| /* mpfr_const_euler -- Euler's constant |
| |
| Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
| Contributed by the Arenaire and Cacao projects, INRIA. |
| |
| This file is part of the GNU MPFR Library. |
| |
| The GNU MPFR Library is free software; you can redistribute it and/or modify |
| it under the terms of the GNU Lesser General Public License as published by |
| the Free Software Foundation; either version 2.1 of the License, or (at your |
| option) any later version. |
| |
| The GNU MPFR Library is distributed in the hope that it will be useful, but |
| WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY |
| or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public |
| License for more details. |
| |
| You should have received a copy of the GNU Lesser General Public License |
| along with the GNU MPFR Library; see the file COPYING.LIB. If not, write to |
| the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, |
| MA 02110-1301, USA. */ |
| |
| #define MPFR_NEED_LONGLONG_H |
| #include "mpfr-impl.h" |
| |
| /* Declare the cache */ |
| MPFR_DECL_INIT_CACHE(__gmpfr_cache_const_euler, mpfr_const_euler_internal); |
| |
| /* Set User Interface */ |
| #undef mpfr_const_euler |
| int |
| mpfr_const_euler (mpfr_ptr x, mp_rnd_t rnd_mode) { |
| return mpfr_cache (x, __gmpfr_cache_const_euler, rnd_mode); |
| } |
| |
| |
| static void mpfr_const_euler_S2 (mpfr_ptr, unsigned long); |
| static void mpfr_const_euler_R (mpfr_ptr, unsigned long); |
| |
| int |
| mpfr_const_euler_internal (mpfr_t x, mp_rnd_t rnd) |
| { |
| mp_prec_t prec = MPFR_PREC(x), m, log2m; |
| mpfr_t y, z; |
| unsigned long n; |
| int inexact; |
| MPFR_ZIV_DECL (loop); |
| |
| log2m = MPFR_INT_CEIL_LOG2 (prec); |
| m = prec + 2 * log2m + 23; |
| |
| mpfr_init2 (y, m); |
| mpfr_init2 (z, m); |
| |
| MPFR_ZIV_INIT (loop, m); |
| for (;;) |
| { |
| mp_exp_t exp_S, err; |
| /* since prec >= 1, we have m >= 24 here, which ensures n >= 9 below */ |
| n = 1 + (unsigned long) ((double) m * LOG2 / 2.0); |
| MPFR_ASSERTD (n >= 9); |
| mpfr_const_euler_S2 (y, n); /* error <= 3 ulps */ |
| exp_S = MPFR_EXP(y); |
| mpfr_set_ui (z, n, GMP_RNDN); |
| mpfr_log (z, z, GMP_RNDD); /* error <= 1 ulp */ |
| mpfr_sub (y, y, z, GMP_RNDN); /* S'(n) - log(n) */ |
| /* the error is less than 1/2 + 3*2^(exp_S-EXP(y)) + 2^(EXP(z)-EXP(y)) |
| <= 1/2 + 2^(exp_S+2-EXP(y)) + 2^(EXP(z)-EXP(y)) |
| <= 1/2 + 2^(1+MAX(exp_S+2,EXP(z))-EXP(y)) */ |
| err = 1 + MAX(exp_S + 2, MPFR_EXP(z)) - MPFR_EXP(y); |
| err = (err >= -1) ? err + 1 : 0; /* error <= 2^err ulp(y) */ |
| exp_S = MPFR_EXP(y); |
| mpfr_const_euler_R (z, n); /* err <= ulp(1/2) = 2^(-m) */ |
| mpfr_sub (y, y, z, GMP_RNDN); |
| /* err <= 1/2 ulp(y) + 2^(-m) + 2^(err + exp_S - EXP(y)) ulp(y). |
| Since the result is between 0.5 and 1, ulp(y) = 2^(-m). |
| So we get 3/2*ulp(y) + 2^(err + exp_S - EXP(y)) ulp(y). |
| 3/2 + 2^e <= 2^(e+1) for e>=1, and <= 2^2 otherwise */ |
| err = err + exp_S - MPFR_EXP(y); |
| err = (err >= 1) ? err + 1 : 2; |
| if (MPFR_LIKELY (MPFR_CAN_ROUND (y, m - err, prec, rnd))) |
| break; |
| MPFR_ZIV_NEXT (loop, m); |
| mpfr_set_prec (y, m); |
| mpfr_set_prec (z, m); |
| } |
| MPFR_ZIV_FREE (loop); |
| |
| inexact = mpfr_set (x, y, rnd); |
| |
| mpfr_clear (y); |
| mpfr_clear (z); |
| |
| return inexact; /* always inexact */ |
| } |
| |
| static void |
| mpfr_const_euler_S2_aux (mpz_t P, mpz_t Q, mpz_t T, unsigned long n, |
| unsigned long a, unsigned long b, int need_P) |
| { |
| if (a + 1 == b) |
| { |
| mpz_set_ui (P, n); |
| if (a > 1) |
| mpz_mul_si (P, P, 1 - (long) a); |
| mpz_set (T, P); |
| mpz_set_ui (Q, a); |
| mpz_mul_ui (Q, Q, a); |
| } |
| else |
| { |
| unsigned long c = (a + b) / 2; |
| mpz_t P2, Q2, T2; |
| mpfr_const_euler_S2_aux (P, Q, T, n, a, c, 1); |
| mpz_init (P2); |
| mpz_init (Q2); |
| mpz_init (T2); |
| mpfr_const_euler_S2_aux (P2, Q2, T2, n, c, b, 1); |
| mpz_mul (T, T, Q2); |
| mpz_mul (T2, T2, P); |
| mpz_add (T, T, T2); |
| if (need_P) |
| mpz_mul (P, P, P2); |
| mpz_mul (Q, Q, Q2); |
| mpz_clear (P2); |
| mpz_clear (Q2); |
| mpz_clear (T2); |
| /* divide by 2 if possible */ |
| { |
| unsigned long v2; |
| v2 = mpz_scan1 (P, 0); |
| c = mpz_scan1 (Q, 0); |
| if (c < v2) |
| v2 = c; |
| c = mpz_scan1 (T, 0); |
| if (c < v2) |
| v2 = c; |
| if (v2) |
| { |
| mpz_tdiv_q_2exp (P, P, v2); |
| mpz_tdiv_q_2exp (Q, Q, v2); |
| mpz_tdiv_q_2exp (T, T, v2); |
| } |
| } |
| } |
| } |
| |
| /* computes S(n) = sum(n^k*(-1)^(k-1)/k!/k, k=1..ceil(4.319136566 * n)) |
| using binary splitting. |
| We have S(n) = sum(f(k), k=1..N) with N=ceil(4.319136566 * n) |
| and f(k) = n^k*(-1)*(k-1)/k!/k, |
| thus f(k)/f(k-1) = -n*(k-1)/k^2 |
| */ |
| static void |
| mpfr_const_euler_S2 (mpfr_t x, unsigned long n) |
| { |
| mpz_t P, Q, T; |
| unsigned long N = (unsigned long) (ALPHA * (double) n + 1.0); |
| mpz_init (P); |
| mpz_init (Q); |
| mpz_init (T); |
| mpfr_const_euler_S2_aux (P, Q, T, n, 1, N + 1, 0); |
| mpfr_set_z (x, T, GMP_RNDN); |
| mpfr_div_z (x, x, Q, GMP_RNDN); |
| mpz_clear (P); |
| mpz_clear (Q); |
| mpz_clear (T); |
| } |
| |
| /* computes R(n) = exp(-n)/n * sum(k!/(-n)^k, k=0..n-2) |
| with error at most 4*ulp(x). Assumes n>=2. |
| Since x <= exp(-n)/n <= 1/8, then 4*ulp(x) <= ulp(1). |
| */ |
| static void |
| mpfr_const_euler_R (mpfr_t x, unsigned long n) |
| { |
| unsigned long k, m; |
| mpz_t a, s; |
| mpfr_t y; |
| |
| MPFR_ASSERTN (n >= 2); /* ensures sum(k!/(-n)^k, k=0..n-2) >= 2/3 */ |
| |
| /* as we multiply the sum by exp(-n), we need only PREC(x) - n/LOG2 bits */ |
| m = MPFR_PREC(x) - (unsigned long) ((double) n / LOG2); |
| |
| mpz_init_set_ui (a, 1); |
| mpz_mul_2exp (a, a, m); |
| mpz_init_set (s, a); |
| |
| for (k = 1; k <= n; k++) |
| { |
| mpz_mul_ui (a, a, k); |
| mpz_div_ui (a, a, n); |
| /* the error e(k) on a is e(k) <= 1 + k/n*e(k-1) with e(0)=0, |
| i.e. e(k) <= k */ |
| if (k % 2) |
| mpz_sub (s, s, a); |
| else |
| mpz_add (s, s, a); |
| } |
| /* the error on s is at most 1+2+...+n = n*(n+1)/2 */ |
| mpz_div_ui (s, s, n); /* err <= 1 + (n+1)/2 */ |
| MPFR_ASSERTN (MPFR_PREC(x) >= mpz_sizeinbase(s, 2)); |
| mpfr_set_z (x, s, GMP_RNDD); /* exact */ |
| mpfr_div_2ui (x, x, m, GMP_RNDD); |
| /* now x = 1/n * sum(k!/(-n)^k, k=0..n-2) <= 1/n */ |
| /* err(x) <= (n+1)/2^m <= (n+1)*exp(n)/2^PREC(x) */ |
| |
| mpfr_init2 (y, m); |
| mpfr_set_si (y, -(long)n, GMP_RNDD); /* assumed exact */ |
| mpfr_exp (y, y, GMP_RNDD); /* err <= ulp(y) <= exp(-n)*2^(1-m) */ |
| mpfr_mul (x, x, y, GMP_RNDD); |
| /* err <= ulp(x) + (n + 1 + 2/n) / 2^prec(x) |
| <= ulp(x) + (n + 1 + 2/n) ulp(x)/x since x*2^(-prec(x)) < ulp(x) |
| <= ulp(x) + (n + 1 + 2/n) 3/(2n) ulp(x) since x >= 2/3*n for n >= 2 |
| <= 4 * ulp(x) for n >= 2 */ |
| mpfr_clear (y); |
| |
| mpz_clear (a); |
| mpz_clear (s); |
| } |