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6ace393821
Similar to various other bugs in this area, pow functions can fail to raise the underflow exception when the result is tiny and inexact but one or more low bits of the intermediate result that is scaled down (or, in the i386 case, converted from a wider evaluation format) are zero. This patch forces the exception in a similar way to previous fixes, thereby concluding the fixes for known bugs with missing underflow exceptions currently filed in Bugzilla. Tested for x86_64, x86, mips64 and powerpc. [BZ #18825] * sysdeps/i386/fpu/i386-math-asm.h (FLT_NARROW_EVAL_UFLOW_NONNAN): New macro. (DBL_NARROW_EVAL_UFLOW_NONNAN): Likewise. (LDBL_CHECK_FORCE_UFLOW_NONNAN): Likewise. * sysdeps/i386/fpu/e_pow.S: Use DEFINE_DBL_MIN. (__ieee754_pow): Use DBL_NARROW_EVAL_UFLOW_NONNAN instead of DBL_NARROW_EVAL, reloading the PIC register as needed. * sysdeps/i386/fpu/e_powf.S: Use DEFINE_FLT_MIN. (__ieee754_powf): Use FLT_NARROW_EVAL_UFLOW_NONNAN instead of FLT_NARROW_EVAL. Use separate return path for case when first argument is NaN. * sysdeps/i386/fpu/e_powl.S: Include <i386-math-asm.h>. Use DEFINE_LDBL_MIN. (__ieee754_powl): Use LDBL_CHECK_FORCE_UFLOW_NONNAN, reloading the PIC register. * sysdeps/ieee754/dbl-64/e_pow.c (__ieee754_pow): Use math_check_force_underflow_nonneg. * sysdeps/ieee754/flt-32/e_powf.c (__ieee754_powf): Force underflow for subnormal result. * sysdeps/ieee754/ldbl-128/e_powl.c (__ieee754_powl): Likewise. * sysdeps/ieee754/ldbl-128ibm/e_powl.c (__ieee754_powl): Use math_check_force_underflow_nonneg. * sysdeps/x86/fpu/powl_helper.c (__powl_helper): Use math_check_force_underflow. * sysdeps/x86_64/fpu/x86_64-math-asm.h (LDBL_CHECK_FORCE_UFLOW_NONNAN): New macro. * sysdeps/x86_64/fpu/e_powl.S: Include <x86_64-math-asm.h>. Use DEFINE_LDBL_MIN. (__ieee754_powl): Use LDBL_CHECK_FORCE_UFLOW_NONNAN. * math/auto-libm-test-in: Add more tests of pow. * math/auto-libm-test-out: Regenerated.
479 lines
13 KiB
C
479 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) 2001-2015 Free Software Foundation, Inc.
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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 <math.h>
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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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/* An ultimate power routine. Given two IEEE double machine numbers y, x it
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computes the correctly rounded (to nearest) value of X^y. */
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double
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SECTION
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__ieee754_pow (double x, double y)
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{
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double z, a, aa, error, t, a1, a2, y1, y2;
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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)
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{ /* of y */
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qx = u.i[HIGH_HALF] & 0x7fffffff;
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/* Is x a NaN? */
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if (((qx == 0x7ff00000) && (u.i[LOW_HALF] != 0)) || (qx > 0x7ff00000))
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return x;
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if (y == 1.0)
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return x;
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if (y == 2.0)
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return x * x;
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if (y == -1.0)
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return 1.0 / x;
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if (y == 0)
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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)
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{ /* if y<-1 or y>1 */
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double retval;
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{
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SET_RESTORE_ROUND (FE_TONEAREST);
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/* Avoid internal underflow for tiny y. The exact value of y does
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not matter if |y| <= 2**-64. */
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if (fabs (y) < 0x1p-64)
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y = y < 0 ? -0x1p-64 : 0x1p-64;
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z = log1 (x, &aa, &error); /* x^y =e^(y log (X)) */
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t = y * CN;
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y1 = t - (t - y);
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y2 = y - y1;
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t = z * CN;
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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 * fabs (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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}
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if (isinf (retval))
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retval = huge * huge;
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else if (retval == 0)
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retval = tiny * tiny;
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else
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math_check_force_underflow_nonneg (retval);
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return retval;
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}
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if (x == 0)
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{
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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) /* NaN */
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return y;
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if (fabs (y) > 1.0e20)
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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)) /* NaN */
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return x;
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if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0)) /* NaN */
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return x == 1.0 ? 1.0 : y;
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/* if x<0 */
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if (u.i[HIGH_HALF] < 0)
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{
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k = checkint (y);
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if (k == 0)
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{
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if (qy == 0x7ff00000)
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{
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if (x == -1.0)
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return 1.0;
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else if (x > -1.0)
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return v.i[HIGH_HALF] < 0 ? INF.x : 0.0;
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else
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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 (x - x) / (x - 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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/* if y even or odd */
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if (k == 1)
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return __ieee754_pow (-x, y);
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else
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{
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double retval;
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{
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SET_RESTORE_ROUND (FE_TONEAREST);
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retval = -__ieee754_pow (-x, y);
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}
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if (isinf (retval))
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retval = -huge * huge;
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else if (retval == 0)
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retval = -tiny * tiny;
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return retval;
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}
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}
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/* x>0 */
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if (qx == 0x7ff00000) /* x= 2^-0x3ff */
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return y > 0 ? x : 0;
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if (qy > 0x45f00000 && qy < 0x7ff00000)
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{
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if (x == 1.0)
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return 1.0;
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if (y > 0)
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return (x > 1.0) ? huge * huge : tiny * tiny;
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if (y < 0)
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return (x < 1.0) ? huge * huge : tiny * tiny;
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}
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if (x == 1.0)
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return 1.0;
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if (y > 0)
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return (x > 1.0) ? INF.x : 0;
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if (y < 0)
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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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/* Compute x^y using more accurate but more slow log routine. */
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static double
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SECTION
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power1 (double x, double y)
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{
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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 * CN;
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y1 = t - (t - y);
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y2 = y - y1;
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t = z * CN;
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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 * fabs (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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/* Compute log(x) (x is left argument). The result is the returned double + the
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parameter DELTA. The result is bounded by ERROR. */
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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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{
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int i, j, m;
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double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
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mynumber u, v;
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#ifdef BIG_ENDI
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mynumber /**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */
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#else
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# ifdef LITTLE_ENDI
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mynumber /**/ 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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{
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x = x * t52.x;
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add = -52.0;
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u.x = x;
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m = u.i[HIGH_HALF];
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}
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if ((m & 0x000fffff) < 0x0006a09e)
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{
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u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000;
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two52.i[LOW_HALF] = (m >> 20);
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}
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else
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{
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u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000;
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two52.i[LOW_HALF] = (m >> 20) + 1;
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}
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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
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* (r7 + t * r8)))))
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- 0.5 * t2 * (t + t1));
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res = e1 + e2;
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*error = 1.0e-21 * fabs (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
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* (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
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* (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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/* Slower but more accurate routine of log. The returned result is double +
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DELTA. The result is bounded by ERROR. */
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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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{
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int i, j, m;
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double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
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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 /**/ two52 = {{0x43300000, 0x00000000}}; /* 2**52 */
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#else
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# ifdef LITTLE_ENDI
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mynumber /**/ 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)
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{ /* x < 2^-1022 */
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x = x * t52.x;
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add = -52.0;
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u.x = x;
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m = u.i[HIGH_HALF];
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}
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if ((m & 0x000fffff) < 0x0006a09e)
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{
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u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000;
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two52.i[LOW_HALF] = (m >> 20);
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}
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else
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{
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u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000;
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two52.i[LOW_HALF] = (m >> 20) + 1;
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}
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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
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* (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 * fabs (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
|
|
{ /*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;
|
|
}
|
|
}
|
|
|
|
/* 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
|
|
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;
|
|
}
|