/* mpfr_erfc -- The Complementary Error Function of a floating-point number Copyright 2005, 2006, 2007 Free Software Foundation, Inc. Contributed by the Arenaire and Cacao projects, INRIA. This file is part of the MPFR Library. The 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 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 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" /* 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). */ static mp_exp_t mpfr_erfc_asympt (mpfr_ptr y, mpfr_srcptr x) { mpfr_t t, xx, err; unsigned long k; mp_prec_t prec = MPFR_PREC(y); mp_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_mul (xx, x, x, GMP_RNDD); /* err <= 1 */ mpfr_ui_div (xx, 1, xx, GMP_RNDU); /* upper bound for 1/(2x^2), err <= 2 */ mpfr_div_2ui (xx, xx, 1, GMP_RNDU); /* exact */ mpfr_set_ui (t, 1, GMP_RNDN); /* current term, exact */ mpfr_set (y, t, GMP_RNDN); /* current sum */ mpfr_set_ui (err, 0, GMP_RNDN); for (k = 1; ; k++) { mpfr_mul_ui (t, t, 2 * k - 1, GMP_RNDU); /* err <= 4k-3 */ mpfr_mul (t, t, xx, GMP_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), GMP_RNDU); mpfr_add_ui (err, err, 14 * k, GMP_RNDU); /* 2^(1-p) * t <= 2 ulp(t) */ mpfr_div_2si (err, err, MPFR_GET_EXP (y) - MPFR_GET_EXP (t), GMP_RNDU); if (MPFR_GET_EXP (t) + (mp_exp_t) prec <= MPFR_GET_EXP (y)) { /* the truncation error is bounded by |t| < ulp(y) */ mpfr_add_ui (err, err, 1, GMP_RNDU); break; } if (k & 1) mpfr_sub (y, y, t, GMP_RNDN); else mpfr_add (y, y, t, GMP_RNDN); } /* the error on y is bounded by err*ulp(y) */ mpfr_mul (t, x, x, GMP_RNDU); /* rel. err <= 2^(1-p) */ mpfr_div_2ui (err, err, 3, GMP_RNDU); /* err/8 */ mpfr_add (err, err, t, GMP_RNDU); /* err/8 + xx */ mpfr_mul_2ui (err, err, 3, GMP_RNDU); /* err + 8*xx */ mpfr_exp (t, t, GMP_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, GMP_RNDN); /* err <= 1/2*ulp(t) + (4*|x*x|+1)*ulp(t) <= (4*|x*x|+3/2)*ulp(t) */ mpfr_const_pi (xx, GMP_RNDZ); /* err <= ulp(Pi) */ mpfr_sqrt (xx, xx, GMP_RNDN); /* err <= 1/2*ulp(xx) + ulp(Pi)/2/sqrt(Pi) <= 3/2*ulp(xx) */ mpfr_mul (t, t, xx, GMP_RNDN); /* err <= (8 |xx| + 13/2) * ulp(t) */ mpfr_div (y, y, t, GMP_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. */ mpfr_add_ui (err, err, 7, GMP_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, mp_rnd_t rnd) { int inex; mpfr_t tmp; mp_exp_t te, err; mp_prec_t prec; MPFR_SAVE_EXPO_DECL (expo); MPFR_ZIV_DECL (loop); MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd), ("y[%#R]=%R inexact=%d", y, 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_SIGN (x) > 0) { /* for x >= 27282, erfc(x) < 2^(-2^30-1) */ if (mpfr_cmp_ui (x, 27282) >= 0) return mpfr_underflow (y, (rnd == GMP_RNDN) ? GMP_RNDZ : rnd, 1); } if (MPFR_SIGN (x) < 0) { /* for x < 0 going to -infinity, erfc(x) tends to 2 by below */ if ((MPFR_PREC(y) <= 7 && mpfr_cmp_si (x, -2) <= 0) || (MPFR_PREC(y) <= 25 && mpfr_cmp_si (x, -4) <= 0) || (MPFR_PREC(y) <= 120 && mpfr_cmp_si (x, -9) <= 0)) { mpfr_set_ui (y, 2, GMP_RNDN); mpfr_set_inexflag (); if (rnd == GMP_RNDZ || rnd == GMP_RNDD) { mpfr_nextbelow (y); return -1; } else return 1; } } /* Init stuff */ MPFR_SAVE_EXPO_MARK (expo); /* 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_SIGN(x) < 0, 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 controler */ for (;;) /* Infinite loop */ { /* use asymptotic formula only whenever x^2 >= p*log(2), otherwise it will not converge */ if (MPFR_SIGN (x) > 0 && 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); else { mpfr_erf (tmp, x, GMP_RNDN); MPFR_ASSERTD (!MPFR_IS_SINGULAR (tmp)); /* FIXME: 0 only for x=0 ? */ te = MPFR_GET_EXP (tmp); mpfr_ui_sub (tmp, 1, tmp, GMP_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 Controler */ 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); }