| /* |
| * IBM Accurate Mathematical Library |
| * written by International Business Machines Corp. |
| * Copyright (C) 2001-2014 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; |
| } |
| /* if y even or odd */ |
| return (k == 1) ? __ieee754_pow (-x, y) : -__ieee754_pow (-x, y); |
| } |
| /* 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 * ABS (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) |
| { |
| 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 */ |
| } |
| |
| /* 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) |
| { |
| 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; |
| } |
| } |
| |
| /* 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; |
| } |
