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416 lines
13 KiB
C
416 lines
13 KiB
C
/*
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* IBM Accurate Mathematical Library
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* written by International Business Machines Corp.
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* Copyright (C) 2001-2012 Free Software Foundation
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU Lesser General Public License as published by
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* the Free Software Foundation; either version 2.1 of the License, or
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* (at your option) any later version.
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*
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* This program is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU Lesser General Public License for more details.
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*
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* You should have received a copy of the GNU Lesser General Public License
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* along with this program; if not, see <http://www.gnu.org/licenses/>.
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*/
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/***************************************************************************/
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/* MODULE_NAME: upow.c */
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/* */
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/* FUNCTIONS: upow */
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/* power1 */
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/* my_log2 */
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/* log1 */
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/* checkint */
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/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h */
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/* halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c */
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/* uexp.c upow.c */
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/* root.tbl uexp.tbl upow.tbl */
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/* An ultimate power routine. Given two IEEE double machine numbers y,x */
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/* it computes the correctly rounded (to nearest) value of x^y. */
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/* Assumption: Machine arithmetic operations are performed in */
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/* round to nearest mode of IEEE 754 standard. */
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/* */
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/***************************************************************************/
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#include "endian.h"
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#include "upow.h"
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#include <dla.h>
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#include "mydefs.h"
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#include "MathLib.h"
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#include "upow.tbl"
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#include <math_private.h>
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#include <fenv.h>
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#ifndef SECTION
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# define SECTION
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#endif
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static const double huge = 1.0e300, tiny = 1.0e-300;
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double __exp1(double x, double xx, double error);
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static double log1(double x, double *delta, double *error);
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static double my_log2(double x, double *delta, double *error);
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double __slowpow(double x, double y,double z);
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static double power1(double x, double y);
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static int checkint(double x);
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/***************************************************************************/
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/* An ultimate power routine. Given two IEEE double machine numbers y,x */
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/* it computes the correctly rounded (to nearest) value of X^y. */
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/***************************************************************************/
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double
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SECTION
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__ieee754_pow(double x, double y) {
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double z,a,aa,error, t,a1,a2,y1,y2;
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#if 0
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double gor=1.0;
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#endif
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mynumber u,v;
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int k;
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int4 qx,qy;
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v.x=y;
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u.x=x;
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if (v.i[LOW_HALF] == 0) { /* of y */
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qx = u.i[HIGH_HALF]&0x7fffffff;
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/* Checking if x is not too small to compute */
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if (((qx==0x7ff00000)&&(u.i[LOW_HALF]!=0))||(qx>0x7ff00000)) return NaNQ.x;
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if (y == 1.0) return x;
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if (y == 2.0) return x*x;
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if (y == -1.0) return 1.0/x;
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if (y == 0) return 1.0;
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}
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/* else */
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if(((u.i[HIGH_HALF]>0 && u.i[HIGH_HALF]<0x7ff00000)|| /* x>0 and not x->0 */
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(u.i[HIGH_HALF]==0 && u.i[LOW_HALF]!=0)) &&
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/* 2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */
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(v.i[HIGH_HALF]&0x7fffffff) < 0x4ff00000) { /* if y<-1 or y>1 */
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double retval;
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SET_RESTORE_ROUND (FE_TONEAREST);
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z = log1(x,&aa,&error); /* x^y =e^(y log (X)) */
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t = y*134217729.0;
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y1 = t - (t-y);
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y2 = y - y1;
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t = z*134217729.0;
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a1 = t - (t-z);
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a2 = (z - a1)+aa;
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a = y1*a1;
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aa = y2*a1 + y*a2;
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a1 = a+aa;
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a2 = (a-a1)+aa;
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error = error*ABS(y);
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t = __exp1(a1,a2,1.9e16*error); /* return -10 or 0 if wasn't computed exactly */
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retval = (t>0)?t:power1(x,y);
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return retval;
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}
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if (x == 0) {
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if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0)
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|| (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000)
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return y;
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if (ABS(y) > 1.0e20) return (y>0)?0:1.0/0.0;
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k = checkint(y);
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if (k == -1)
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return y < 0 ? 1.0/x : x;
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else
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return y < 0 ? 1.0/0.0 : 0.0; /* return 0 */
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}
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qx = u.i[HIGH_HALF]&0x7fffffff; /* no sign */
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qy = v.i[HIGH_HALF]&0x7fffffff; /* no sign */
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if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0)) return NaNQ.x;
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if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0))
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return x == 1.0 ? 1.0 : NaNQ.x;
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/* if x<0 */
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if (u.i[HIGH_HALF] < 0) {
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k = checkint(y);
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if (k==0) {
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if (qy == 0x7ff00000) {
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if (x == -1.0) return 1.0;
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else if (x > -1.0) return v.i[HIGH_HALF] < 0 ? INF.x : 0.0;
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else return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x;
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}
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else if (qx == 0x7ff00000)
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return y < 0 ? 0.0 : INF.x;
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return NaNQ.x; /* y not integer and x<0 */
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}
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else if (qx == 0x7ff00000)
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{
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if (k < 0)
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return y < 0 ? nZERO.x : nINF.x;
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else
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return y < 0 ? 0.0 : INF.x;
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}
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return (k==1)?__ieee754_pow(-x,y):-__ieee754_pow(-x,y); /* if y even or odd */
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}
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/* x>0 */
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if (qx == 0x7ff00000) /* x= 2^-0x3ff */
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{if (y == 0) return NaNQ.x;
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return (y>0)?x:0; }
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if (qy > 0x45f00000 && qy < 0x7ff00000) {
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if (x == 1.0) return 1.0;
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if (y>0) return (x>1.0)?huge*huge:tiny*tiny;
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if (y<0) return (x<1.0)?huge*huge:tiny*tiny;
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}
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if (x == 1.0) return 1.0;
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if (y>0) return (x>1.0)?INF.x:0;
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if (y<0) return (x<1.0)?INF.x:0;
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return 0; /* unreachable, to make the compiler happy */
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}
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#ifndef __ieee754_pow
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strong_alias (__ieee754_pow, __pow_finite)
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#endif
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/**************************************************************************/
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/* Computing x^y using more accurate but more slow log routine */
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/**************************************************************************/
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static double
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SECTION
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power1(double x, double y) {
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double z,a,aa,error, t,a1,a2,y1,y2;
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z = my_log2(x,&aa,&error);
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t = y*134217729.0;
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y1 = t - (t-y);
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y2 = y - y1;
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t = z*134217729.0;
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a1 = t - (t-z);
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a2 = z - a1;
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a = y*z;
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aa = ((y1*a1-a)+y1*a2+y2*a1)+y2*a2+aa*y;
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a1 = a+aa;
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a2 = (a-a1)+aa;
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error = error*ABS(y);
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t = __exp1(a1,a2,1.9e16*error);
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return (t >= 0)?t:__slowpow(x,y,z);
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}
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/****************************************************************************/
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/* Computing log(x) (x is left argument). The result is the returned double */
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/* + the parameter delta. */
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/* The result is bounded by error (rightmost argument) */
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/****************************************************************************/
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static double
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SECTION
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log1(double x, double *delta, double *error) {
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int i,j,m;
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#if 0
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int n;
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#endif
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double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0;
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#if 0
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double cor;
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#endif
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mynumber u,v;
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#ifdef BIG_ENDI
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mynumber
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/**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */
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#else
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#ifdef LITTLE_ENDI
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mynumber
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/**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */
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#endif
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#endif
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u.x = x;
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m = u.i[HIGH_HALF];
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*error = 0;
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*delta = 0;
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if (m < 0x00100000) /* 1<x<2^-1007 */
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{ x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF];}
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if ((m&0x000fffff) < 0x0006a09e)
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{u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); }
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else
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{u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; }
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v.x = u.x + bigu.x;
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uu = v.x - bigu.x;
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i = (v.i[LOW_HALF]&0x000003ff)<<2;
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if (two52.i[LOW_HALF] == 1023) /* nx = 0 */
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{
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if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10 */
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{
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t = x - 1.0;
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t1 = (t+5.0e6)-5.0e6;
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t2 = t-t1;
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e1 = t - 0.5*t1*t1;
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e2 = t*t*t*(r3+t*(r4+t*(r5+t*(r6+t*(r7+t*r8)))))-0.5*t2*(t+t1);
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res = e1+e2;
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*error = 1.0e-21*ABS(t);
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*delta = (e1-res)+e2;
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return res;
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} /* |x-1| < 1.5*2**-10 */
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else
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{
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v.x = u.x*(ui.x[i]+ui.x[i+1])+bigv.x;
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vv = v.x-bigv.x;
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j = v.i[LOW_HALF]&0x0007ffff;
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j = j+j+j;
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eps = u.x - uu*vv;
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e1 = eps*ui.x[i];
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e2 = eps*(ui.x[i+1]+vj.x[j]*(ui.x[i]+ui.x[i+1]));
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e = e1+e2;
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e2 = ((e1-e)+e2);
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t=ui.x[i+2]+vj.x[j+1];
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t1 = t+e;
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t2 = (((t-t1)+e)+(ui.x[i+3]+vj.x[j+2]))+e2+e*e*(p2+e*(p3+e*p4));
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res=t1+t2;
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*error = 1.0e-24;
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*delta = (t1-res)+t2;
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return res;
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}
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} /* nx = 0 */
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else /* nx != 0 */
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{
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eps = u.x - uu;
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nx = (two52.x - two52e.x)+add;
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e1 = eps*ui.x[i];
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e2 = eps*ui.x[i+1];
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e=e1+e2;
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e2 = (e1-e)+e2;
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t=nx*ln2a.x+ui.x[i+2];
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t1=t+e;
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t2=(((t-t1)+e)+nx*ln2b.x+ui.x[i+3]+e2)+e*e*(q2+e*(q3+e*(q4+e*(q5+e*q6))));
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res = t1+t2;
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*error = 1.0e-21;
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*delta = (t1-res)+t2;
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return res;
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} /* nx != 0 */
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}
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/****************************************************************************/
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/* More slow but more accurate routine of log */
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/* Computing log(x)(x is left argument).The result is return double + delta.*/
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/* The result is bounded by error (right argument) */
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/****************************************************************************/
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static double
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SECTION
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my_log2(double x, double *delta, double *error) {
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int i,j,m;
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#if 0
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int n;
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#endif
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double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0;
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#if 0
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double cor;
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#endif
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double ou1,ou2,lu1,lu2,ov,lv1,lv2,a,a1,a2;
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double y,yy,z,zz,j1,j2,j7,j8;
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#ifndef DLA_FMS
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double j3,j4,j5,j6;
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#endif
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mynumber u,v;
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#ifdef BIG_ENDI
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mynumber
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/**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */
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#else
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#ifdef LITTLE_ENDI
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mynumber
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/**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */
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#endif
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#endif
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u.x = x;
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m = u.i[HIGH_HALF];
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*error = 0;
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*delta = 0;
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add=0;
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if (m<0x00100000) { /* x < 2^-1022 */
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x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF]; }
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if ((m&0x000fffff) < 0x0006a09e)
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{u.i[HIGH_HALF] = (m&0x000fffff)|0x3ff00000; two52.i[LOW_HALF]=(m>>20); }
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else
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{u.i[HIGH_HALF] = (m&0x000fffff)|0x3fe00000; two52.i[LOW_HALF]=(m>>20)+1; }
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v.x = u.x + bigu.x;
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uu = v.x - bigu.x;
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i = (v.i[LOW_HALF]&0x000003ff)<<2;
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/*------------------------------------- |x-1| < 2**-11------------------------------- */
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if ((two52.i[LOW_HALF] == 1023) && (i == 1200))
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{
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t = x - 1.0;
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EMULV(t,s3,y,yy,j1,j2,j3,j4,j5);
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ADD2(-0.5,0,y,yy,z,zz,j1,j2);
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MUL2(t,0,z,zz,y,yy,j1,j2,j3,j4,j5,j6,j7,j8);
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MUL2(t,0,y,yy,z,zz,j1,j2,j3,j4,j5,j6,j7,j8);
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e1 = t+z;
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e2 = (((t-e1)+z)+zz)+t*t*t*(ss3+t*(s4+t*(s5+t*(s6+t*(s7+t*s8)))));
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res = e1+e2;
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*error = 1.0e-25*ABS(t);
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*delta = (e1-res)+e2;
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return res;
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}
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/*----------------------------- |x-1| > 2**-11 -------------------------- */
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else
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{ /*Computing log(x) according to log table */
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nx = (two52.x - two52e.x)+add;
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ou1 = ui.x[i];
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ou2 = ui.x[i+1];
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lu1 = ui.x[i+2];
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lu2 = ui.x[i+3];
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v.x = u.x*(ou1+ou2)+bigv.x;
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vv = v.x-bigv.x;
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j = v.i[LOW_HALF]&0x0007ffff;
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j = j+j+j;
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eps = u.x - uu*vv;
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ov = vj.x[j];
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lv1 = vj.x[j+1];
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lv2 = vj.x[j+2];
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a = (ou1+ou2)*(1.0+ov);
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a1 = (a+1.0e10)-1.0e10;
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a2 = a*(1.0-a1*uu*vv);
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e1 = eps*a1;
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e2 = eps*a2;
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e = e1+e2;
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e2 = (e1-e)+e2;
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t=nx*ln2a.x+lu1+lv1;
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t1 = t+e;
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t2 = (((t-t1)+e)+(lu2+lv2+nx*ln2b.x+e2))+e*e*(p2+e*(p3+e*p4));
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res=t1+t2;
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*error = 1.0e-27;
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*delta = (t1-res)+t2;
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return res;
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}
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}
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/**********************************************************************/
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/* Routine receives a double x and checks if it is an integer. If not */
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/* it returns 0, else it returns 1 if even or -1 if odd. */
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/**********************************************************************/
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static int
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SECTION
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checkint(double x) {
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union {int4 i[2]; double x;} u;
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int k,m,n;
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#if 0
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int l;
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#endif
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u.x = x;
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m = u.i[HIGH_HALF]&0x7fffffff; /* no sign */
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if (m >= 0x7ff00000) return 0; /* x is +/-inf or NaN */
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if (m >= 0x43400000) return 1; /* |x| >= 2**53 */
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if (m < 0x40000000) return 0; /* |x| < 2, can not be 0 or 1 */
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n = u.i[LOW_HALF];
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k = (m>>20)-1023; /* 1 <= k <= 52 */
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if (k == 52) return (n&1)? -1:1; /* odd or even*/
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if (k>20) {
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if (n<<(k-20)) return 0; /* if not integer */
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return (n<<(k-21))?-1:1;
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}
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if (n) return 0; /*if not integer*/
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if (k == 20) return (m&1)? -1:1;
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if (m<<(k+12)) return 0;
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return (m<<(k+11))?-1:1;
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}
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