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121
Task/Eulers-constant-0.5772.../C/eulers-constant-0.5772...-1.c
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121
Task/Eulers-constant-0.5772.../C/eulers-constant-0.5772...-1.c
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/*********************************************
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Subject: Comparing five methods for
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computing Euler's constant 0.5772...
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tested : tcc-0.9.27
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--------------------------------------------*/
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#include <math.h>
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#include <stdio.h>
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#define eps 1e-6
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int main(void) {
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double a, b, h, n2, r, u, v;
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int k, k2, m, n;
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printf("From the definition, err. 3e-10\n");
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n = 400;
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h = 1;
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for (k = 2; k <= n; k++) {
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h += 1.0 / k;
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}
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//faster convergence: Negoi, 1997
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a = log(n +.5 + 1.0 / (24*n));
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printf("Hn %.16f\n", h);
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printf("gamma %.16f\nk = %d\n\n", h - a, n);
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printf("Sweeney, 1963, err. idem\n");
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n = 21;
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double s[] = {0, n};
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r = n;
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k = 1;
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do {
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k += 1;
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r *= (double) n / k;
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s[k & 1] += r / k;
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} while (r > eps);
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printf("gamma %.16f\nk = %d\n\n", s[1] - s[0] - log(n), k);
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printf("Bailey, 1988\n");
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n = 5;
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a = 1;
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h = 1;
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n2 = pow(2,n);
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r = 1;
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k = 1;
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do {
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k += 1;
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r *= n2 / k;
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h += 1.0 / k;
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b = a; a += r * h;
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} while (fabs(b - a) > eps);
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a *= n2 / exp(n2);
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printf("gamma %.16f\nk = %d\n\n", a - n * log(2), k);
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printf("Brent-McMillan, 1980\n");
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n = 13;
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a = -log(n);
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b = 1;
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u = a;
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v = b;
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n2 = n * n;
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k2 = 0;
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k = 0;
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do {
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k2 += 2*k + 1;
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k += 1;
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a *= n2 / k;
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b *= n2 / k2;
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a = (a + b) / k;
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u += a;
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v += b;
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} while (fabs(a) > eps);
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printf("gamma %.16f\nk = %d\n\n", u / v, k);
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printf("How Euler did it in 1735\n");
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//Bernoulli numbers with even indices
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double B2[] = {1.0,1.0/6,-1.0/30,1.0/42,-1.0/30,\
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5.0/66,-691.0/2730,7.0/6,-3617.0/510,43867.0/798};
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m = 7;
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if (m > 9) return(0);
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n = 10;
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//n-th harmonic number
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h = 1;
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for (k = 2; k <= n; k++) {
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h += 1.0 / k;
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}
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printf("Hn %.16f\n", h);
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h -= log(n);
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printf(" -ln %.16f\n", h);
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//expansion C = -digamma(1)
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a = -1.0 / (2*n);
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n2 = n * n;
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r = 1;
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for (k = 1; k <= m; k++) {
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r *= n2;
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a += B2[k] / (2*k * r);
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}
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printf("err %.16f\ngamma %.16f\nk = %d", a, h + a, n + m);
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printf("\n\nC = 0.57721566490153286...\n");
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}
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178
Task/Eulers-constant-0.5772.../C/eulers-constant-0.5772...-2.c
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178
Task/Eulers-constant-0.5772.../C/eulers-constant-0.5772...-2.c
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@ -0,0 +1,178 @@
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/**************************************************
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Subject: Computation of Euler's constant 0.5772...
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with the Brent-McMillan algorithm B1,
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Math. Comp. 34 (1980), 305-312
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tested : tcc-0.9.27 with gmp 6.2.0
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-------------------------------------------------*/
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#include <gmp.h>
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#include <stdio.h>
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#include <stdlib.h>
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#include <time.h>
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//multi-precision float pointers
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mpf_ptr u, v, k2;
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//precision parameters
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unsigned long e10, e2;
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long e;
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double f;
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//log(x/y) with the Taylor series for atanh(x-y/x+y)
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void ln (mpf_ptr s, unsigned long x, unsigned long y) {
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mpf_ptr d = u, q = v;
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unsigned long k;
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//Möbius transformation
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k = x; x -= y; y += k;
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if (x != 1) {
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printf ("ln: illegal argument x - y != 1");
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exit;
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}
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//s = 1 / (x + y)
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mpf_set_ui (s, y);
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mpf_ui_div (s, 1, s);
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//k2 = s * s
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mpf_mul (k2, s, s);
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mpf_set (d, s);
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k = 1;
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do {
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k += 2;
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//d *= k2
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mpf_mul (d, d, k2);
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//q = d / k
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mpf_div_ui (q, d, k);
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//s += q
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mpf_add (s, s, q);
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f = mpf_get_d_2exp (&e, q);
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} while (abs(e) < e2);
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//s *= 2
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mpf_mul_2exp (s, s, 1);
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}
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int main (void) {
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mpf_ptr a = malloc(sizeof(__mpf_struct));
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mpf_ptr b = malloc(sizeof(__mpf_struct));
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u = malloc(sizeof(__mpf_struct));
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v = malloc(sizeof(__mpf_struct));
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k2 = malloc(sizeof(__mpf_struct));
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//unsigned long integers
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unsigned long k, n, n2, r, s, t;
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clock_t tim = clock();
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// n = 2^i * 3^j * 5^k
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// log(n) = r * log(16/15) + s * log(25/24) + t * log(81/80)
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// solve linear system for r, s, t
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// 4 -3 -4| i
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// -1 -1 4| j
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// -1 2 -1| k
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//examples
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t = 1;
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switch (t) {
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case 1 :
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n = 60;
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r = 41;
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s = 30;
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t = 18;
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//100 digits
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break;
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case 2 :
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n = 4800;
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r = 85;
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s = 62;
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t = 37;
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//8000 digits, 0.6 s
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break;
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case 3 :
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n = 9375;
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r = 91;
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s = 68;
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t = 40;
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//15625 digits, 2.5 s
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break;
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default :
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n = 18750;
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r = 98;
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s = 73;
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t = 43;
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//31250 digits, 12 s. @2.00GHz
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}
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//decimal precision
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e10 = n / .6;
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//binary precision
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e2 = (1 + e10) / .30103;
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//initialize mpf's
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mpf_set_default_prec (e2);
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mpf_inits (a, b, u, v, k2, (mpf_ptr)0);
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//Compute log terms
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ln (b, 16, 15);
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//a = r * b
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mpf_mul_ui (a, b, r);
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ln (b, 25, 24);
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//a += s * b
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mpf_mul_ui (u, b, s);
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mpf_add (a, a, u);
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ln (b, 81, 80);
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//a += t * b
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mpf_mul_ui (u, b, t);
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mpf_add (a, a, u);
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//gmp_printf ("log(%lu) %.*Ff\n", n, e10, a);
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//B&M, algorithm B1
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//a = -a, b = 1
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mpf_neg (a, a);
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mpf_set_ui (b, 1);
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mpf_set (u, a);
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mpf_set (v, b);
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k = 0;
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n2 = n * n;
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//k2 = k * k
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mpf_set_ui (k2, 0);
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do {
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//k2 += 2k + 1
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mpf_add_ui (k2, k2, (k << 1) + 1);
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k += 1;
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//b = b * n2 / k2
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mpf_div (b, b, k2);
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mpf_mul_ui (b, b, n2);
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//a = (a * n2 / k + b) / k
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mpf_div_ui (a, a, k);
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mpf_mul_ui (a, a, n2);
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mpf_add (a, a, b);
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mpf_div_ui (a, a, k);
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//u += a, v += b
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mpf_add (u, u, a);
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mpf_add (v, v, b);
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f = mpf_get_d_2exp (&e, a);
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} while (abs(e) < e2);
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mpf_div (u, u, v);
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gmp_printf ("gamma %.*Ff (maxerr. 1e-%lu)\n", e10, u, e10);
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gmp_printf ("k = %lu\n\n", k);
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tim = clock() - tim;
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printf("time: %.7f s\n",((double)tim)/CLOCKS_PER_SEC);
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}
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@ -0,0 +1,28 @@
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/*******************************************
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Subject: Euler's constant 0.5772...
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tested : tcc-0.9.27 with mpfr 4.1.0
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------------------------------------------*/
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#include <gmp.h>
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#include <mpfr.h>
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#include <stdio.h>
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#include <stdlib.h>
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#include <time.h>
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int main (void) {
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mpfr_ptr a = malloc(sizeof(__mpfr_struct));
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unsigned long e2, e10;
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clock_t tim = clock();
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//decimal precision
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e10 = 100;
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//binary precision
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e2 = (1 + e10) / .30103;
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mpfr_init2 (a, e2);
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mpfr_const_euler (a, MPFR_RNDN);
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mpfr_printf ("gamma %.*Rf\n\n", e10, a);
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tim = clock() - tim;
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gmp_printf ("time: %.7f s\n",((double)tim)/CLOCKS_PER_SEC);
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}
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