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glibc/sysdeps/ieee754/dbl-64/e_pow.c

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/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2013 Free Software Foundation, Inc.
*
* 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 2.1 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 Lesser 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, see <http://www.gnu.org/licenses/>.
*/
/***************************************************************************/
/* MODULE_NAME: upow.c */
/* */
/* FUNCTIONS: upow */
/* power1 */
/* my_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_private.h>
#include <fenv.h>
#ifndef SECTION
# define SECTION
#endif
static const double huge = 1.0e300, tiny = 1.0e-300;
double __exp1(double x, double xx, double error);
static double log1(double x, double *delta, double *error);
static double my_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);
/***************************************************************************/
/* An ultimate power routine. Given two IEEE double machine numbers y,x */
/* it computes the correctly rounded (to nearest) value of X^y. */
/***************************************************************************/
double
SECTION
__ieee754_pow(double x, double y) {
double z,a,aa,error, t,a1,a2,y1,y2;
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;
/* Is x a NaN? */
if (((qx == 0x7ff00000) && (u.i[LOW_HALF] != 0)) || (qx > 0x7ff00000))
return x;
if (y == 1.0) return x;
if (y == 2.0) return x*x;
if (y == -1.0) return 1.0/x;
if (y == 0) return 1.0;
}
/* 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 */
double retval;
SET_RESTORE_ROUND (FE_TONEAREST);
/* Avoid internal underflow for tiny y. The exact value of y does
not matter if |y| <= 2**-64. */
if (ABS (y) < 0x1p-64)
y = y < 0 ? -0x1p-64 : 0x1p-64;
z = log1(x,&aa,&error); /* x^y =e^(y log (X)) */
t = y*CN;
y1 = t - (t-y);
y2 = y - y1;
t = z*CN;
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 */
retval = (t>0)?t:power1(x,y);
return retval;
}
if (x == 0) {
if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0)
|| (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000) /* NaN */
return y;
if (ABS(y) > 1.0e20) return (y>0)?0:1.0/0.0;
k = checkint(y);
if (k == -1)
return y < 0 ? 1.0/x : x;
else
return y < 0 ? 1.0/0.0 : 0.0; /* return 0 */
}
qx = u.i[HIGH_HALF]&0x7fffffff; /* no sign */
qy = v.i[HIGH_HALF]&0x7fffffff; /* no sign */
if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0)) /* NaN */
return x;
if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0)) /* NaN */
return x == 1.0 ? 1.0 : y;
/* if x<0 */
if (u.i[HIGH_HALF] < 0) {
k = checkint(y);
if (k==0) {
if (qy == 0x7ff00000) {
if (x == -1.0) return 1.0;
else if (x > -1.0) return v.i[HIGH_HALF] < 0 ? INF.x : 0.0;
else return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x;
}
else if (qx == 0x7ff00000)
return y < 0 ? 0.0 : INF.x;
return (x - x) / (x - x); /* y not integer and x<0 */
}
else if (qx == 0x7ff00000)
{
if (k < 0)
return y < 0 ? nZERO.x : nINF.x;
else
return y < 0 ? 0.0 : INF.x;
}
return (k==1)?__ieee754_pow(-x,y):-__ieee754_pow(-x,y); /* if y even or odd */
}
/* x>0 */
if (qx == 0x7ff00000) /* x= 2^-0x3ff */
return y > 0 ? x : 0;
if (qy > 0x45f00000 && qy < 0x7ff00000) {
if (x == 1.0) return 1.0;
if (y>0) return (x>1.0)?huge*huge:tiny*tiny;
if (y<0) return (x<1.0)?huge*huge:tiny*tiny;
}
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;
return 0; /* unreachable, to make the compiler happy */
}
#ifndef __ieee754_pow
strong_alias (__ieee754_pow, __pow_finite)
#endif
/**************************************************************************/
/* Computing x^y using more accurate but more slow log routine */
/**************************************************************************/
static double
SECTION
power1(double x, double y) {
double z,a,aa,error, t,a1,a2,y1,y2;
z = my_log2(x,&aa,&error);
t = y*CN;
y1 = t - (t-y);
y2 = y - y1;
t = z*CN;
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);
}
/****************************************************************************/
/* 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
SECTION
log1(double x, double *delta, double *error) {
int i,j,m;
double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0;
mynumber u,v;
#ifdef BIG_ENDI
mynumber
/**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */
#else
#ifdef LITTLE_ENDI
mynumber
/**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */
#endif
#endif
u.x = x;
m = u.i[HIGH_HALF];
*error = 0;
*delta = 0;
if (m < 0x00100000) /* 1<x<2^-1007 */
{ x = x*t52.x; add = -52.0; u.x = x; m = u.i[HIGH_HALF];}
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;
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
SECTION
my_log2(double x, double *delta, double *error) {
int i,j,m;
double uu,vv,eps,nx,e,e1,e2,t,t1,t2,res,add=0;
double ou1,ou2,lu1,lu2,ov,lv1,lv2,a,a1,a2;
double y,yy,z,zz,j1,j2,j7,j8;
#ifndef DLA_FMS
double j3,j4,j5,j6;
#endif
mynumber u,v;
#ifdef BIG_ENDI
mynumber
/**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */
#else
#ifdef LITTLE_ENDI
mynumber
/**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */
#endif
#endif
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]; }
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;
}
}
/**********************************************************************/
/* 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
SECTION
checkint(double x) {
union {int4 i[2]; double x;} u;
int k,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;
}
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