setInterval(function(){
console.log(); // put a new line every second
},1000);
var x = calcPi({
write: async function(phrase:string){
return new Promise(function(resolve){
setTimeout(function(){
process.stdout.write(phrase);
resolve();
},1);
});
}
});
console.log('.'); //start!
Here the calculation does not continue if the consumer does not consume the character.
Visual Basic
Option Explicit
Sub Main()
Const VECSIZE As Long = 3350
Const BUFSIZE As Long = 201
Dim buffer(1 To BUFSIZE) As Long
Dim vect(1 To VECSIZE) As Long
Dim more As