From e4d8276142b9c07b23043ef44b0fe8fa7bcc3121 Mon Sep 17 00:00:00 2001 From: Ulrich Drepper Date: Mon, 12 Mar 2001 00:04:52 +0000 Subject: Update. 2001-03-11 Ulrich Drepper Last-bit accurate math library implementation by IBM Haifa. Contributed by Abraham Ziv , Moshe Olshansky , Ealan Henis , and Anna Reitman . * math/Makefile (dbl-only-routines): New variable. (libm-routines): Add $(dbl-only-routines). * sysdeps/ieee754/dbl-64/e_acos.c: Empty, definition is in e_asin.c. * sysdeps/ieee754/dbl-64/e_asin.c: Replaced with accurate asin implementation. * sysdeps/ieee754/dbl-64/e_atan2.c: Replaced with accurate atan2 implementation. * sysdeps/ieee754/dbl-64/e_exp.c: Replaced with accurate exp implementation. * sysdeps/ieee754/dbl-64/e_lgamma_r.c: Don't use __kernel_sin and __kernel_cos. * sysdeps/ieee754/dbl-64/e_log.c: Replaced with accurate log implementation. * sysdeps/ieee754/dbl-64/e_remainder.c: Replaced with accurate remainder implementation. * sysdeps/ieee754/dbl-64/e_pow.c: Replaced with accurate pow implementation. * sysdeps/ieee754/dbl-64/e_sqrt.c: Replaced with accurate sqrt implementation. * sysdeps/ieee754/dbl-64/k_cos.c: Empty, definition is in s_sin.c. * sysdeps/ieee754/dbl-64/k_sin.c: Empty, definition is in s_sin.c. * sysdeps/ieee754/dbl-64/s_atan.c: Replaced with accurate atan implementation. * sysdeps/ieee754/dbl-64/s_cos.c: Empty, definition is in s_sin.c. * sysdeps/ieee754/dbl-64/s_sin.c: Replaced with accurate sin/cos implementation. * sysdeps/ieee754/dbl-64/s_sincos.c: Rewritten to not use __kernel_sin and __kernel_cos. * sysdeps/ieee754/dbl-64/s_tan.c: Replaced with accurate tan implementation. * sysdeps/ieee754/dbl-64/Dist: Add new non-code files. * sysdeps/ieee754/dbl-64/MathLib.h: New file. * sysdeps/ieee754/dbl-64/asincos.tbl: New file. * sysdeps/ieee754/dbl-64/atnat.h: New file. * sysdeps/ieee754/dbl-64/atnat2.h: New file. * sysdeps/ieee754/dbl-64/branred.c: New file. * sysdeps/ieee754/dbl-64/branred.h: New file. * sysdeps/ieee754/dbl-64/dla.h: New file. * sysdeps/ieee754/dbl-64/doasin.c: New file. * sysdeps/ieee754/dbl-64/doasin.h: New file. * sysdeps/ieee754/dbl-64/dosincos.c: New file. * sysdeps/ieee754/dbl-64/dosincos.h: New file. * sysdeps/ieee754/dbl-64/endian.h: New file. * sysdeps/ieee754/dbl-64/halfulp.c: New file. * sysdeps/ieee754/dbl-64/mpa.c: New file. * sysdeps/ieee754/dbl-64/mpa.h: New file. * sysdeps/ieee754/dbl-64/mpa2.h: New file. * sysdeps/ieee754/dbl-64/mpatan.c: New file. * sysdeps/ieee754/dbl-64/mpatan.h: New file. * sysdeps/ieee754/dbl-64/mpatan2.c: New file. * sysdeps/ieee754/dbl-64/mpexp.c: New file. * sysdeps/ieee754/dbl-64/mpexp.h: New file. * sysdeps/ieee754/dbl-64/mplog.c: New file. * sysdeps/ieee754/dbl-64/mplog.h: New file. * sysdeps/ieee754/dbl-64/mpsqrt.c: New file. * sysdeps/ieee754/dbl-64/mpsqrt.h: New file. * sysdeps/ieee754/dbl-64/mptan.c: New file. * sysdeps/ieee754/dbl-64/mydefs.h: New file. * sysdeps/ieee754/dbl-64/powtwo.tbl: New file. * sysdeps/ieee754/dbl-64/root.tbl: New file. * sysdeps/ieee754/dbl-64/sincos.tbl: New file. * sysdeps/ieee754/dbl-64/sincos32.c: New file. * sysdeps/ieee754/dbl-64/sincos32.h: New file. * sysdeps/ieee754/dbl-64/slowexp.c: New file. * sysdeps/ieee754/dbl-64/slowpow.c: New file. * sysdeps/ieee754/dbl-64/uasncs.h: New file. * sysdeps/ieee754/dbl-64/uatan.tbl: New file. * sysdeps/ieee754/dbl-64/uexp.h: New file. * sysdeps/ieee754/dbl-64/uexp.tbl: New file. * sysdeps/ieee754/dbl-64/ulog.h: New file. * sysdeps/ieee754/dbl-64/ulog.tbl: New file. * sysdeps/ieee754/dbl-64/upow.h: New file. * sysdeps/ieee754/dbl-64/upow.tbl: New file. * sysdeps/ieee754/dbl-64/urem.h: New file. * sysdeps/ieee754/dbl-64/uroot.h: New file. * sysdeps/ieee754/dbl-64/usncs.h: New file. * sysdeps/ieee754/dbl-64/utan.h: New file. * sysdeps/ieee754/dbl-64/utan.tbl: New file. * sysdeps/i386/fpu/branred.c: New file. * sysdeps/i386/fpu/doasin.c: New file. * sysdeps/i386/fpu/dosincos.c: New file. * sysdeps/i386/fpu/halfulp.c: New file. * sysdeps/i386/fpu/mpa.c: New file. * sysdeps/i386/fpu/mpatan.c: New file. * sysdeps/i386/fpu/mpatan2.c: New file. * sysdeps/i386/fpu/mpexp.c: New file. * sysdeps/i386/fpu/mplog.c: New file. * sysdeps/i386/fpu/mpsqrt.c: New file. * sysdeps/i386/fpu/mptan.c: New file. * sysdeps/i386/fpu/sincos32.c: New file. * sysdeps/i386/fpu/slowexp.c: New file. * sysdeps/i386/fpu/slowpow.c: New file. * sysdeps/ia64/fpu/branred.c: New file. * sysdeps/ia64/fpu/doasin.c: New file. * sysdeps/ia64/fpu/dosincos.c: New file. * sysdeps/ia64/fpu/halfulp.c: New file. * sysdeps/ia64/fpu/mpa.c: New file. * sysdeps/ia64/fpu/mpatan.c: New file. * sysdeps/ia64/fpu/mpatan2.c: New file. * sysdeps/ia64/fpu/mpexp.c: New file. * sysdeps/ia64/fpu/mplog.c: New file. * sysdeps/ia64/fpu/mpsqrt.c: New file. * sysdeps/ia64/fpu/mptan.c: New file. * sysdeps/ia64/fpu/sincos32.c: New file. * sysdeps/ia64/fpu/slowexp.c: New file. * sysdeps/ia64/fpu/slowpow.c: New file. * sysdeps/m68k/fpu/branred.c: New file. * sysdeps/m68k/fpu/doasin.c: New file. * sysdeps/m68k/fpu/dosincos.c: New file. * sysdeps/m68k/fpu/halfulp.c: New file. * sysdeps/m68k/fpu/mpa.c: New file. * sysdeps/m68k/fpu/mpatan.c: New file. * sysdeps/m68k/fpu/mpatan2.c: New file. * sysdeps/m68k/fpu/mpexp.c: New file. * sysdeps/m68k/fpu/mplog.c: New file. * sysdeps/m68k/fpu/mpsqrt.c: New file. * sysdeps/m68k/fpu/mptan.c: New file. * sysdeps/m68k/fpu/sincos32.c: New file. * sysdeps/m68k/fpu/slowexp.c: New file. * sysdeps/m68k/fpu/slowpow.c: New file. * iconvdata/gconv-modules: Add a number of alias, mostly for IBM codepages. --- sysdeps/ieee754/dbl-64/e_pow.c | 642 ++++++++++++++++++++--------------------- 1 file changed, 307 insertions(+), 335 deletions(-) (limited to 'sysdeps/ieee754/dbl-64/e_pow.c') diff --git a/sysdeps/ieee754/dbl-64/e_pow.c b/sysdeps/ieee754/dbl-64/e_pow.c index 73e8f471f6..5c5bbb58d7 100644 --- a/sysdeps/ieee754/dbl-64/e_pow.c +++ b/sysdeps/ieee754/dbl-64/e_pow.c @@ -1,356 +1,328 @@ -/* @(#)e_pow.c 5.1 93/09/24 */ /* - * ==================================================== - * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * IBM Accurate Mathematical Library + * Copyright (c) International Business Machines Corp., 2001 * - * Developed at SunPro, a Sun Microsystems, Inc. business. - * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice - * is preserved. - * ==================================================== - */ -/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, - for performance improvement on pipelined processors. -*/ - -#if defined(LIBM_SCCS) && !defined(lint) -static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $"; -#endif - -/* __ieee754_pow(x,y) return x**y - * - * n - * Method: Let x = 2 * (1+f) - * 1. Compute and return log2(x) in two pieces: - * log2(x) = w1 + w2, - * where w1 has 53-24 = 29 bit trailing zeros. - * 2. Perform y*log2(x) = n+y' by simulating muti-precision - * arithmetic, where |y'|<=0.5. - * 3. Return x**y = 2**n*exp(y'*log2) + * 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. * - * Special cases: - * 1. (anything) ** 0 is 1 - * 2. (anything) ** 1 is itself - * 3. (anything) ** NAN is NAN - * 4. NAN ** (anything except 0) is NAN - * 5. +-(|x| > 1) ** +INF is +INF - * 6. +-(|x| > 1) ** -INF is +0 - * 7. +-(|x| < 1) ** +INF is +0 - * 8. +-(|x| < 1) ** -INF is +INF - * 9. +-1 ** +-INF is NAN - * 10. +0 ** (+anything except 0, NAN) is +0 - * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 - * 12. +0 ** (-anything except 0, NAN) is +INF - * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF - * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) - * 15. +INF ** (+anything except 0,NAN) is +INF - * 16. +INF ** (-anything except 0,NAN) is +0 - * 17. -INF ** (anything) = -0 ** (-anything) - * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) - * 19. (-anything except 0 and inf) ** (non-integer) is NAN + * 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. * - * Accuracy: - * pow(x,y) returns x**y nearly rounded. In particular - * pow(integer,integer) - * always returns the correct integer provided it is - * representable. - * - * Constants : - * The hexadecimal values are the intended ones for the following - * constants. The decimal values may be used, provided that the - * compiler will convert from decimal to binary accurately enough - * to produce the hexadecimal values shown. + * 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: upow.c */ +/* */ +/* FUNCTIONS: upow */ +/* power1 */ +/* log2 */ +/* log1 */ +/* checkint */ +/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h */ +/* halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c */ +/* uexp.c upow.c */ +/* root.tbl uexp.tbl upow.tbl */ +/* An ultimate power routine. Given two IEEE double machine numbers y,x */ +/* it computes the correctly rounded (to nearest) value of x^y. */ +/* Assumption: Machine arithmetic operations are performed in */ +/* round to nearest mode of IEEE 754 standard. */ +/* */ +/***************************************************************************/ +#include "endian.h" +#include "upow.h" +#include "dla.h" +#include "mydefs.h" +#include "MathLib.h" +#include "upow.tbl" -#include "math.h" -#include "math_private.h" -#define zero C[0] -#define one C[1] -#define two C[2] -#define two53 C[3] -#define huge C[4] -#define tiny C[5] -#define L1 C[6] -#define L2 C[7] -#define L3 C[8] -#define L4 C[9] -#define L5 C[10] -#define L6 C[11] -#define P1 C[12] -#define P2 C[13] -#define P3 C[14] -#define P4 C[15] -#define P5 C[16] -#define lg2 C[17] -#define lg2_h C[18] -#define lg2_l C[19] -#define ovt C[20] -#define cp C[21] -#define cp_h C[22] -#define cp_l C[23] -#define ivln2 C[24] -#define ivln2_h C[25] -#define ivln2_l C[26] -#ifdef __STDC__ -static const double -#else -static double -#endif -bp[] = {1.0, 1.5,}, -dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ -dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ -C[] = { -0.0, -1.0, -2.0, -9007199254740992.0 , -1.0e300, -1.0e-300, -5.99999999999994648725e-01 , -4.28571428578550184252e-01 , -3.33333329818377432918e-01 , -2.72728123808534006489e-01 , -2.30660745775561754067e-01 , -2.06975017800338417784e-01 , -1.66666666666666019037e-01 , --2.77777777770155933842e-03 , -6.61375632143793436117e-05 , --1.65339022054652515390e-06 , -4.13813679705723846039e-08 , -6.93147180559945286227e-01 , -6.93147182464599609375e-01 , --1.90465429995776804525e-09 , -8.0085662595372944372e-0017 , -9.61796693925975554329e-01 , -9.61796700954437255859e-01 , --7.02846165095275826516e-09 , -1.44269504088896338700e+00 , -1.44269502162933349609e+00 , -1.92596299112661746887e-08 }; +double __exp1(double x, double xx, double error); +static double log1(double x, double *delta, double *error); +static double log2(double x, double *delta, double *error); +double slowpow(double x, double y,double z); +static double power1(double x, double y); +static int checkint(double x); -#ifdef __STDC__ - double __ieee754_pow(double x, double y) -#else - double __ieee754_pow(x,y) - double x, y; -#endif -{ - double z,ax,z_h,z_l,p_h,p_l; - double y1,t1,t2,r,s,t,u,v,w, t12,t14,r_1,r_2,r_3; - int32_t i,j,k,yisint,n; - int32_t hx,hy,ix,iy; - u_int32_t lx,ly; +/***************************************************************************/ +/* An ultimate power routine. Given two IEEE double machine numbers y,x */ +/* it computes the correctly rounded (to nearest) value of X^y. */ +/***************************************************************************/ +double __ieee754_upow(double x, double y) { + double z,a,aa,error, t,a1,a2,y1,y2,gor=1.0; + mynumber u,v; + int k; + int4 qx,qy; + v.x=y; + u.x=x; + if (v.i[LOW_HALF] == 0) { /* of y */ + qx = u.i[HIGH_HALF]&0x7fffffff; + /* Checking if x is not too small to compute */ + if (((qx==0x7ff00000)&&(u.i[LOW_HALF]!=0))||(qx>0x7ff00000)) return NaNQ.x; + if (y == 1.0) return x; + if (y == 2.0) return x*x; + if (y == -1.0) return (x!=0)?1.0/x:NaNQ.x; + if (y == 0) return ((x>0)&&(qx<0x7ff00000))?1.0:NaNQ.x; + } + /* else */ + if(((u.i[HIGH_HALF]>0 && u.i[HIGH_HALF]<0x7ff00000)|| /* x>0 and not x->0 */ + (u.i[HIGH_HALF]==0 && u.i[LOW_HALF]!=0)) && + /* 2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */ + (v.i[HIGH_HALF]&0x7fffffff) < 0x4ff00000) { /* if y<-1 or y>1 */ + z = log1(x,&aa,&error); /* x^y =e^(y log (X)) */ + t = y*134217729.0; + y1 = t - (t-y); + y2 = y - y1; + t = z*134217729.0; + a1 = t - (t-z); + a2 = (z - a1)+aa; + a = y1*a1; + aa = y2*a1 + y*a2; + a1 = a+aa; + a2 = (a-a1)+aa; + error = error*ABS(y); + t = __exp1(a1,a2,1.9e16*error); /* return -10 or 0 if wasn't computed exactly */ + return (t>0)?t:power1(x,y); + } - EXTRACT_WORDS(hx,lx,x); - EXTRACT_WORDS(hy,ly,y); - ix = hx&0x7fffffff; iy = hy&0x7fffffff; + if (x == 0) { + if (ABS(y) > 1.0e20) return (y>0)?0:NaNQ.x; + k = checkint(y); + if (k == 0 || y < 0) return NaNQ.x; + else return (k==1)?0:x; /* return 0 */ + } + /* if x<0 */ + if (u.i[HIGH_HALF] < 0) { + k = checkint(y); + if (k==0) return NaNQ.x; /* y not integer and x<0 */ + return (k==1)?upow(-x,y):-upow(-x,y); /* if y even or odd */ + } + /* x>0 */ + qx = u.i[HIGH_HALF]&0x7fffffff; /* no sign */ + qy = v.i[HIGH_HALF]&0x7fffffff; /* no sign */ - /* y==zero: x**0 = 1 */ - if((iy|ly)==0) return C[1]; + if (qx > 0x7ff00000 || (qx == 0x7ff00000 && u.i[LOW_HALF] != 0)) return NaNQ.x; + /* if 0 0x7ff00000 || (qy == 0x7ff00000 && v.i[LOW_HALF] != 0)) return NaNQ.x; + /* if y<2^-0x7fe */ - /* x==+-1 */ - if(x == 1.0) return C[1]; - if(x == -1.0 && isinf(y)) return C[1]; + if (qx == 0x7ff00000) /* x= 2^-0x3ff */ + {if (y == 0) return NaNQ.x; + return (y>0)?x:0; } - /* +-NaN return x+y */ - if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || - iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) - return x+y; + if (qy > 0x45f00000 && qy < 0x7ff00000) { + if (x == 1.0) return 1.0; + if (y>0) return (x>1.0)?INF.x:0; + if (y<0) return (x<1.0)?INF.x:0; + } - /* determine if y is an odd int when x < 0 - * yisint = 0 ... y is not an integer - * yisint = 1 ... y is an odd int - * yisint = 2 ... y is an even int - */ - yisint = 0; - if(hx<0) { - if(iy>=0x43400000) yisint = 2; /* even integer y */ - else if(iy>=0x3ff00000) { - k = (iy>>20)-0x3ff; /* exponent */ - if(k>20) { - j = ly>>(52-k); - if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1); - } else if(ly==0) { - j = iy>>(20-k); - if((int32_t)(j<<(20-k))==iy) yisint = 2-(j&1); - } - } - } + if (x == 1.0) return NaNQ.x; + if (y>0) return (x>1.0)?INF.x:0; + if (y<0) return (x<1.0)?INF.x:0; + return 0; /* unreachable, to make the compiler happy */ +} - /* special value of y */ - if(ly==0) { - if (iy==0x7ff00000) { /* y is +-inf */ - if(((ix-0x3ff00000)|lx)==0) - return y - y; /* inf**+-1 is NaN */ - else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ - return (hy>=0)? y: C[0]; - else /* (|x|<1)**-,+inf = inf,0 */ - return (hy<0)?-y: C[0]; - } - if(iy==0x3ff00000) { /* y is +-1 */ - if(hy<0) return C[1]/x; else return x; - } - if(hy==0x40000000) return x*x; /* y is 2 */ - if(hy==0x3fe00000) { /* y is 0.5 */ - if(hx>=0) /* x >= +0 */ - return __ieee754_sqrt(x); - } - } +/**************************************************************************/ +/* Computing x^y using more accurate but more slow log routine */ +/**************************************************************************/ +static double power1(double x, double y) { + double z,a,aa,error, t,a1,a2,y1,y2; + z = log2(x,&aa,&error); + t = y*134217729.0; + y1 = t - (t-y); + y2 = y - y1; + t = z*134217729.0; + a1 = t - (t-z); + a2 = z - a1; + a = y*z; + aa = ((y1*a1-a)+y1*a2+y2*a1)+y2*a2+aa*y; + a1 = a+aa; + a2 = (a-a1)+aa; + error = error*ABS(y); + t = __exp1(a1,a2,1.9e16*error); + return (t >= 0)?t:slowpow(x,y,z); +} - ax = fabs(x); - /* special value of x */ - if(lx==0) { - if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ - z = ax; /*x is +-0,+-inf,+-1*/ - if(hy<0) z = C[1]/z; /* z = (1/|x|) */ - if(hx<0) { - if(((ix-0x3ff00000)|yisint)==0) { - z = (z-z)/(z-z); /* (-1)**non-int is NaN */ - } else if(yisint==1) - z = -z; /* (x<0)**odd = -(|x|**odd) */ - } - return z; - } - } +/****************************************************************************/ +/* Computing log(x) (x is left argument). The result is the returned double */ +/* + the parameter delta. */ +/* The result is bounded by error (rightmost argument) */ +/****************************************************************************/ +static double log1(double x, double *delta, double *error) { + int i,j,m,n; + double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,cor,add=0; + mynumber u,v; - /* (x<0)**(non-int) is NaN */ - if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); + u.x = x; + m = u.i[HIGH_HALF]; + *error = 0; + *delta = 0; + if (m < 0x00100000) /* 10x41e00000) { /* if |y| > 2**31 */ - if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ - if(ix<=0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5]; - if(ix>=0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5]; - } - /* over/underflow if x is not close to one */ - if(ix<0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5]; - if(ix>0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5]; - /* now |1-x| is tiny <= 2**-20, suffice to compute - log(x) by x-x^2/2+x^3/3-x^4/4 */ - t = x-1; /* t has 20 trailing zeros */ - w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); - u = C[25]*t; /* ivln2_h has 21 sig. bits */ - v = t*C[26]-w*C[24]; - t1 = u+v; - SET_LOW_WORD(t1,0); - t2 = v-(t1-u); - } else { - double s2,s_h,s_l,t_h,t_l,s22,s24,s26,r1,r2,r3; - n = 0; - /* take care subnormal number */ - if(ix<0x00100000) - {ax *= C[3]; n -= 53; GET_HIGH_WORD(ix,ax); } - n += ((ix)>>20)-0x3ff; - j = ix&0x000fffff; - /* determine interval */ - ix = j|0x3ff00000; /* normalize ix */ - if(j<=0x3988E) k=0; /* |x|>20); } + else + {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; } - /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ - u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ - v = C[1]/(ax+bp[k]); - s = u*v; - s_h = s; - SET_LOW_WORD(s_h,0); - /* t_h=ax+bp[k] High */ - t_h = C[0]; - SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); - t_l = ax - (t_h-bp[k]); - s_l = v*((u-s_h*t_h)-s_h*t_l); - /* compute log(ax) */ - s2 = s*s; -#ifdef DO_NOT_USE_THIS - r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); -#else - r1 = C[10]+s2*C[11]; s22=s2*s2; - r2 = C[8]+s2*C[9]; s24=s22*s22; - r3 = C[6]+s2*C[7]; s26=s24*s22; - r = r3*s22 + r2*s24 + r1*s26; -#endif - r += s_l*(s_h+s); - s2 = s_h*s_h; - t_h = 3.0+s2+r; - SET_LOW_WORD(t_h,0); - t_l = r-((t_h-3.0)-s2); - /* u+v = s*(1+...) */ - u = s_h*t_h; - v = s_l*t_h+t_l*s; - /* 2/(3log2)*(s+...) */ - p_h = u+v; - SET_LOW_WORD(p_h,0); - p_l = v-(p_h-u); - z_h = C[22]*p_h; /* cp_h+cp_l = 2/(3*log2) */ - z_l = C[23]*p_h+p_l*C[21]+dp_l[k]; - /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ - t = (double)n; - t1 = (((z_h+z_l)+dp_h[k])+t); - SET_LOW_WORD(t1,0); - t2 = z_l-(((t1-t)-dp_h[k])-z_h); - } + v.x = u.x + bigu.x; + uu = v.x - bigu.x; + i = (v.i[LOW_HALF]&0x000003ff)<<2; + if (two52.i[LOW_HALF] == 1023) /* nx = 0 */ + { + if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10 */ + { + t = x - 1.0; + t1 = (t+5.0e6)-5.0e6; + t2 = t-t1; + e1 = t - 0.5*t1*t1; + e2 = t*t*t*(r3+t*(r4+t*(r5+t*(r6+t*(r7+t*r8)))))-0.5*t2*(t+t1); + res = e1+e2; + *error = 1.0e-21*ABS(t); + *delta = (e1-res)+e2; + return res; + } /* |x-1| < 1.5*2**-10 */ + else + { + v.x = u.x*(ui.x[i]+ui.x[i+1])+bigv.x; + vv = v.x-bigv.x; + j = v.i[LOW_HALF]&0x0007ffff; + j = j+j+j; + eps = u.x - uu*vv; + e1 = eps*ui.x[i]; + e2 = eps*(ui.x[i+1]+vj.x[j]*(ui.x[i]+ui.x[i+1])); + e = e1+e2; + e2 = ((e1-e)+e2); + t=ui.x[i+2]+vj.x[j+1]; + t1 = t+e; + t2 = (((t-t1)+e)+(ui.x[i+3]+vj.x[j+2]))+e2+e*e*(p2+e*(p3+e*p4)); + res=t1+t2; + *error = 1.0e-24; + *delta = (t1-res)+t2; + return res; + } + } /* nx = 0 */ + else /* nx != 0 */ + { + eps = u.x - uu; + nx = (two52.x - two52e.x)+add; + e1 = eps*ui.x[i]; + e2 = eps*ui.x[i+1]; + e=e1+e2; + e2 = (e1-e)+e2; + t=nx*ln2a.x+ui.x[i+2]; + t1=t+e; + t2=(((t-t1)+e)+nx*ln2b.x+ui.x[i+3]+e2)+e*e*(q2+e*(q3+e*(q4+e*(q5+e*q6)))); + res = t1+t2; + *error = 1.0e-21; + *delta = (t1-res)+t2; + return res; + } /* nx != 0 */ +} + +/****************************************************************************/ +/* More slow but more accurate routine of log */ +/* Computing log(x)(x is left argument).The result is return double + delta.*/ +/* The result is bounded by error (right argument) */ +/****************************************************************************/ +static double log2(double x, double *delta, double *error) { + int i,j,m,n; + double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,cor,add=0; + double ou1,ou2,lu1,lu2,ov,lv1,lv2,a,a1,a2; + double y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8; + mynumber u,v; + + u.x = x; + m = u.i[HIGH_HALF]; + *error = 0; + *delta = 0; + add=0; + if (m<0x00100000) { /* x < 2^-1022 */ + x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF]; } - s = C[1]; /* s (sign of result -ve**odd) = -1 else = 1 */ - if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) - s = -C[1];/* (-ve)**(odd int) */ + if ((m&0x000fffff) < 0x0006a09e) + {u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); } + else + {u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; } + + v.x = u.x + bigu.x; + uu = v.x - bigu.x; + i = (v.i[LOW_HALF]&0x000003ff)<<2; + /*------------------------------------- |x-1| < 2**-11------------------------------- */ + if ((two52.i[LOW_HALF] == 1023) && (i == 1200)) + { + t = x - 1.0; + EMULV(t,s3,y,yy,j1,j2,j3,j4,j5); + ADD2(-0.5,0,y,yy,z,zz,j1,j2); + MUL2(t,0,z,zz,y,yy,j1,j2,j3,j4,j5,j6,j7,j8); + MUL2(t,0,y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8); + + e1 = t+z; + e2 = (((t-e1)+z)+zz)+t*t*t*(ss3+t*(s4+t*(s5+t*(s6+t*(s7+t*s8))))); + res = e1+e2; + *error = 1.0e-25*ABS(t); + *delta = (e1-res)+e2; + return res; + } + /*----------------------------- |x-1| > 2**-11 -------------------------- */ + else + { /*Computing log(x) according to log table */ + nx = (two52.x - two52e.x)+add; + ou1 = ui.x[i]; + ou2 = ui.x[i+1]; + lu1 = ui.x[i+2]; + lu2 = ui.x[i+3]; + v.x = u.x*(ou1+ou2)+bigv.x; + vv = v.x-bigv.x; + j = v.i[LOW_HALF]&0x0007ffff; + j = j+j+j; + eps = u.x - uu*vv; + ov = vj.x[j]; + lv1 = vj.x[j+1]; + lv2 = vj.x[j+2]; + a = (ou1+ou2)*(1.0+ov); + a1 = (a+1.0e10)-1.0e10; + a2 = a*(1.0-a1*uu*vv); + e1 = eps*a1; + e2 = eps*a2; + e = e1+e2; + e2 = (e1-e)+e2; + t=nx*ln2a.x+lu1+lv1; + t1 = t+e; + t2 = (((t-t1)+e)+(lu2+lv2+nx*ln2b.x+e2))+e*e*(p2+e*(p3+e*p4)); + res=t1+t2; + *error = 1.0e-27; + *delta = (t1-res)+t2; + return res; + } +} - /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ - y1 = y; - SET_LOW_WORD(y1,0); - p_l = (y-y1)*t1+y*t2; - p_h = y1*t1; - z = p_l+p_h; - EXTRACT_WORDS(j,i,z); - if (j>=0x40900000) { /* z >= 1024 */ - if(((j-0x40900000)|i)!=0) /* if z > 1024 */ - return s*C[4]*C[4]; /* overflow */ - else { - if(p_l+C[20]>z-p_h) return s*C[4]*C[4]; /* overflow */ - } - } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ - if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ - return s*C[5]*C[5]; /* underflow */ - else { - if(p_l<=z-p_h) return s*C[5]*C[5]; /* underflow */ - } - } - /* - * compute 2**(p_h+p_l) - */ - i = j&0x7fffffff; - k = (i>>20)-0x3ff; - n = 0; - if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ - n = j+(0x00100000>>(k+1)); - k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ - t = C[0]; - SET_HIGH_WORD(t,n&~(0x000fffff>>k)); - n = ((n&0x000fffff)|0x00100000)>>(20-k); - if(j<0) n = -n; - p_h -= t; - } - t = p_l+p_h; - SET_LOW_WORD(t,0); - u = t*C[18]; - v = (p_l-(t-p_h))*C[17]+t*C[19]; - z = u+v; - w = v-(z-u); - t = z*z; -#ifdef DO_NOT_USE_THIS - t1 = z - t*(C[12]+t*(C[13]+t*(C[14]+t*(C[15]+t*C[16])))); -#else - r_1 = C[15]+t*C[16]; t12 = t*t; - r_2 = C[13]+t*C[14]; t14 = t12*t12; - r_3 = t*C[12]; - t1 = z - r_3 - t12*r_2 - t14*r_1; -#endif - r = (z*t1)/(t1-C[2])-(w+z*w); - z = C[1]-(r-z); - GET_HIGH_WORD(j,z); - j += (n<<20); - if((j>>20)<=0) z = __scalbn(z,n); /* subnormal output */ - else SET_HIGH_WORD(z,j); - return s*z; +/**********************************************************************/ +/* Routine receives a double x and checks if it is an integer. If not */ +/* it returns 0, else it returns 1 if even or -1 if odd. */ +/**********************************************************************/ +static int checkint(double x) { + union {int4 i[2]; double x;} u; + int k,l,m,n; + u.x = x; + m = u.i[HIGH_HALF]&0x7fffffff; /* no sign */ + if (m >= 0x7ff00000) return 0; /* x is +/-inf or NaN */ + if (m >= 0x43400000) return 1; /* |x| >= 2**53 */ + if (m < 0x40000000) return 0; /* |x| < 2, can not be 0 or 1 */ + n = u.i[LOW_HALF]; + k = (m>>20)-1023; /* 1 <= k <= 52 */ + if (k == 52) return (n&1)? -1:1; /* odd or even*/ + if (k>20) { + if (n<<(k-20)) return 0; /* if not integer */ + return (n<<(k-21))?-1:1; + } + if (n) return 0; /*if not integer*/ + if (k == 20) return (m&1)? -1:1; + if (m<<(k+12)) return 0; + return (m<<(k+11))?-1:1; } -- cgit v1.2.3