Check Machin-like formulas: Difference between revisions

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Yes! 'pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)' is true

No! 'pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)' not true

</pre>

=={{header|XPL0}}==

<lang XPL0>code ChOut=8, Text=12; \intrinsic routines

int Number(18); \numbers from equations

def LF=$0A; \ASCII line feed (end-of-line character)

func Parse(S); \Convert numbers in string S to binary in Number array

char S;

int I, Neg;

proc GetNum; \Get number from string S

int N;

[while S(0)<^0 ! S(0)>^9 do S:= S+1;

N:= S(0)-^0; S:= S+1;

while S(0)>=^0 & S(0)<=^9 do

[N:= N*10 + S(0) - ^0; S:= S+1];

Number(I):= N; I:= I+1;

];

[while S(0)#^= do S:= S+1; \skip to "="

I:= 0;

loop [Neg:= false; \assume positive term

loop [S:= S+1; \next char

case S(0) of

LF: [Number(I):= 0; return S+1]; \mark end of array

^-: Neg:= true; \term is negative

^a: [Number(I):= 1; I:= I+1; quit] \no coefficient so use 1

other if S(0)>=^0 & S(0)<=^9 then \if digit

[S:= S-1; GetNum; quit]; \backup and get number

];

GetNum; \numerator

if Neg then Number(I-1):= -Number(I-1); \tan(-a) = -tan(a)

GetNum; \denominator

];

func GCD(U, V); \Return the greatest common divisor of U and V

int U, V;

int T;

[while V do \Euclid's method

[T:= U; U:= V; V:= rem(T/V)];

return abs(U);

];

proc Verify; \Verify that tangent of equation = 1 (i.e: E = F)

int E, F, I, J;

proc Machin(A, B, C, D);

int A, B, C, D;

int Div;

\tan(a+b) = (tan(a) + tan(b)) / (1 - tan(a)*tan(b))

\tan(arctan(A/B) + arctan(C/D))

\ = (tan(arctan(A/B)) + tan(arctan(C/D))) / (1 - tan(arctan(A/B))*tan(arctan(C/D)))

\ = (A/B + C/D) / (1 - A/B*C/D)

\ = (A*D/B*D + B*C/B*D) / (B*D/B*D - A*C/B*D)

\ = (A*D + B*C) / (B*D - A*C)

[E:= A*D + B*C; F:= B*D - A*C;

Div:= GCD(E, F); \keep integers from getting too big

E:= E/Div; F:= F/Div;

];

[E:= 0; F:= 1; I:= 0;

while Number(I) do

[for J:= 1 to Number(I) do

Machin(E, F, Number(I+1), Number(I+2));

I:= I+3;

];

Text(0, if E=F then "Yes " else "No ");

];

char S, SS; int I;

[S:= "pi/4 = arctan(1/2) + arctan(1/3)

pi/4 = 2*arctan(1/3) + arctan(1/7)

pi/4 = 4*arctan(1/5) - arctan(1/239)

pi/4 = 5*arctan(1/7) + 2*arctan(3/79)

pi/4 = 5*arctan(29/278) + 7*arctan(3/79)

pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)

pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)

pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)

pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)

pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)

pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)

pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)

pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)

pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)

pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)

pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)

pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)

"; \Python version of equations (thanks!)

for I:= 1 to 17 do

[SS:= S; \save start of string line

S:= Parse(S); \returns start of next line

Verify; \correct Machin equation? Yes or No

repeat ChOut(0, SS(0)); SS:= SS+1 until SS(0)=LF; ChOut(0, LF); \show equation

];

]</lang>

<pre>

Yes pi/4 = arctan(1/2) + arctan(1/3)

Yes pi/4 = 2*arctan(1/3) + arctan(1/7)

Yes pi/4 = 4*arctan(1/5) - arctan(1/239)

Yes pi/4 = 5*arctan(1/7) + 2*arctan(3/79)

Yes pi/4 = 5*arctan(29/278) + 7*arctan(3/79)

Yes pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)

Yes pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)

Yes pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)

Yes pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)

Yes pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)

Yes pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)

Yes pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)

Yes pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)

Yes pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)

Yes pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)

Yes pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)

No pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)

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