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Diffstat (limited to 'Build/source/libs/mpfr/mpfr-src/src/erfc.c')
| -rw-r--r-- | Build/source/libs/mpfr/mpfr-src/src/erfc.c | 277 |
1 files changed, 277 insertions, 0 deletions
diff --git a/Build/source/libs/mpfr/mpfr-src/src/erfc.c b/Build/source/libs/mpfr/mpfr-src/src/erfc.c new file mode 100644 index 00000000000..09111039e41 --- /dev/null +++ b/Build/source/libs/mpfr/mpfr-src/src/erfc.c @@ -0,0 +1,277 @@ +/* mpfr_erfc -- The Complementary Error Function of a floating-point number + +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" + +/* erfc(x) = 1 - erf(x) */ + +/* Put in y an approximation of erfc(x) for large x, using formulae 7.1.23 and + 7.1.24 from Abramowitz and Stegun. + Returns e such that the error is bounded by 2^e ulp(y), + or returns 0 in case of underflow. +*/ +static mpfr_exp_t +mpfr_erfc_asympt (mpfr_ptr y, mpfr_srcptr x) +{ + mpfr_t t, xx, err; + unsigned long k; + mpfr_prec_t prec = MPFR_PREC(y); + mpfr_exp_t exp_err; + + mpfr_init2 (t, prec); + mpfr_init2 (xx, prec); + mpfr_init2 (err, 31); + /* let u = 2^(1-p), and let us represent the error as (1+u)^err + with a bound for err */ + mpfr_sqr (xx, x, MPFR_RNDD); /* err <= 1 */ + mpfr_ui_div (xx, 1, xx, MPFR_RNDU); /* upper bound for 1/(2x^2), err <= 2 */ + mpfr_div_2ui (xx, xx, 1, MPFR_RNDU); /* exact */ + mpfr_set_ui (t, 1, MPFR_RNDN); /* current term, exact */ + mpfr_set (y, t, MPFR_RNDN); /* current sum */ + mpfr_set_ui (err, 0, MPFR_RNDN); + for (k = 1; ; k++) + { + mpfr_mul_ui (t, t, 2 * k - 1, MPFR_RNDU); /* err <= 4k-3 */ + mpfr_mul (t, t, xx, MPFR_RNDU); /* err <= 4k */ + /* for -1 < x < 1, and |nx| < 1, we have |(1+x)^n| <= 1+7/4|nx|. + Indeed, for x>=0: log((1+x)^n) = n*log(1+x) <= n*x. Let y=n*x < 1, + then exp(y) <= 1+7/4*y. + For x<=0, let x=-x, we can prove by induction that (1-x)^n >= 1-n*x.*/ + mpfr_mul_2si (err, err, MPFR_GET_EXP (y) - MPFR_GET_EXP (t), MPFR_RNDU); + mpfr_add_ui (err, err, 14 * k, MPFR_RNDU); /* 2^(1-p) * t <= 2 ulp(t) */ + mpfr_div_2si (err, err, MPFR_GET_EXP (y) - MPFR_GET_EXP (t), MPFR_RNDU); + if (MPFR_GET_EXP (t) + (mpfr_exp_t) prec <= MPFR_GET_EXP (y)) + { + /* the truncation error is bounded by |t| < ulp(y) */ + mpfr_add_ui (err, err, 1, MPFR_RNDU); + break; + } + if (k & 1) + mpfr_sub (y, y, t, MPFR_RNDN); + else + mpfr_add (y, y, t, MPFR_RNDN); + } + /* the error on y is bounded by err*ulp(y) */ + mpfr_sqr (t, x, MPFR_RNDU); /* rel. err <= 2^(1-p) */ + mpfr_div_2ui (err, err, 3, MPFR_RNDU); /* err/8 */ + mpfr_add (err, err, t, MPFR_RNDU); /* err/8 + xx */ + mpfr_mul_2ui (err, err, 3, MPFR_RNDU); /* err + 8*xx */ + mpfr_exp (t, t, MPFR_RNDU); /* err <= 1/2*ulp(t) + err(x*x)*t + <= 1/2*ulp(t)+2*|x*x|*ulp(t) + <= (2*|x*x|+1/2)*ulp(t) */ + mpfr_mul (t, t, x, MPFR_RNDN); /* err <= 1/2*ulp(t) + (4*|x*x|+1)*ulp(t) + <= (4*|x*x|+3/2)*ulp(t) */ + mpfr_const_pi (xx, MPFR_RNDZ); /* err <= ulp(Pi) */ + mpfr_sqrt (xx, xx, MPFR_RNDN); /* err <= 1/2*ulp(xx) + ulp(Pi)/2/sqrt(Pi) + <= 3/2*ulp(xx) */ + mpfr_mul (t, t, xx, MPFR_RNDN); /* err <= (8 |xx| + 13/2) * ulp(t) */ + mpfr_div (y, y, t, MPFR_RNDN); /* the relative error on input y is bounded + by (1+u)^err with u = 2^(1-p), that on + t is bounded by (1+u)^(8 |xx| + 13/2), + thus that on output y is bounded by + 8 |xx| + 7 + err. */ + + if (MPFR_IS_ZERO(y)) + { + /* If y is zero, most probably we have underflow. We check it directly + using the fact that erfc(x) <= exp(-x^2)/sqrt(Pi)/x for x >= 0. + We compute an upper approximation of exp(-x^2)/sqrt(Pi)/x. + */ + mpfr_sqr (t, x, MPFR_RNDD); /* t <= x^2 */ + mpfr_neg (t, t, MPFR_RNDU); /* -x^2 <= t */ + mpfr_exp (t, t, MPFR_RNDU); /* exp(-x^2) <= t */ + mpfr_const_pi (xx, MPFR_RNDD); /* xx <= sqrt(Pi), cached */ + mpfr_mul (xx, xx, x, MPFR_RNDD); /* xx <= sqrt(Pi)*x */ + mpfr_div (y, t, xx, MPFR_RNDN); /* if y is zero, this means that the upper + approximation of exp(-x^2)/sqrt(Pi)/x + is nearer from 0 than from 2^(-emin-1), + thus we have underflow. */ + exp_err = 0; + } + else + { + mpfr_add_ui (err, err, 7, MPFR_RNDU); + exp_err = MPFR_GET_EXP (err); + } + + mpfr_clear (t); + mpfr_clear (xx); + mpfr_clear (err); + return exp_err; +} + +int +mpfr_erfc (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd) +{ + int inex; + mpfr_t tmp; + mpfr_exp_t te, err; + mpfr_prec_t prec; + mpfr_exp_t emin = mpfr_get_emin (); + 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; + } + /* erfc(+inf) = 0+, erfc(-inf) = 2 erfc (0) = 1 */ + else if (MPFR_IS_INF (x)) + return mpfr_set_ui (y, MPFR_IS_POS (x) ? 0 : 2, rnd); + else + return mpfr_set_ui (y, 1, rnd); + } + + if (MPFR_IS_POS (x)) + { + /* by default, emin = 1-2^30, thus the smallest representable + number is 1/2*2^emin = 2^(-2^30): + for x >= 27282, erfc(x) < 2^(-2^30-1), and + for x >= 1787897414, erfc(x) < 2^(-2^62-1). + */ + if ((emin >= -1073741823 && mpfr_cmp_ui (x, 27282) >= 0) || + mpfr_cmp_ui (x, 1787897414) >= 0) + { + /* May be incorrect if MPFR_EMAX_MAX >= 2^62. */ + MPFR_STAT_STATIC_ASSERT ((MPFR_EMAX_MAX >> 31) >> 31 == 0); + return mpfr_underflow (y, (rnd == MPFR_RNDN) ? MPFR_RNDZ : rnd, 1); + } + } + + /* Init stuff */ + MPFR_SAVE_EXPO_MARK (expo); + + if (MPFR_IS_NEG (x)) + { + mpfr_exp_t e = MPFR_EXP(x); + /* For x < 0 going to -infinity, erfc(x) tends to 2 by below. + More precisely, we have 2 + 1/sqrt(Pi)/x/exp(x^2) < erfc(x) < 2. + Thus log2 |2 - erfc(x)| <= -log2|x| - x^2 / log(2). + If |2 - erfc(x)| < 2^(-PREC(y)) then the result is either 2 or + nextbelow(2). + For x <= -27282, -log2|x| - x^2 / log(2) <= -2^30. + */ + if ((MPFR_PREC(y) <= 7 && e >= 2) || /* x <= -2 */ + (MPFR_PREC(y) <= 25 && e >= 3) || /* x <= -4 */ + (MPFR_PREC(y) <= 120 && mpfr_cmp_si (x, -9) <= 0) || + mpfr_cmp_si (x, -27282) <= 0) + { + near_two: + mpfr_set_ui (y, 2, MPFR_RNDN); + MPFR_SET_INEXFLAG (); + if (rnd == MPFR_RNDZ || rnd == MPFR_RNDD) + { + mpfr_nextbelow (y); + inex = -1; + } + else + inex = 1; + goto end; + } + else if (e >= 3) /* more accurate test */ + { + mpfr_t t, u; + int near_2; + mpfr_init2 (t, 32); + mpfr_init2 (u, 32); + /* the following is 1/log(2) rounded to zero on 32 bits */ + mpfr_set_str_binary (t, "1.0111000101010100011101100101001"); + mpfr_sqr (u, x, MPFR_RNDZ); + mpfr_mul (t, t, u, MPFR_RNDZ); /* t <= x^2/log(2) */ + mpfr_neg (u, x, MPFR_RNDZ); /* 0 <= u <= |x| */ + mpfr_log2 (u, u, MPFR_RNDZ); /* u <= log2(|x|) */ + mpfr_add (t, t, u, MPFR_RNDZ); /* t <= log2|x| + x^2 / log(2) */ + /* Taking into account that mpfr_exp_t >= mpfr_prec_t */ + mpfr_set_exp_t (u, MPFR_PREC (y), MPFR_RNDU); + near_2 = mpfr_cmp (t, u) >= 0; /* 1 if PREC(y) <= u <= t <= ... */ + mpfr_clear (t); + mpfr_clear (u); + if (near_2) + goto near_two; + } + } + + /* erfc(x) ~ 1, with error < 2^(EXP(x)+1) */ + MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, __gmpfr_one, - MPFR_GET_EXP (x) - 1, + 0, MPFR_IS_NEG (x), + rnd, inex = _inexact; goto end); + + prec = MPFR_PREC (y) + MPFR_INT_CEIL_LOG2 (MPFR_PREC (y)) + 3; + if (MPFR_GET_EXP (x) > 0) + prec += 2 * MPFR_GET_EXP(x); + + mpfr_init2 (tmp, prec); + + MPFR_ZIV_INIT (loop, prec); /* Initialize the ZivLoop controller */ + for (;;) /* Infinite loop */ + { + /* use asymptotic formula only whenever x^2 >= p*log(2), + otherwise it will not converge */ + if (MPFR_IS_POS (x) && + 2 * MPFR_GET_EXP (x) - 2 >= MPFR_INT_CEIL_LOG2 (prec)) + /* we have x^2 >= p in that case */ + { + err = mpfr_erfc_asympt (tmp, x); + if (err == 0) /* underflow case */ + { + mpfr_clear (tmp); + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_underflow (y, (rnd == MPFR_RNDN) ? MPFR_RNDZ : rnd, 1); + } + } + else + { + mpfr_erf (tmp, x, MPFR_RNDN); + MPFR_ASSERTD (!MPFR_IS_SINGULAR (tmp)); /* FIXME: 0 only for x=0 ? */ + te = MPFR_GET_EXP (tmp); + mpfr_ui_sub (tmp, 1, tmp, MPFR_RNDN); + /* See error analysis in algorithms.tex for details */ + if (MPFR_IS_ZERO (tmp)) + { + prec *= 2; + err = prec; /* ensures MPFR_CAN_ROUND fails */ + } + else + err = MAX (te - MPFR_GET_EXP (tmp), 0) + 1; + } + 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_ZIV_FREE (loop); /* Free the ZivLoop Controller */ + + inex = mpfr_set (y, tmp, rnd); /* Set y to the computed value */ + mpfr_clear (tmp); + + end: + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (y, inex, rnd); +} |