diff options
Diffstat (limited to 'sysdeps/ieee754/dbl-64/e_exp.c')
| -rw-r--r-- | sysdeps/ieee754/dbl-64/e_exp.c | 400 |
1 files changed, 240 insertions, 160 deletions
diff --git a/sysdeps/ieee754/dbl-64/e_exp.c b/sysdeps/ieee754/dbl-64/e_exp.c index 6486cc3cfe..30d3941887 100644 --- a/sysdeps/ieee754/dbl-64/e_exp.c +++ b/sysdeps/ieee754/dbl-64/e_exp.c @@ -1,163 +1,243 @@ -/* Double-precision floating point e^x. - Copyright (C) 1997, 1998, 2000 Free Software Foundation, Inc. - This file is part of the GNU C Library. - Contributed by Geoffrey Keating <geoffk@ozemail.com.au> - - The GNU C Library is free software; you can redistribute it and/or - modify it under the terms of the GNU Library General Public License as - published by the Free Software Foundation; either version 2 of the - License, or (at your option) any later version. - - The GNU C 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 - Library General Public License for more details. - - You should have received a copy of the GNU Library General Public - License along with the GNU C Library; see the file COPYING.LIB. If not, - write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, - Boston, MA 02111-1307, USA. */ - -/* How this works: - The basic design here is from - Shmuel Gal and Boris Bachelis, "An Accurate Elementary Mathematical - Library for the IEEE Floating Point Standard", ACM Trans. Math. Soft., - 17 (1), March 1991, pp. 26-45. - - The input value, x, is written as - - x = n * ln(2)_0 + t/512 + delta[t] + x + n * ln(2)_1 - - where: - - n is an integer, 1024 >= n >= -1075; - - ln(2)_0 is the first 43 bits of ln(2), and ln(2)_1 is the remainder, so - that |ln(2)_1| < 2^-32; - - t is an integer, 177 >= t >= -177 - - delta is based on a table entry, delta[t] < 2^-28 - - x is whatever is left, |x| < 2^-10 - - Then e^x is approximated as - - e^x = 2^n_1 ( 2^n_0 e^(t/512 + delta[t]) - + ( 2^n_0 e^(t/512 + delta[t]) - * ( p(x + n * ln(2)_1) - - n*ln(2)_1 - - n*ln(2)_1 * p(x + n * ln(2)_1) ) ) ) - - where - - p(x) is a polynomial approximating e(x)-1; - - e^(t/512 + delta[t]) is obtained from a table; - - n_1 + n_0 = n, so that |n_0| < DBL_MIN_EXP-1. - - If it happens that n_1 == 0 (this is the usual case), that multiplication - is omitted. - */ -#ifndef _GNU_SOURCE -#define _GNU_SOURCE -#endif -#include <stdlib.h> -#include <float.h> -#include <ieee754.h> -#include <math.h> -#include <fenv.h> -#include <inttypes.h> -#include <math_private.h> - -extern const float __exp_deltatable[178]; -extern const double __exp_atable[355] /* __attribute__((mode(DF))) */; - -static const volatile double TWO1023 = 8.988465674311579539e+307; -static const volatile double TWOM1000 = 9.3326361850321887899e-302; - -double -__ieee754_exp (double x) -{ - static const double himark = 709.7827128933840868; - static const double lomark = -745.1332191019412221; - /* Check for usual case. */ - if (isless (x, himark) && isgreater (x, lomark)) - { - static const double THREEp42 = 13194139533312.0; - static const double THREEp51 = 6755399441055744.0; - /* 1/ln(2). */ - static const double M_1_LN2 = 1.442695040888963387; - /* ln(2), part 1 */ - static const double M_LN2_0 = .6931471805598903302; - /* ln(2), part 2 */ - static const double M_LN2_1 = 5.497923018708371155e-14; - - int tval, unsafe, n_i; - double x22, n, t, dely, result; - union ieee754_double ex2_u, scale_u; - fenv_t oldenv; - - feholdexcept (&oldenv); -#ifdef FE_TONEAREST - fesetround (FE_TONEAREST); -#endif - - /* Calculate n. */ - n = x * M_1_LN2 + THREEp51; - n -= THREEp51; - x = x - n*M_LN2_0; - - /* Calculate t/512. */ - t = x + THREEp42; - t -= THREEp42; - x -= t; - - /* Compute tval = t. */ - tval = (int) (t * 512.0); - - if (t >= 0) - x -= __exp_deltatable[tval]; - else - x += __exp_deltatable[-tval]; - - /* Now, the variable x contains x + n*ln(2)_1. */ - dely = n*M_LN2_1; - - /* Compute ex2 = 2^n_0 e^(t/512+delta[t]). */ - ex2_u.d = __exp_atable[tval+177]; - n_i = (int)n; - /* 'unsafe' is 1 iff n_1 != 0. */ - unsafe = abs(n_i) >= -DBL_MIN_EXP - 1; - ex2_u.ieee.exponent += n_i >> unsafe; - - /* Compute scale = 2^n_1. */ - scale_u.d = 1.0; - scale_u.ieee.exponent += n_i - (n_i >> unsafe); - - /* Approximate e^x2 - 1, using a fourth-degree polynomial, - with maximum error in [-2^-10-2^-28,2^-10+2^-28] - less than 4.9e-19. */ - x22 = (((0.04166666898464281565 - * x + 0.1666666766008501610) - * x + 0.499999999999990008) - * x + 0.9999999999999976685) * x; - /* Allow for impact of dely. */ - x22 -= dely + dely*x22; - - /* Return result. */ - fesetenv (&oldenv); - - result = x22 * ex2_u.d + ex2_u.d; - if (!unsafe) - return result; - else - return result * scale_u.d; +/* + * IBM Accurate Mathematical Library + * Copyright (c) International Business Machines Corp., 2001 + * + * This program 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 of the License, or + * (at your option) any later version. + * + * This program 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 General Public License for more details. + * + * You should have received a copy of the GNU Lesser General Public License + * along with this program; if not, write to the Free Software + * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. + */ +/***************************************************************************/ +/* MODULE_NAME:uexp.c */ +/* */ +/* FUNCTION:uexp */ +/* exp1 */ +/* */ +/* FILES NEEDED:dla.h endian.h mpa.h mydefs.h uexp.h */ +/* mpa.c mpexp.x slowexp.c */ +/* */ +/* An ultimate exp routine. Given an IEEE double machine number x */ +/* it computes the correctly rounded (to nearest) value of e^x */ +/* Assumption: Machine arithmetic operations are performed in */ +/* round to nearest mode of IEEE 754 standard. */ +/* */ +/***************************************************************************/ + +#include "endian.h" +#include "uexp.h" +#include "mydefs.h" +#include "MathLib.h" +#include "uexp.tbl" +double __slowexp(double); + +/***************************************************************************/ +/* An ultimate exp routine. Given an IEEE double machine number x */ +/* it computes the correctly rounded (to nearest) value of e^x */ +/***************************************************************************/ +double __ieee754_exp(double x) { + double bexp, t, eps, del, base, y, al, bet, res, rem, cor; + mynumber junk1, junk2, binexp = {0,0}; + int4 k,i,j,m,n,ex; + + junk1.x = x; + m = junk1.i[HIGH_HALF]; + n = m&hugeint; + + if (n > smallint && n < bigint) { + + y = x*log2e.x + three51.x; + bexp = y - three51.x; /* multiply the result by 2**bexp */ + + junk1.x = y; + + eps = bexp*ln_two2.x; /* x = bexp*ln(2) + t - eps */ + t = x - bexp*ln_two1.x; + + y = t + three33.x; + base = y - three33.x; /* t rounded to a multiple of 2**-18 */ + junk2.x = y; + del = (t - base) - eps; /* x = bexp*ln(2) + base + del */ + eps = del + del*del*(p3.x*del + p2.x); + + binexp.i[HIGH_HALF] =(junk1.i[LOW_HALF]+1023)<<20; + + i = ((junk2.i[LOW_HALF]>>8)&0xfffffffe)+356; + j = (junk2.i[LOW_HALF]&511)<<1; + + al = coar.x[i]*fine.x[j]; + bet =(coar.x[i]*fine.x[j+1] + coar.x[i+1]*fine.x[j]) + coar.x[i+1]*fine.x[j+1]; + + rem=(bet + bet*eps)+al*eps; + res = al + rem; + cor = (al - res) + rem; + if (res == (res+cor*err_0)) return res*binexp.x; + else return __slowexp(x); /*if error is over bound */ + } + + if (n <= smallint) return 1.0; + + if (n >= badint) { + if (n > infint) return(zero/zero); /* x is NaN, return invalid */ + if (n < infint) return ( (x>0) ? (hhuge*hhuge) : (tiny*tiny) ); + /* x is finite, cause either overflow or underflow */ + if (junk1.i[LOW_HALF] != 0) return (zero/zero); /* x is NaN */ + return ((x>0)?inf.x:zero ); /* |x| = inf; return either inf or 0 */ + } + + y = x*log2e.x + three51.x; + bexp = y - three51.x; + junk1.x = y; + eps = bexp*ln_two2.x; + t = x - bexp*ln_two1.x; + y = t + three33.x; + base = y - three33.x; + junk2.x = y; + del = (t - base) - eps; + eps = del + del*del*(p3.x*del + p2.x); + i = ((junk2.i[LOW_HALF]>>8)&0xfffffffe)+356; + j = (junk2.i[LOW_HALF]&511)<<1; + al = coar.x[i]*fine.x[j]; + bet =(coar.x[i]*fine.x[j+1] + coar.x[i+1]*fine.x[j]) + coar.x[i+1]*fine.x[j+1]; + rem=(bet + bet*eps)+al*eps; + res = al + rem; + cor = (al - res) + rem; + if (m>>31) { + ex=junk1.i[LOW_HALF]; + if (res < 1.0) {res+=res; cor+=cor; ex-=1;} + if (ex >=-1022) { + binexp.i[HIGH_HALF] = (1023+ex)<<20; + if (res == (res+cor*err_0)) return res*binexp.x; + else return __slowexp(x); /*if error is over bound */ } - /* Exceptional cases: */ - else if (isless (x, himark)) - { - if (__isinf (x)) - /* e^-inf == 0, with no error. */ - return 0; - else - /* Underflow */ - return TWOM1000 * TWOM1000; + ex = -(1022+ex); + binexp.i[HIGH_HALF] = (1023-ex)<<20; + res*=binexp.x; + cor*=binexp.x; + eps=1.0000000001+err_0*binexp.x; + t=1.0+res; + y = ((1.0-t)+res)+cor; + res=t+y; + cor = (t-res)+y; + if (res == (res + eps*cor)) + { binexp.i[HIGH_HALF] = 0x00100000; + return (res-1.0)*binexp.x; } - else - /* Return x, if x is a NaN or Inf; or overflow, otherwise. */ - return TWO1023*x; + else return __slowexp(x); /* if error is over bound */ + } + else { + binexp.i[HIGH_HALF] =(junk1.i[LOW_HALF]+767)<<20; + if (res == (res+cor*err_0)) return res*binexp.x*t256.x; + else return __slowexp(x); + } +} + +/************************************************************************/ +/* Compute e^(x+xx)(Double-Length number) .The routine also receive */ +/* bound of error of previous calculation .If after computing exp */ +/* error bigger than allows routine return non positive number */ +/*else return e^(x + xx) (always positive ) */ +/************************************************************************/ + +double __exp1(double x, double xx, double error) { + double bexp, t, eps, del, base, y, al, bet, res, rem, cor; + mynumber junk1, junk2, binexp = {0,0}; + int4 k,i,j,m,n,ex; + + junk1.x = x; + m = junk1.i[HIGH_HALF]; + n = m&hugeint; /* no sign */ + + if (n > smallint && n < bigint) { + y = x*log2e.x + three51.x; + bexp = y - three51.x; /* multiply the result by 2**bexp */ + + junk1.x = y; + + eps = bexp*ln_two2.x; /* x = bexp*ln(2) + t - eps */ + t = x - bexp*ln_two1.x; + + y = t + three33.x; + base = y - three33.x; /* t rounded to a multiple of 2**-18 */ + junk2.x = y; + del = (t - base) + (xx-eps); /* x = bexp*ln(2) + base + del */ + eps = del + del*del*(p3.x*del + p2.x); + + binexp.i[HIGH_HALF] =(junk1.i[LOW_HALF]+1023)<<20; + + i = ((junk2.i[LOW_HALF]>>8)&0xfffffffe)+356; + j = (junk2.i[LOW_HALF]&511)<<1; + + al = coar.x[i]*fine.x[j]; + bet =(coar.x[i]*fine.x[j+1] + coar.x[i+1]*fine.x[j]) + coar.x[i+1]*fine.x[j+1]; + + rem=(bet + bet*eps)+al*eps; + res = al + rem; + cor = (al - res) + rem; + if (res == (res+cor*(1.0+error+err_1))) return res*binexp.x; + else return -10.0; + } + + if (n <= smallint) return 1.0; /* if x->0 e^x=1 */ + + if (n >= badint) { + if (n > infint) return(zero/zero); /* x is NaN, return invalid */ + if (n < infint) return ( (x>0) ? (hhuge*hhuge) : (tiny*tiny) ); + /* x is finite, cause either overflow or underflow */ + if (junk1.i[LOW_HALF] != 0) return (zero/zero); /* x is NaN */ + return ((x>0)?inf.x:zero ); /* |x| = inf; return either inf or 0 */ + } + + y = x*log2e.x + three51.x; + bexp = y - three51.x; + junk1.x = y; + eps = bexp*ln_two2.x; + t = x - bexp*ln_two1.x; + y = t + three33.x; + base = y - three33.x; + junk2.x = y; + del = (t - base) + (xx-eps); + eps = del + del*del*(p3.x*del + p2.x); + i = ((junk2.i[LOW_HALF]>>8)&0xfffffffe)+356; + j = (junk2.i[LOW_HALF]&511)<<1; + al = coar.x[i]*fine.x[j]; + bet =(coar.x[i]*fine.x[j+1] + coar.x[i+1]*fine.x[j]) + coar.x[i+1]*fine.x[j+1]; + rem=(bet + bet*eps)+al*eps; + res = al + rem; + cor = (al - res) + rem; + if (m>>31) { + ex=junk1.i[LOW_HALF]; + if (res < 1.0) {res+=res; cor+=cor; ex-=1;} + if (ex >=-1022) { + binexp.i[HIGH_HALF] = (1023+ex)<<20; + if (res == (res+cor*(1.0+error+err_1))) return res*binexp.x; + else return -10.0; + } + ex = -(1022+ex); + binexp.i[HIGH_HALF] = (1023-ex)<<20; + res*=binexp.x; + cor*=binexp.x; + eps=1.00000000001+(error+err_1)*binexp.x; + t=1.0+res; + y = ((1.0-t)+res)+cor; + res=t+y; + cor = (t-res)+y; + if (res == (res + eps*cor)) + {binexp.i[HIGH_HALF] = 0x00100000; return (res-1.0)*binexp.x;} + else return -10.0; + } + else { + binexp.i[HIGH_HALF] =(junk1.i[LOW_HALF]+767)<<20; + if (res == (res+cor*(1.0+error+err_1))) + return res*binexp.x*t256.x; + else return -10.0; + } } |