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Diffstat (limited to 'REORG.TODO/sysdeps/ieee754/dbl-64/e_pow.c')
| -rw-r--r-- | REORG.TODO/sysdeps/ieee754/dbl-64/e_pow.c | 481 |
1 files changed, 481 insertions, 0 deletions
diff --git a/REORG.TODO/sysdeps/ieee754/dbl-64/e_pow.c b/REORG.TODO/sysdeps/ieee754/dbl-64/e_pow.c new file mode 100644 index 0000000000..9f6439ee42 --- /dev/null +++ b/REORG.TODO/sysdeps/ieee754/dbl-64/e_pow.c @@ -0,0 +1,481 @@ +/* + * IBM Accurate Mathematical Library + * written by International Business Machines Corp. + * Copyright (C) 2001-2017 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 <math.h> +#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)) + && (y != 0 || issignaling (x))) + return x + 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 (fabs (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 * 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); + } + + if (isinf (retval)) + retval = huge * huge; + else if (retval == 0) + retval = tiny * tiny; + else + math_check_force_underflow_nonneg (retval); + 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 + y; + if (fabs (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 + y; + if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0)) /* NaN */ + return x == 1.0 && !issignaling (y) ? 1.0 : y + 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; + } + /* if y even or odd */ + if (k == 1) + return __ieee754_pow (-x, y); + else + { + double retval; + { + SET_RESTORE_ROUND (FE_TONEAREST); + retval = -__ieee754_pow (-x, y); + } + if (isinf (retval)) + retval = -huge * huge; + else if (retval == 0) + retval = -tiny * tiny; + return retval; + } + } + /* 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 + +/* 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. */ +static double +SECTION +log1 (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; + 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 * fabs (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 */ +} + +/* 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; + 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) != 0) + return 0; /* if not integer */ + 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) != 0) + return 0; + return (m << (k + 11) != 0) ? -1 : 1; +} |