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-rw-r--r--sysdeps/ieee754/dbl-64/e_pow.c197
1 files changed, 38 insertions, 159 deletions
diff --git a/sysdeps/ieee754/dbl-64/e_pow.c b/sysdeps/ieee754/dbl-64/e_pow.c
index 663fa392c2..96d5b23ccc 100644
--- a/sysdeps/ieee754/dbl-64/e_pow.c
+++ b/sysdeps/ieee754/dbl-64/e_pow.c
@@ -1,7 +1,7 @@
/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
- * Copyright (C) 2001-2016 Free Software Foundation, Inc.
+ * Copyright (C) 2001-2018 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
@@ -20,13 +20,9 @@
/* 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. */
@@ -42,6 +38,7 @@
#include "MathLib.h"
#include "upow.tbl"
#include <math_private.h>
+#include <math-underflow.h>
#include <fenv.h>
#ifndef SECTION
@@ -50,11 +47,8 @@
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);
+double __exp1 (double x, double xx);
+static double log1 (double x, double *delta);
static int checkint (double x);
/* An ultimate power routine. Given two IEEE double machine numbers y, x it
@@ -63,7 +57,7 @@ double
SECTION
__ieee754_pow (double x, double y)
{
- double z, a, aa, error, t, a1, a2, y1, y2;
+ double z, a, aa, t, a1, a2, y1, y2;
mynumber u, v;
int k;
int4 qx, qy;
@@ -73,8 +67,9 @@ __ieee754_pow (double x, double y)
{ /* 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 ((((qx == 0x7ff00000) && (u.i[LOW_HALF] != 0)) || (qx > 0x7ff00000))
+ && (y != 0 || issignaling (x)))
+ return x + x;
if (y == 1.0)
return x;
if (y == 2.0)
@@ -99,7 +94,7 @@ __ieee754_pow (double x, double y)
not matter if |y| <= 2**-64. */
if (fabs (y) < 0x1p-64)
y = y < 0 ? -0x1p-64 : 0x1p-64;
- z = log1 (x, &aa, &error); /* x^y =e^(y log (X)) */
+ z = log1 (x, &aa); /* x^y =e^(y log (X)) */
t = y * CN;
y1 = t - (t - y);
y2 = y - y1;
@@ -110,9 +105,16 @@ __ieee754_pow (double x, double y)
aa = y2 * a1 + y * a2;
a1 = a + aa;
a2 = (a - a1) + aa;
- error = error * fabs (y);
- t = __exp1 (a1, a2, 1.9e16 * error); /* return -10 or 0 if wasn't computed exactly */
- retval = (t > 0) ? t : power1 (x, y);
+
+ /* Maximum relative error RElog of log1 is 1.0e-21 (69.7 bits).
+ Maximum relative error REexp of __exp1 is 8.8e-22 (69.9 bits).
+ We actually compute exp ((1 + RElog) * log (x) * y) * (1 + REexp).
+ Since RElog/REexp are tiny and log (x) * y is at most log (DBL_MAX),
+ this is equivalent to pow (x, y) * (1 + 710 * RElog + REexp).
+ So the relative error is 710 * 1.0e-21 + 8.8e-22 = 7.1e-19
+ (60.2 bits). The worst-case ULP error is 0.5064. */
+
+ retval = __exp1 (a1, a2);
}
if (isinf (retval))
@@ -128,7 +130,7 @@ __ieee754_pow (double x, double y)
{
if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0)
|| (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000) /* NaN */
- return y;
+ return y + y;
if (fabs (y) > 1.0e20)
return (y > 0) ? 0 : 1.0 / 0.0;
k = checkint (y);
@@ -142,9 +144,9 @@ __ieee754_pow (double x, double y)
qy = v.i[HIGH_HALF] & 0x7fffffff; /* no sign */
if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0)) /* NaN */
- return x;
+ return x + y;
if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0)) /* NaN */
- return x == 1.0 ? 1.0 : y;
+ return x == 1.0 && !issignaling (y) ? 1.0 : y + y;
/* if x<0 */
if (u.i[HIGH_HALF] < 0)
@@ -217,33 +219,11 @@ __ieee754_pow (double x, double y)
strong_alias (__ieee754_pow, __pow_finite)
#endif
-/* Compute 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 * fabs (y);
- t = __exp1 (a1, a2, 1.9e16 * error);
- return (t >= 0) ? t : __slowpow (x, y, z);
-}
-
/* Compute log(x) (x is left argument). The result is the returned double + the
- parameter DELTA. The result is bounded by ERROR. */
+ parameter DELTA. */
static double
SECTION
-log1 (double x, double *delta, double *error)
+log1 (double x, double *delta)
{
unsigned int i, j;
int m;
@@ -259,9 +239,7 @@ log1 (double x, double *delta, double *error)
u.x = x;
m = u.i[HIGH_HALF];
- *error = 0;
- *delta = 0;
- if (m < 0x00100000) /* 1<x<2^-1007 */
+ if (m < 0x00100000) /* Handle denormal x. */
{
x = x * t52.x;
add = -52.0;
@@ -283,7 +261,7 @@ log1 (double x, double *delta, double *error)
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 (two52.i[LOW_HALF] == 1023) /* Exponent of x is 0. */
{
if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10 */
{
@@ -295,8 +273,8 @@ log1 (double x, double *delta, double *error)
* (r7 + t * r8)))))
- 0.5 * t2 * (t + t1));
res = e1 + e2;
- *error = 1.0e-21 * fabs (t);
*delta = (e1 - res) + e2;
+ /* Max relative error is 1.464844e-24, so accurate to 79.1 bits. */
return res;
} /* |x-1| < 1.5*2**-10 */
else
@@ -315,12 +293,12 @@ log1 (double x, double *delta, double *error)
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;
+ /* Max relative error is 1.0e-24, so accurate to 79.7 bits. */
return res;
}
- } /* nx = 0 */
- else /* nx != 0 */
+ }
+ else /* Exponent of x != 0. */
{
eps = u.x - uu;
nx = (two52.x - two52e.x) + add;
@@ -333,113 +311,13 @@ log1 (double x, double *delta, double *error)
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 */
-}
-
-/* Slower but more accurate routine of log. The returned result is double +
- DELTA. The result is bounded by ERROR. */
-static double
-SECTION
-my_log2 (double x, double *delta, double *error)
-{
- unsigned int i, j;
- int 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 * fabs (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;
+ /* Max relative error is 1.0e-21, so accurate to 69.7 bits. */
return res;
}
}
+
/* This function 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
@@ -451,7 +329,8 @@ checkint (double x)
int4 i[2];
double x;
} u;
- int k, m, n;
+ int k;
+ unsigned int m, n;
u.x = x;
m = u.i[HIGH_HALF] & 0x7fffffff; /* no sign */
if (m >= 0x7ff00000)
@@ -466,15 +345,15 @@ checkint (double x)
return (n & 1) ? -1 : 1; /* odd or even */
if (k > 20)
{
- if (n << (k - 20))
+ if (n << (k - 20) != 0)
return 0; /* if not integer */
- return (n << (k - 21)) ? -1 : 1;
+ return (n << (k - 21) != 0) ? -1 : 1;
}
if (n)
return 0; /*if not integer */
if (k == 20)
return (m & 1) ? -1 : 1;
- if (m << (k + 12))
+ if (m << (k + 12) != 0)
return 0;
- return (m << (k + 11)) ? -1 : 1;
+ return (m << (k + 11) != 0) ? -1 : 1;
}
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