diff options
Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/eint.c')
-rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/eint.c | 382 |
1 files changed, 0 insertions, 382 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/eint.c b/Build/source/libs/mpfr/mpfr-src/src/eint.c deleted file mode 100644 index 4df65821d6f..00000000000 --- a/Build/source/libs/mpfr/mpfr-src/src/eint.c +++ /dev/null @@ -1,382 +0,0 @@ -/* mpfr_eint, mpfr_eint1 -- the exponential integral - -Copyright 2005-2020 Free Software Foundation, Inc. -Contributed by the AriC and Caramba 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 3 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.LESSER. If not, see -https://www.gnu.org/licenses/ or 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" - -/* eint1(x) = -gamma - log(x) - sum((-1)^k*z^k/k/k!, k=1..infinity) for x > 0 - = - eint(-x) for x < 0 - where - eint (x) = gamma + log(x) + sum(z^k/k/k!, k=1..infinity) for x > 0 - eint (x) is undefined for x < 0. -*/ - -/* Compute in y an approximation of sum(x^k/k/k!, k=1..infinity), - assuming x != 0, and return e such that the absolute error is - bounded by 2^e ulp(y). - Return PREC(y) when the truncated series does not converge. -*/ -static mpfr_exp_t -mpfr_eint_aux (mpfr_t y, mpfr_srcptr x) -{ - mpfr_t eps; /* dynamic (absolute) error bound on t */ - mpfr_t erru, errs; - mpz_t m, s, t, u; - mpfr_exp_t e, sizeinbase; - mpfr_prec_t w = MPFR_PREC(y); - unsigned long k; - MPFR_GROUP_DECL (group); - - MPFR_LOG_FUNC ( - ("x[%Pu]=%.*Rg", mpfr_get_prec (x), mpfr_log_prec, x), - ("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y)); - - /* for |x| <= 1, we have S := sum(x^k/k/k!, k=1..infinity) = x + R(x) - where |R(x)| <= (x/2)^2/(1-|x|/2) <= 2*(x/2)^2 - thus |R(x)/x| <= |x|/2 - thus if |x| <= 2^(-PREC(y)) we have |S - o(x)| <= ulp(y) */ - - if (MPFR_GET_EXP(x) <= - (mpfr_exp_t) w) - { - mpfr_set (y, x, MPFR_RNDN); - return 0; - } - - mpz_init (s); /* initializes to 0 */ - mpz_init (t); - mpz_init (u); - mpz_init (m); - MPFR_GROUP_INIT_3 (group, 31, eps, erru, errs); - e = mpfr_get_z_2exp (m, x); /* x = m * 2^e with m != 0 */ - MPFR_LOG_MSG (("e=%" MPFR_EXP_FSPEC "d\n", (mpfr_eexp_t) e)); - MPFR_ASSERTD (mpz_sizeinbase (m, 2) == MPFR_PREC (x)); /* since m != 0 */ - if (MPFR_PREC (x) > w) - { - e += MPFR_PREC (x) - w; - mpz_tdiv_q_2exp (m, m, MPFR_PREC (x) - w); /* one still has m != 0 */ - MPFR_LOG_MSG (("e=%" MPFR_EXP_FSPEC "d\n", (mpfr_eexp_t) e)); - } - /* Remove trailing zeroes from m: this will speed up much cases where - x is a small integer divided by a power of 2. - Note: As shown above, m != 0. This is needed for the "e += ..." below, - otherwise n would take the largest value of mp_bitcnt_t and could be - too large. */ - { - mp_bitcnt_t n = mpz_scan1 (m, 0); - mpz_tdiv_q_2exp (m, m, n); - /* Since one initially has mpz_sizeinbase (m, 2) == MPFR_PREC (x) - and m has not increased, one can deduce that n <= MPFR_PREC (x), - so that the cast to mpfr_prec_t is valid. This cast is needed to - ensure that the operand e of the addition below is not converted - to an unsigned integer type, which could yield incorrect results - with some C implementations. */ - MPFR_ASSERTD (n <= MPFR_PREC (x)); - e += (mpfr_prec_t) n; - } - /* initialize t to 2^w */ - mpz_set_ui (t, 1); - mpz_mul_2exp (t, t, w); - mpfr_set_ui (eps, 0, MPFR_RNDN); /* eps[0] = 0 */ - mpfr_set_ui (errs, 0, MPFR_RNDN); /* maximal error on s */ - for (k = 1;; k++) - { - /* let t[k] = x^k/k/k!, and eps[k] be the absolute error on t[k]: - since t[k] = trunc(t[k-1]*m*2^e/k), we have - eps[k+1] <= 1 + eps[k-1]*|m|*2^e/k + |t[k-1]|*|m|*2^(1-w)*2^e/k - = 1 + (eps[k-1] + |t[k-1]|*2^(1-w))*|m|*2^e/k - = 1 + (eps[k-1]*2^(w-1) + |t[k-1]|)*2^(1-w)*|m|*2^e/k */ - mpfr_mul_2ui (eps, eps, w - 1, MPFR_RNDU); - if (mpz_sgn (t) >= 0) - mpfr_add_z (eps, eps, t, MPFR_RNDU); - else - mpfr_sub_z (eps, eps, t, MPFR_RNDU); - MPFR_MPZ_SIZEINBASE2 (sizeinbase, m); - mpfr_mul_2si (eps, eps, sizeinbase - (w - 1) + e, MPFR_RNDU); - mpfr_div_ui (eps, eps, k, MPFR_RNDU); - mpfr_add_ui (eps, eps, 1, MPFR_RNDU); - mpz_mul (t, t, m); - if (e < 0) - mpz_tdiv_q_2exp (t, t, -e); - else - mpz_mul_2exp (t, t, e); - mpz_tdiv_q_ui (t, t, k); - mpz_tdiv_q_ui (u, t, k); - mpz_add (s, s, u); - /* the absolute error on u is <= 1 + eps[k]/k */ - mpfr_div_ui (erru, eps, k, MPFR_RNDU); - mpfr_add_ui (erru, erru, 1, MPFR_RNDU); - /* and that on s is the sum of all errors on u */ - mpfr_add (errs, errs, erru, MPFR_RNDU); - /* we are done when t is smaller than errs */ - if (mpz_sgn (t) == 0) - sizeinbase = 0; - else - MPFR_MPZ_SIZEINBASE2 (sizeinbase, t); - if (sizeinbase < MPFR_GET_EXP (errs)) - break; - } - /* the truncation error is bounded by (|t|+eps)/k*(|x|/k + |x|^2/k^2 + ...) - <= (|t|+eps)/k*|x|/(k-|x|) */ - mpz_abs (t, t); - mpfr_add_z (eps, eps, t, MPFR_RNDU); - mpfr_div_ui (eps, eps, k, MPFR_RNDU); - mpfr_abs (erru, x, MPFR_RNDU); /* |x| */ - mpfr_mul (eps, eps, erru, MPFR_RNDU); - mpfr_ui_sub (erru, k, erru, MPFR_RNDD); - if (MPFR_IS_NEG (erru)) - { - /* the truncated series does not converge, return fail */ - e = w; - } - else - { - mpfr_div (eps, eps, erru, MPFR_RNDU); - mpfr_add (errs, errs, eps, MPFR_RNDU); - mpfr_set_z (y, s, MPFR_RNDN); - mpfr_div_2ui (y, y, w, MPFR_RNDN); - /* errs was an absolute error bound on s. We must convert it to an error - in terms of ulp(y). Since ulp(y) = 2^(EXP(y)-PREC(y)), we must - divide the error by 2^(EXP(y)-PREC(y)), but since we divided also - y by 2^w = 2^PREC(y), we must simply divide by 2^EXP(y). */ - e = MPFR_GET_EXP (errs) - MPFR_GET_EXP (y); - } - MPFR_GROUP_CLEAR (group); - mpz_clear (s); - mpz_clear (t); - mpz_clear (u); - mpz_clear (m); - MPFR_LOG_MSG (("e=%" MPFR_EXP_FSPEC "d\n", (mpfr_eexp_t) e)); - return e; -} - -/* Return in y an approximation of Ei(x) using the asymptotic expansion: - Ei(x) = exp(x)/x * (1 + 1/x + 2/x^2 + ... + k!/x^k + ...) - Assumes |x| >= PREC(y) * log(2). - Returns the error bound in terms of ulp(y). -*/ -static mpfr_exp_t -mpfr_eint_asympt (mpfr_ptr y, mpfr_srcptr x) -{ - mpfr_prec_t p = MPFR_PREC(y); - mpfr_t invx, t, err; - unsigned long k; - mpfr_exp_t err_exp; - - MPFR_LOG_FUNC ( - ("x[%Pu]=%.*Rg", mpfr_get_prec (x), mpfr_log_prec, x), - ("err_exp=%" MPFR_EXP_FSPEC "d", (mpfr_eexp_t) err_exp)); - - mpfr_init2 (t, p); - mpfr_init2 (invx, p); - mpfr_init2 (err, 31); /* error in ulps on y */ - mpfr_ui_div (invx, 1, x, MPFR_RNDN); /* invx = 1/x*(1+u) with |u|<=2^(1-p) */ - mpfr_set_ui (t, 1, MPFR_RNDN); /* exact */ - mpfr_set (y, t, MPFR_RNDN); - mpfr_set_ui (err, 0, MPFR_RNDN); - for (k = 1; MPFR_GET_EXP(t) + (mpfr_exp_t) p > MPFR_GET_EXP(y); k++) - { - mpfr_mul (t, t, invx, MPFR_RNDN); /* 2 more roundings */ - mpfr_mul_ui (t, t, k, MPFR_RNDN); /* 1 more rounding: t = k!/x^k*(1+u)^e - with u=2^{-p} and |e| <= 3*k */ - /* we use the fact that |(1+u)^n-1| <= 2*|n*u| for |n*u| <= 1, thus - the error on t is less than 6*k*2^{-p}*t <= 6*k*ulp(t) */ - /* err is in terms of ulp(y): transform it in terms of ulp(t) */ - mpfr_mul_2si (err, err, MPFR_GET_EXP(y) - MPFR_GET_EXP(t), MPFR_RNDU); - mpfr_add_ui (err, err, 6 * k, MPFR_RNDU); - /* transform back in terms of ulp(y) */ - mpfr_div_2si (err, err, MPFR_GET_EXP(y) - MPFR_GET_EXP(t), MPFR_RNDU); - mpfr_add (y, y, t, MPFR_RNDN); - } - /* add the truncation error bounded by ulp(y): 1 ulp */ - mpfr_mul (y, y, invx, MPFR_RNDN); /* err <= 2*err + 3/2 */ - mpfr_exp (t, x, MPFR_RNDN); /* err(t) <= 1/2*ulp(t) */ - mpfr_mul (y, y, t, MPFR_RNDN); /* again: err <= 2*err + 3/2 */ - mpfr_mul_2ui (err, err, 2, MPFR_RNDU); - mpfr_add_ui (err, err, 8, MPFR_RNDU); - err_exp = MPFR_GET_EXP(err); - mpfr_clear (t); - mpfr_clear (invx); - mpfr_clear (err); - return err_exp; -} - -/* mpfr_eint returns Ei(x) for x >= 0, - and -E1(-x) for x < 0, following http://dlmf.nist.gov/6.2 */ -int -mpfr_eint (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd) -{ - int inex; - mpfr_t tmp, ump, x_abs; - mpfr_exp_t err, te; - mpfr_prec_t prec; - MPFR_SAVE_EXPO_DECL (expo); - MPFR_ZIV_DECL (loop); - - MPFR_LOG_FUNC ( - ("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd), - ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inex)); - - if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) - { - if (MPFR_IS_NAN (x)) - { - MPFR_SET_NAN (y); - MPFR_RET_NAN; - } - else if (MPFR_IS_INF (x)) - { - /* eint(+inf) = +inf and eint(-inf) = -0 */ - if (MPFR_IS_POS (x)) - { - MPFR_SET_INF(y); - MPFR_SET_POS(y); - } - else - { - MPFR_SET_ZERO(y); - MPFR_SET_NEG(y); - } - MPFR_RET(0); - } - else /* eint(+/-0) = -Inf */ - { - MPFR_SET_INF(y); - MPFR_SET_NEG(y); - MPFR_SET_DIVBY0 (); - MPFR_RET(0); - } - } - - MPFR_TMP_INIT_ABS (x_abs, x); - - MPFR_SAVE_EXPO_MARK (expo); - - /* Init stuff */ - prec = MPFR_PREC (y) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (y)) + 6; - mpfr_init2 (tmp, 64); - mpfr_init2 (ump, 64); - - /* Since eint(x) >= exp(x)/x, we have log2(eint(x)) >= (x-log(x))/log(2). - Let's compute k <= (x-log(x))/log(2) in a low precision. If k >= emax, - then log2(eint(x)) >= emax, and eint(x) >= 2^emax, i.e. it overflows. */ - if (MPFR_IS_POS(x)) - { - mpfr_log (tmp, x, MPFR_RNDU); - mpfr_sub (ump, x, tmp, MPFR_RNDD); - mpfr_div (ump, ump, __gmpfr_const_log2_RNDU, MPFR_RNDD); - /* FIXME: We really need a mpfr_cmp_exp_t function. */ - MPFR_ASSERTN (MPFR_EMAX_MAX <= LONG_MAX); - if (mpfr_cmp_ui (ump, __gmpfr_emax) >= 0) - { - mpfr_clear (tmp); - mpfr_clear (ump); - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_overflow (y, rnd, 1); - } - } - - /* Since E1(x) <= exp(-x) for x >= 1, we have log2(E1(x)) <= -x/log(2). - Let's compute k >= -x/log(2) in a low precision. If k < emin - then log2(E1(x)) <= emin-1, and E1(x) <= 2^(emin-1): it underflows. */ - if (MPFR_IS_NEG(x) && MPFR_GET_EXP(x) >= 1) - { - mpfr_div (ump, x, __gmpfr_const_log2_RNDD, MPFR_RNDU); - MPFR_ASSERTN (MPFR_EMIN_MIN >= LONG_MIN); - if (mpfr_cmp_si (ump, __gmpfr_emin) < 0) - { - mpfr_clear (tmp); - mpfr_clear (ump); - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_underflow (y, rnd, -1); - } - } - - /* eint() has a root 0.37250741078136663446..., - so if x is near, already take more bits */ - if (MPFR_IS_POS(x) && MPFR_GET_EXP(x) == -1) /* 1/4 <= x < 1/2 */ - { - mpfr_t y; - mpfr_init2 (y, 32); - /* 1599907147/2^32 is a 32-bit approximation of 0.37250741078136663446 */ - mpfr_set_ui_2exp (y, 1599907147UL, -32, MPFR_RNDN); - mpfr_sub (y, x, y, MPFR_RNDN); - prec += (mpfr_zero_p (y)) ? 32 - : mpfr_get_exp (y) < 0 ? -mpfr_get_exp (y) : 0; - mpfr_clear (y); - } - - mpfr_set_prec (tmp, prec); - mpfr_set_prec (ump, prec); - - MPFR_ZIV_INIT (loop, prec); /* Initialize the ZivLoop controller */ - for (;;) /* Infinite loop */ - { - /* For the asymptotic expansion to work, we need that the smallest - value of k!/|x|^k is smaller than 2^(-p). The minimum is obtained for - x=k, and it is smaller than e*sqrt(x)/e^x for x>=1. */ - if (MPFR_GET_EXP (x) > 0 && - mpfr_cmp_d (x_abs, ((double) prec + - 0.5 * (double) MPFR_GET_EXP (x)) * LOG2 + 1.0) > 0) - err = mpfr_eint_asympt (tmp, x); - else - { - err = mpfr_eint_aux (tmp, x); /* error <= 2^err ulp(tmp) */ - te = MPFR_GET_EXP(tmp); - mpfr_const_euler (ump, MPFR_RNDN); /* 0.577 -> EXP(ump)=0 */ - mpfr_add (tmp, tmp, ump, MPFR_RNDN); - /* If tmp <> 0: - error <= 1/2 + 1/2*2^(EXP(ump)-EXP(tmp)) + 2^(te-EXP(tmp)+err) - <= 1/2 + 2^(MAX(EXP(ump), te+err+1) - EXP(tmp)) - <= 2^(MAX(0, 1 + MAX(EXP(ump), te+err+1) - EXP(tmp))). - If tmp = 0 we can use the same bound, replacing - EXP(tmp) by EXP(ump). */ - err = MAX(1, te + err + 2); - te = MPFR_IS_ZERO(tmp) ? MPFR_GET_EXP(ump) : MPFR_GET_EXP(tmp); - err = err - te; - err = MAX(0, err); - mpfr_log (ump, x_abs, MPFR_RNDN); - mpfr_add (tmp, tmp, ump, MPFR_RNDN); - /* same formula as above, except now EXP(ump) is not 0 */ - err += te + 1; - if (MPFR_LIKELY (!MPFR_IS_ZERO (ump))) - err = MAX (MPFR_GET_EXP (ump), err); - /* if tmp is zero, we surely cannot round correctly */ - err = (MPFR_IS_ZERO(tmp)) ? prec : MAX(0, err - MPFR_GET_EXP (tmp)); - } - /* Note: we assume here that MPFR_CAN_ROUND returns the same result - for rnd and MPFR_INVERT_RND(rnd) */ - if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - err, MPFR_PREC (y), rnd))) - break; - MPFR_ZIV_NEXT (loop, prec); /* Increase used precision */ - mpfr_set_prec (tmp, prec); - mpfr_set_prec (ump, prec); - } - MPFR_ZIV_FREE (loop); /* Free the ZivLoop Controller */ - - /* Set y to the computed value */ - inex = mpfr_set (y, tmp, rnd); - mpfr_clear (tmp); - mpfr_clear (ump); - - MPFR_SAVE_EXPO_FREE (expo); - return mpfr_check_range (y, inex, rnd); -} |