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Euler's constant 0.5772...

From Rosetta Code
Task
Euler's constant 0.5772...
You are encouraged to solve this task according to the task description, using any language you may know.


Task.

Compute the Euler constant 0.5772...

Discovered by Leonhard Euler around 1730, it is the most ubiquitous mathematical constant after pi and e, but appears more arcane than these.

Denoted gamma (γ), it measures the amount by which the partial sums of the harmonic series (the simplest diverging series) differ from the logarithmic function (its approximating integral): lim n → ∞ (1 + 1/2 + 1/3 + … + 1/n − log(n)).

The definition of γ converges too slowly to be numerically useful, but in 1735 Euler himself applied his recently discovered summation formula to compute ‘the notable number’ accurate to 15 places. For a single-precision implementation this is still the most economic algorithm.

In 1961, the young Donald Knuth used Euler's method to evaluate γ to 1271 places. Knuth found that the computation of the Bernoulli numbers required in the Euler-Maclaurin formula was the most time-consuming part of the procedure.

The next year Dura Sweeney computed 3566 places, using a formula based on the expansion of the exponential integral which didn't need Bernoulli numbers. It's a bit-hungry method though: 2d digits of working precision obtain d correct places only.

This was remedied in 1988 by David Bailey; meanwhile Richard Brent and Ed McMillan had published an even more efficient algorithm based on Bessel function identities and found 30100 places in 20 hours time.

Nowadays the old records have far been exceeded: over 6·1011 decimal places are already known. These massive computations suggest that γ is neither rational nor algebraic, but this is yet to be proven.


References.

[1] Gourdon and Sebah, The Euler constant γ. (for all formulas)

[2] Euler's original journal article translated from the latin (p. 9)



C[edit]

Single precision[edit]

/*********************************************
Subject: Comparing five methods for
computing Euler's constant 0.5772...
tested : tcc-0.9.27
--------------------------------------------*/

#include <math.h>
#include <stdio.h>
 
#define eps 1e-6
 
int main(void) {
double a, b, h, n2, r, u, v;
int k, k2, m, n;
 
printf("From the definition, err. 3e-10\n");
 
n = 400;
 
h = 1;
for (k = 2; k <= n; k++) {
h += 1.0 / k;
}
//faster convergence: Negoi, 1997
a = log(n +.5 + 1.0 / (24*n));
 
printf("Hn  %.16f\n", h);
printf("gamma %.16f\nk = %d\n\n", h - a, n);
 
 
printf("Sweeney, 1963, err. idem\n");
 
n = 21;
 
double s[] = {0, n};
r = n;
k = 1;
do {
k += 1;
r *= (double) n / k;
s[k & 1] += r / k;
} while (r > eps);
 
printf("gamma %.16f\nk = %d\n\n", s[1] - s[0] - log(n), k);
 
 
printf("Bailey, 1988\n");
 
n = 5;
 
a = 1;
h = 1;
n2 = pow(2,n);
r = 1;
k = 1;
do {
k += 1;
r *= n2 / k;
h += 1.0 / k;
b = a; a += r * h;
} while (fabs(b - a) > eps);
a *= n2 / exp(n2);
 
printf("gamma %.16f\nk = %d\n\n", a - n * log(2), k);
 
 
printf("Brent-McMillan, 1980\n");
 
n = 13;
 
a = -log(n);
b = 1;
u = a;
v = b;
n2 = n * n;
k2 = 0;
k = 0;
do {
k2 += 2*k + 1;
k += 1;
a *= n2 / k;
b *= n2 / k2;
a = (a + b) / k;
u += a;
v += b;
} while (fabs(a) > eps);
 
printf("gamma %.16f\nk = %d\n\n", u / v, k);
 
 
printf("How Euler did it in 1735\n");
//Bernoulli numbers with even indices
double B2[] = {1.0,1.0/6,-1.0/30,1.0/42,-1.0/30,\
5.0/66,-691.0/2730,7.0/6,-3617.0/510,43867.0/798};
m = 7;
if (m > 9) return(0);
 
n = 10;
 
//n-th harmonic number
h = 1;
for (k = 2; k <= n; k++) {
h += 1.0 / k;
}
printf("Hn  %.16f\n", h);
 
h -= log(n);
printf(" -ln %.16f\n", h);
 
//expansion C = -digamma(1)
a = -1.0 / (2*n);
n2 = n * n;
r = 1;
for (k = 1; k <= m; k++) {
r *= n2;
a += B2[k] / (2*k * r);
}
 
printf("err  %.16f\ngamma %.16f\nk = %d", a, h + a, n + m);
 
printf("\n\nC = 0.57721566490153286...\n");
}
output:
From the definition, err. 3e-10
Hn    6.5699296911765055
gamma 0.5772156645765731
k = 400

Sweeney, 1963, err. idem
gamma 0.5772156645636311
k = 68

Bailey, 1988
gamma 0.5772156649015341
k = 89

Brent-McMillan, 1980
gamma 0.5772156649015329
k = 40

How Euler did it in 1735
Hn    2.9289682539682538
  -ln 0.6263831609742079
err  -0.0491674960726754
gamma 0.5772156649015325
k = 17

C  =  0.57721566490153286...

Multi precision[edit]

From first principles

/**************************************************
Subject: Computation of Euler's constant 0.5772...
with the Brent-McMillan algorithm B1,
Math. Comp. 34 (1980), 305-312
tested : tcc-0.9.27 with gmp 6.2.0
-------------------------------------------------*/

#include <gmp.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
 
//multi-precision float pointers
mpf_ptr u, v, k2;
 
//precision parameters
unsigned long e10, e2;
long e;
double f;
 
//log(x/y) with the Taylor series for atanh(x-y/x+y)
void ln (mpf_ptr s, unsigned long x, unsigned long y) {
mpf_ptr d = u, q = v;
unsigned long k;
//Möbius transformation
k = x; x -= y; y += k;
 
if (x != 1) {
printf ("ln: illegal argument x - y != 1");
exit;
}
 
//s = 1 / (x + y)
mpf_set_ui (s, y);
mpf_ui_div (s, 1, s);
//k2 = s * s
mpf_mul (k2, s, s);
mpf_set (d, s);
 
k = 1;
do {
k += 2;
//d *= k2
mpf_mul (d, d, k2);
//q = d / k
mpf_div_ui (q, d, k);
//s += q
mpf_add (s, s, q);
 
f = mpf_get_d_2exp (&e, q);
} while (abs(e) < e2);
 
//s *= 2
mpf_mul_2exp (s, s, 1);
}
 
int main (void) {
mpf_ptr a = malloc(sizeof(__mpf_struct));
mpf_ptr b = malloc(sizeof(__mpf_struct));
u = malloc(sizeof(__mpf_struct));
v = malloc(sizeof(__mpf_struct));
k2 = malloc(sizeof(__mpf_struct));
//unsigned long integers
unsigned long k, n, n2, r, s, t;
 
clock_t tim = clock();
 
// n = 2^i * 3^j * 5^k
 
// log(n) = r * log(16/15) + s * log(25/24) + t * log(81/80)
 
// solve linear system for r, s, t
// 4 -3 -4| i
// -1 -1 4| j
// -1 2 -1| k
 
//examples
t = 1;
switch (t) {
case 1 :
n = 60;
r = 41;
s = 30;
t = 18;
//100 digits
break;
case 2 :
n = 4800;
r = 85;
s = 62;
t = 37;
//8000 digits, 0.6 s
break;
case 3 :
n = 9375;
r = 91;
s = 68;
t = 40;
//15625 digits, 2.5 s
break;
default :
n = 18750;
r = 98;
s = 73;
t = 43;
//31250 digits, 12 s. @2.00GHz
}
 
//decimal precision
e10 = n / .6;
//binary precision
e2 = (1 + e10) / .30103;
 
//initialize mpf's
mpf_set_default_prec (e2);
mpf_inits (a, b, u, v, k2, (mpf_ptr)0);
 
//Compute log terms
 
ln (b, 16, 15);
 
//a = r * b
mpf_mul_ui (a, b, r);
 
ln (b, 25, 24);
 
//a += s * b
mpf_mul_ui (u, b, s);
mpf_add (a, a, u);
 
ln (b, 81, 80);
 
//a += t * b
mpf_mul_ui (u, b, t);
mpf_add (a, a, u);
 
//gmp_printf ("log(%lu) %.*Ff\n", n, e10, a);
 
//B&M, algorithm B1
 
//a = -a, b = 1
mpf_neg (a, a);
mpf_set_ui (b, 1);
mpf_set (u, a);
mpf_set (v, b);
 
k = 0;
n2 = n * n;
//k2 = k * k
mpf_set_ui (k2, 0);
do {
//k2 += 2k + 1
mpf_add_ui (k2, k2, (k << 1) + 1);
k += 1;
 
//b = b * n2 / k2
mpf_div (b, b, k2);
mpf_mul_ui (b, b, n2);
//a = (a * n2 / k + b) / k
mpf_div_ui (a, a, k);
mpf_mul_ui (a, a, n2);
mpf_add (a, a, b);
mpf_div_ui (a, a, k);
 
//u += a, v += b
mpf_add (u, u, a);
mpf_add (v, v, b);
 
f = mpf_get_d_2exp (&e, a);
} while (abs(e) < e2);
 
mpf_div (u, u, v);
gmp_printf ("gamma %.*Ff (maxerr. 1e-%lu)\n", e10, u, e10);
 
gmp_printf ("k = %lu\n\n", k);
 
tim = clock() - tim;
printf("time: %.7f s\n",((double)tim)/CLOCKS_PER_SEC);
}
output:
gamma 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 (maxerr. 1e-100)
k = 255

The easy way[edit]

/*******************************************
Subject: Euler's constant 0.5772...
tested : tcc-0.9.27 with mpfr 4.1.0
------------------------------------------*/

#include <gmp.h>
#include <mpfr.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
 
int main (void) {
mpfr_ptr a = malloc(sizeof(__mpfr_struct));
unsigned long e2, e10;
clock_t tim = clock();
 
//decimal precision
e10 = 100;
 
//binary precision
e2 = (1 + e10) / .30103;
mpfr_init2 (a, e2);
 
mpfr_const_euler (a, MPFR_RNDN);
mpfr_printf ("gamma %.*Rf\n\n", e10, a);
 
tim = clock() - tim;
gmp_printf ("time: %.7f s\n",((double)tim)/CLOCKS_PER_SEC);
}


FreeBASIC[edit]

Single precision[edit]

'**********************************************
'Subject: Comparing five methods for
' computing Euler's constant 0.5772...
'tested : FreeBasic 1.08.1
'----------------------------------------------
const eps = 1e-6
dim as double a, b, h, n2, r, u, v
dim as integer k, k2, m, n
 
? "From the definition, err. 3e-10"
 
n = 400
 
h = 1
for k = 2 to n
h += 1 / k
next k
'faster convergence: Negoi, 1997
a = log(n +.5 + 1 / (24*n))
 
? "Hn "; h
? "gamma"; h - a; !"\nk ="; n
?
 
 
? "Sweeney, 1963, err. idem"
 
n = 21
 
dim as double s(1) = {0, n}
r = n
k = 1
do
k += 1
r *= n / k
s(k and 1) += r / k
loop until r < eps
 
? "gamma"; s(1) - s(0) - log(n); !"\nk ="; k
?
 
 
? "Bailey, 1988"
 
n = 5
 
a = 1
h = 1
n2 = 2^n
r = 1
k = 1
do
k += 1
r *= n2 / k
h += 1 / k
b = a: a += r * h
loop until abs(b - a) < eps
a *= n2 / exp(n2)
 
? "gamma"; a - n * log(2); !"\nk ="; k
?
 
 
? "Brent-McMillan, 1980"
 
n = 13
 
a = -log(n)
b = 1
u = a
v = b
n2 = n * n
k2 = 0
k = 0
do
k2 += 2*k + 1
k += 1
a *= n2 / k
b *= n2 / k2
a = (a + b) / k
u += a
v += b
loop until abs(a) < eps
 
? "gamma"; u / v; !"\nk ="; k
?
 
 
? "How Euler did it in 1735"
'Bernoulli numbers with even indices
dim as double B2(9) = {1,1/6,-1/30,1/42,_
-1/30,5/66,-691/2730,7/6,-3617/510,43867/798}
m = 7
if m > 9 then end
 
n = 10
 
'n-th harmonic number
h = 1
for k = 2 to n
h += 1 / k
next k
? "Hn "; h
 
h -= log(n)
? " -ln"; h
 
'expansion C = -digamma(1)
a = -1 / (2*n)
n2 = n * n
r = 1
for k = 1 to m
r *= n2
a += B2(k) / (2*k * r)
next k
 
? "err "; a; !"\ngamma"; h + a; !"\nk ="; n + m
?
? "C = 0.57721566490153286..."
end
output:
From the definition, err. 3e-10
Hn    6.569929691176506
gamma 0.5772156645765731
k = 400

Sweeney, 1963, err. idem
gamma 0.5772156645636311
k = 68

Bailey, 1988
gamma 0.5772156649015341
k = 89

Brent-McMillan, 1980
gamma 0.5772156649015329
k = 40

How Euler did it in 1735
Hn    2.928968253968254
  -ln 0.6263831609742079
err  -0.04916749607267539
gamma 0.5772156649015325
k = 17

C  =  0.57721566490153286...

Multi precision[edit]

From first principles

'***************************************************
'Subject: Computation of Euler's constant 0.5772...
' with the Brent-McMillan algorithm B1,
' Math. Comp. 34 (1980), 305-312
'tested : FreeBasic 1.08.1 with gmp 6.2.0
'---------------------------------------------------
#include "gmp.bi"
 
'multi-precision float pointers
Dim as mpf_ptr a, b
Dim shared as mpf_ptr k2, u, v
'unsigned long integers
Dim as ulong k, n, n2, r, s, t
'precision parameters
Dim shared as ulong e10, e2
Dim shared e as clong
Dim shared f as double
Dim as double tim = TIMER
CLS
 
a = allocate(len(__mpf_struct))
b = allocate(len(__mpf_struct))
u = allocate(len(__mpf_struct))
v = allocate(len(__mpf_struct))
k2 = allocate(len(__mpf_struct))
 
'log(x/y) with the Taylor series for atanh(x-y/x+y)
Sub ln (byval s as mpf_ptr, byval x as ulong, byval y as ulong)
Dim as mpf_ptr d = u, q = v
Dim k as ulong
'Möbius transformation
k = x: x -= y: y += k
 
If x <> 1 Then
Print "ln: illegal argument x - y <> 1"
End
End If
 
's = 1 / (x + y)
mpf_set_ui (s, y)
mpf_ui_div (s, 1, s)
'k2 = s * s
mpf_mul (k2, s, s)
mpf_set (d, s)
 
k = 1
Do
k += 2
'd *= k2
mpf_mul (d, d, k2)
'q = d / k
mpf_div_ui (q, d, k)
's += q
mpf_add (s, s, q)
 
f = mpf_get_d_2exp (@e, q)
Loop until abs(e) > e2
 
's *= 2
mpf_mul_2exp (s, s, 1)
End Sub
 
'Main
 
'n = 2^i * 3^j * 5^k
 
'log(n) = r * log(16/15) + s * log(25/24) + t * log(81/80)
 
'solve linear system for r, s, t
' 4 -3 -4| i
'-1 -1 4| j
'-1 2 -1| k
 
'examples
t = 1
select case t
case 1
n = 60
r = 41
s = 30
t = 18
'100 digits
case 2
n = 4800
r = 85
s = 62
t = 37
'8000 digits, 0.6 s
case 3
n = 9375
r = 91
s = 68
t = 40
'15625 digits, 2.5 s
case else
n = 18750
r = 98
s = 73
t = 43
'31250 digits, 12 s. @2.00GHz
end select
 
'decimal precision
e10 = n / .6
'binary precision
e2 = (1 + e10) / .30103
 
'initialize mpf's
mpf_set_default_prec (e2)
mpf_inits (a, b, u, v, k2, Cptr(mpf_ptr, 0))
 
'Compute log terms
 
ln b, 16, 15
 
'a = r * b
mpf_mul_ui (a, b, r)
 
ln b, 25, 24
 
'a += s * b
mpf_mul_ui (u, b, s)
mpf_add (a, a, u)
 
ln b, 81, 80
 
'a += t * b
mpf_mul_ui (u, b, t)
mpf_add (a, a, u)
 
''gmp_printf (!"log(%lu) %.*Ff\n", n, e10, a)
 
'B&M, algorithm B1
 
'a = -a, b = 1
mpf_neg (a, a)
mpf_set_ui (b, 1)
mpf_set (u, a)
mpf_set (v, b)
 
k = 0
n2 = n * n
'k2 = k * k
mpf_set_ui (k2, 0)
do
'k2 += 2k + 1
mpf_add_ui (k2, k2, (k shl 1) + 1)
k += 1
 
'b = b * n2 / k2
mpf_div (b, b, k2)
mpf_mul_ui (b, b, n2)
'a = (a * n2 / k + b) / k
mpf_div_ui (a, a, k)
mpf_mul_ui (a, a, n2)
mpf_add (a, a, b)
mpf_div_ui (a, a, k)
 
'u += a, v += b
mpf_add (u, u, a)
mpf_add (v, v, b)
 
f = mpf_get_d_2exp (@e, a)
Loop until abs(e) > e2
 
mpf_div (u, u, v)
gmp_printf (!"gamma %.*Ff (maxerr. 1e-%lu)\n", e10, u, e10)
 
gmp_printf (!"k = %lu\n\n", k)
 
gmp_printf (!"time: %.7f s\n", TIMER - tim)
end
output:
gamma 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 (maxerr. 1e-100)
k = 255

The easy way[edit]

' ******************************************
'Subject: Euler's constant 0.5772...
'tested : FreeBasic 1.08.1 with mpfr 4.1.0
'-------------------------------------------
#include "gmp.bi"
#include "mpfr.bi"
 
dim as mpfr_ptr a = allocate(len(__mpfr_struct))
dim as ulong e2, e10
dim as double tim = TIMER
 
'decimal precision
e10 = 100
 
'binary precision
e2 = (1 + e10) / .30103
mpfr_init2 (a, e2)
 
mpfr_const_euler (a, MPFR_RNDN)
mpfr_printf (!"gamma %.*Rf\n\n", e10, a)
 
gmp_printf (!"time: %.7f s\n", TIMER - tim)
end


Julia[edit]

Translation of: PARI/GP
display(MathConstants.γ)  # γ = 0.5772156649015...
 

PARI/GP[edit]

built-in:

\l "euler_const.log"
\p 100
print("gamma ", Euler);
\q

Perl[edit]

#!/usr/bin/perl
 
use strict; # https://en.wikipedia.org/wiki/Euler%27s_constant
use warnings;
use List::Util qw( sum );
 
print sum( map 1 / $_, 1 .. 1e6) - log 1e6, "\n";
Output:
0.577216164900715

Phix[edit]

-- demo\rosetta\Eulers_constant.exw
without js  -- no mpfr_get_d_2exp() or mpfr_const_euler() in mpfr.js as yet

-- part 1, translation of Perl, with the same inaccuracy
constant C = sum(sq_div(1,tagset(1e6)))-log(1e6)
printf(1,"gamma %.12f  (max 12d.p. of accuracy)\n",C)

-- part 2, translation of C, from first principles.
requires("1.0.1") -- mpfr_get_d_2exp(), mpfr_const_euler()
include mpfr.e
mpfr u, v, k2;
integer e, e10, e2
atom f
 
//log(x/y) with the Taylor series for atanh(x-y/x+y)
procedure ln(mpfr s, integer x, y)
    mpfr d = u, q = v;
    assert((x-y)==1) 
    y += x
    mpfr_set_si(s, y)
    mpfr_si_div(s, 1, s)            // s = 1 / (x + y)

    mpfr_mul(k2, s, s)              // k2 = s * s
    mpfr_set(d, s)
 
    integer k = 1
    while true do
        k += 2;
        mpfr_mul(d, d, k2)          // d *= k2
        mpfr_div_si(q, d, k)        // q = d / k
        mpfr_add(s, s, q)           // s += q
        {f,e} = mpfr_get_d_2exp(q)
        if abs(e)>=e2 then exit end if
    end while 

    mpfr_mul_si(s, s, 2)            //s *= 2
end procedure
 
mpfr a, b
integer k, 
        n = 60,     -- (required precision in decimal dp *6/10)
        n2, 
        r = 41,
        s = 30,
        t = 18;

// n = 2^i * 3^j * 5^k
 
// log(n) = r * log(16/15) + s * log(25/24) + t * log(81/80)
 
// solve linear system for r, s, t
//  4 -3 -4| i
// -1 -1  4| j
// -1  2 -1| k
 
//decimal precision
e10 = floor(n/0.6)
//binary precision
e2 = floor((1 + e10) / 0.30103)
 
mpfr_set_default_precision(e2)
{a, b, u, v, k2} = mpfr_inits(5)
 
//Compute log terms
 
ln(b, 16, 15)
 
mpfr_mul_si(a, b, r)    // a = r * b
 
ln(b, 25, 24)
 
mpfr_mul_si(u, b, s)    
mpfr_add (a, a, u)      // a += s * b
 
ln(b, 81, 80)
 
mpfr_mul_si(u, b, t)
mpfr_add (a, a, u)      // a += t * b
 
mpfr_neg(a, a)          // a = -a
mpfr_set_si(b, 1)       // b = 1
mpfr_set (u, a)
mpfr_set (v, b)
 
k = 0;
n2 = n * n;
//k2 = k * k
mpfr_set_si(k2, 0)
while true do
    mpfr_add_si(k2, k2, k*2+1)      // k2 += 2k + 1
    k += 1;
 
    mpfr_div(b, b, k2)
    mpfr_mul_si(b, b, n2)           // b = b * n2 / k2
   
    mpfr_div_si(a, a, k)
    mpfr_mul_si(a, a, n2)
    mpfr_add (a, a, b)
    mpfr_div_si(a, a, k)            // a = (a * n2 / k + b) / k
 
    mpfr_add(u, u, a)               // u += a
    mpfr_add(v, v, b)               // v += b
 
    {f,e} = mpfr_get_d_2exp (a)
    if abs(e)>=e2 then exit end if
end while
 
mpfr_div(u, u, v)
string su = mpfr_get_fixed(u,e10)
printf(1,"gamma %s (maxerr. 1e-%d)\n", {su, e10})
 
-- part 3, the easy way
mpfr gamma = mpfr_init(0,-100)
mpfr_const_euler(gamma)
printf(1,"gamma %s (mpfr_const_euler)\n",{mpfr_get_fixed(gamma,100)})
Output:
gamma 0.577216164901  (max 12d.p. of accuracy)
gamma 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467494 (maxerr. 1e-100)
gamma 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 (mpfr_const_euler)

Raku[edit]

# 20211124 Raku programming solution 
 
sub gamma (\N) {
 
# my $sum = (1/2 - 1/3).FatRat; # for arbitrary precision
my $sum = (1/2 - 1/3);
 
return $sum + [+] (2..N).race.map: -> \n {
my $power = 2**n;
my $sign = -1;
 
# my FatRat $term; # for arbitrary precision
# for ($power..^2*$power) { $term += (1 / (($sign *= -1)*$_)).FatRat }
 
my $term;
for ($power..^2*$power) { $term += ($sign = -$sign) / $_ }
 
n*$term
}
}
 
say gamma 23 ;
Output:
0.5772149198434515

Wren[edit]

Translation of: C

Single precision (Cli)[edit]

Library: Wren-fmt

Note that, whilst internally double arithmetic is carried out to the same precision as C (Wren is written in C), printing doubles is effectively limited to a maximum of 14 decimal places.

import "./fmt" for Fmt
 
var eps = 1e-6
 
System.print("From the definition, err. 3e-10")
var n = 400
var h = 1
for (k in 2..n) h = h + 1/k
//faster convergence: Negoi, 1997
var a = (n + 0.5 + 1/(24*n)).log
 
Fmt.print("Hn $0.14f", h)
Fmt.print("gamma $0.14f\nk = $d\n", h - a, n)
 
System.print("Sweeney, 1963, err. idem")
n = 21
var s = [0, n]
var r = n
var k = 1
while (true) {
k = k + 1
r = r * n / k
s[k & 1] = s[k & 1] + r/k
if (r <= eps) break
}
Fmt.print("gamma $0.14f\nk = $d\n", s[1] - s[0] - n.log, k)
 
System.print("Bailey, 1988")
n = 5
a = 1
h = 1
var n2 = 2.pow(n)
r = 1
k = 1
while (true) {
k = k + 1
r = r * n2 / k
h = h + 1/k
var b = a
a = a + r * h
if ((b-a).abs <= eps) break
}
a = a * n2 / n2.exp
Fmt.print("gamma $0.14f\nk = $d\n", a - n * 2.log, k)
 
System.print("Brent-McMillan, 1980")
n = 13
a = -n.log
var b = 1
var u = a
var v = b
n2 = n * n
var k2 = 0
k = 0
while (true) {
k2 = k2 + 2*k + 1
k = k + 1
a = a * n2 / k
b = b * n2 / k2
a = (a + b)/k
u = u + a
v = v + b
if (a.abs <= eps) break
}
Fmt.print("gamma $0.14f\nk = $d\n", u / v, k)
 
System.print("How Euler did it in 1735")
// Bernoulli numbers with even indices
var b2 = [1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, 43867/798]
var m = 7
n = 10
// n'th harmonic number
h = 1
for (k in 2..n) h = h + 1/k
Fmt.print("Hn $0.14f", h)
h = h - n.log
Fmt.print(" -ln $0.14f", h)
// expansion C = -digamma(1)
a = -1 / (2*n)
n2 = n * n
r = 1
for (k in 1..m) {
r = r * n2
a = a + b2[k] / (2*k*r)
}
Fmt.print("err $0.14f\ngamma $0.14f\nk = $d", a, h + a, n + m)
System.print("\nC = 0.57721566490153286...")
Output:
From the definition, err. 3e-10
Hn    6.56992969117651
gamma 0.57721566457657
k = 400

Sweeney, 1963, err. idem
gamma 0.57721566456363
k = 68

Bailey, 1988
gamma 0.57721566490154
k = 89

Brent-McMillan, 1980
gamma 0.57721566490153
k = 40

How Euler did it in 1735
Hn    2.92896825396825
  -ln 0.62638316097421
err  -0.04916749607268
gamma 0.57721566490153
k = 17

C = 0.57721566490153286...

Multi precision (Embedded)[edit]

Library: Wren-gmp

The display is limited to 100 digits for all four examples as I couldn't see much point in showing them all.

import "./gmp" for Mpf
 
var euler = Fn.new { |n, r, s, t|
// decimal precision
var e10 = (n/0.6).floor
 
// binary precision
var e2 = ((1 + n/0.6)/0.30103).round
Mpf.defaultPrec = e2
 
var b = Mpf.new().log(Mpf.from(16).div(15))
var a = b.mul(r)
b = Mpf.new().log(Mpf.from(25).div(24))
a.add(b.mul(s))
b = Mpf.new().log(Mpf.from(81).div(80))
var u = b * t
a.add(u).neg
b.set(1)
u.set(a)
var v = Mpf.from(b)
var k = 0
var n2 = n * n
var k2 = Mpf.zero
while (true) {
k2.add((k << 1) + 1)
k = k + 1
b.mul(n2).div(k2)
a.mul(n2).div(k).add(b).div(k)
u.add(a)
v.add(b)
var e = Mpf.frexp(a)[1]
if (e.abs >= e2) break
}
u.div(v)
System.print("gamma %(u.toString(10, 100)) (maxerr. 1e-%(e10))")
System.print("k = %(k)")
}
 
var start = System.clock
euler.call(60, 41, 30, 18)
euler.call(4800, 85, 62, 37)
euler.call(9375, 91, 68, 40)
euler.call(18750, 98, 73, 43)
System.print("\nTook %(System.clock - start) seconds.")
Output:
gamma 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 (maxerr. 1e-100)
k = 255
gamma 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 (maxerr. 1e-8000)
k = 20462
gamma 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 (maxerr. 1e-15625)
k = 39967
gamma 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 (maxerr. 1e-31250)
k = 79936

Took 2.330538 seconds.

The easy way (Embedded)[edit]

import "./gmp" for Mpf
 
var prec = (101/0.30103).round
var gamma = Mpf.euler(prec)
System.print(gamma.toString(10, 100))
Output:
0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495