diff options
Diffstat (limited to 'sysdeps/ieee754/dbl-64/e_pow.c')
| -rw-r--r-- | sysdeps/ieee754/dbl-64/e_pow.c | 658 |
1 files changed, 339 insertions, 319 deletions
diff --git a/sysdeps/ieee754/dbl-64/e_pow.c b/sysdeps/ieee754/dbl-64/e_pow.c index 9bf29e5cb3..ba38bfefcb 100644 --- a/sysdeps/ieee754/dbl-64/e_pow.c +++ b/sysdeps/ieee754/dbl-64/e_pow.c @@ -1,360 +1,380 @@ -/* - * IBM Accurate Mathematical Library - * written by International Business Machines Corp. - * 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 - * 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 */ -/* log1 */ -/* checkint */ -/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h */ -/* 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. */ -/* */ -/***************************************************************************/ +/* Double-precision x^y function. + Copyright (C) 2018 Free Software Foundation, Inc. + This file is part of the GNU C Library. + + The GNU C Library 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. + + The GNU C Library 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 the GNU C Library; if not, see + <http://www.gnu.org/licenses/>. */ + #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_private.h> -#include <math-underflow.h> -#include <fenv.h> +#include <stdint.h> +#include <math-barriers.h> +#include <math-narrow-eval.h> +#include "math_config.h" -#ifndef SECTION -# define SECTION -#endif +/* +Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53) +relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma) +ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma) +*/ -static const double huge = 1.0e300, tiny = 1.0e-300; +#define T __pow_log_data.tab +#define A __pow_log_data.poly +#define Ln2hi __pow_log_data.ln2hi +#define Ln2lo __pow_log_data.ln2lo +#define N (1 << POW_LOG_TABLE_BITS) +#define OFF 0x3fe6955500000000 -double __exp1 (double x, double xx); -static double log1 (double x, double *delta); -static int checkint (double x); +/* Top 12 bits of a double (sign and exponent bits). */ +static inline uint32_t +top12 (double x) +{ + return asuint64 (x) >> 52; +} -/* 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) +/* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about + additional 15 bits precision. IX is the bit representation of x, but + normalized in the subnormal range using the sign bit for the exponent. */ +static inline double_t +log_inline (uint64_t ix, double_t *tail) { - double z, a, aa, 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; + /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ + double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p; + uint64_t iz, tmp; + int k, i; - { - SET_RESTORE_ROUND (FE_TONEAREST); + /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. + The range is split into N subintervals. + The ith subinterval contains z and c is near its center. */ + tmp = ix - OFF; + i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N; + k = (int64_t) tmp >> 52; /* arithmetic shift */ + iz = ix - (tmp & 0xfffULL << 52); + z = asdouble (iz); + kd = (double_t) k; - /* 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); /* 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; + /* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */ + invc = T[i].invc; + logc = T[i].logc; + logctail = T[i].logctail; - /* Maximum relative error RElog of log1 is 1.0e-21 (69.7 bits). - Maximum relative error REexp of __exp1 is 1.0e-18 (59.8 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 + 1.0e-18 = 1.7e-18 - (59 bits). The worst-case ULP error is 0.515. */ + /* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and + |z/c - 1| < 1/N, so r = z/c - 1 is exactly representible. */ +#ifdef __FP_FAST_FMA + r = __builtin_fma (z, invc, -1.0); +#else + /* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */ + double_t zhi = asdouble ((iz + (1ULL << 31)) & (-1ULL << 32)); + double_t zlo = z - zhi; + double_t rhi = zhi * invc - 1.0; + double_t rlo = zlo * invc; + r = rhi + rlo; +#endif - retval = __exp1 (a1, a2); - } + /* k*Ln2 + log(c) + r. */ + t1 = kd * Ln2hi + logc; + t2 = t1 + r; + lo1 = kd * Ln2lo + logctail; + lo2 = t1 - t2 + r; - if (isinf (retval)) - retval = huge * huge; - else if (retval == 0) - retval = tiny * tiny; - else - math_check_force_underflow_nonneg (retval); - return retval; - } + /* Evaluation is optimized assuming superscalar pipelined execution. */ + double_t ar, ar2, ar3, lo3, lo4; + ar = A[0] * r; /* A[0] = -0.5. */ + ar2 = r * ar; + ar3 = r * ar2; + /* k*Ln2 + log(c) + r + A[0]*r*r. */ +#ifdef __FP_FAST_FMA + hi = t2 + ar2; + lo3 = __builtin_fma (ar, r, -ar2); + lo4 = t2 - hi + ar2; +#else + double_t arhi = A[0] * rhi; + double_t arhi2 = rhi * arhi; + hi = t2 + arhi2; + lo3 = rlo * (ar + arhi); + lo4 = t2 - hi + arhi2; +#endif + /* p = log1p(r) - r - A[0]*r*r. */ + p = (ar3 + * (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6])))); + lo = lo1 + lo2 + lo3 + lo4 + p; + y = hi + lo; + *tail = hi - y + lo; + return y; +} + +#undef N +#undef T +#define N (1 << EXP_TABLE_BITS) +#define InvLn2N __exp_data.invln2N +#define NegLn2hiN __exp_data.negln2hiN +#define NegLn2loN __exp_data.negln2loN +#define Shift __exp_data.shift +#define T __exp_data.tab +#define C2 __exp_data.poly[5 - EXP_POLY_ORDER] +#define C3 __exp_data.poly[6 - EXP_POLY_ORDER] +#define C4 __exp_data.poly[7 - EXP_POLY_ORDER] +#define C5 __exp_data.poly[8 - EXP_POLY_ORDER] +#define C6 __exp_data.poly[9 - EXP_POLY_ORDER] - if (x == 0) +/* Handle cases that may overflow or underflow when computing the result that + is scale*(1+TMP) without intermediate rounding. The bit representation of + scale is in SBITS, however it has a computed exponent that may have + overflown into the sign bit so that needs to be adjusted before using it as + a double. (int32_t)KI is the k used in the argument reduction and exponent + adjustment of scale, positive k here means the result may overflow and + negative k means the result may underflow. */ +static inline double +specialcase (double_t tmp, uint64_t sbits, uint64_t ki) +{ + double_t scale, y; + + if ((ki & 0x80000000) == 0) + { + /* k > 0, the exponent of scale might have overflowed by <= 460. */ + sbits -= 1009ull << 52; + scale = asdouble (sbits); + y = 0x1p1009 * (scale + scale * tmp); + return check_oflow (y); + } + /* k < 0, need special care in the subnormal range. */ + sbits += 1022ull << 52; + /* Note: sbits is signed scale. */ + scale = asdouble (sbits); + y = scale + scale * tmp; + if (fabs (y) < 1.0) { - 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 */ + /* Round y to the right precision before scaling it into the subnormal + range to avoid double rounding that can cause 0.5+E/2 ulp error where + E is the worst-case ulp error outside the subnormal range. So this + is only useful if the goal is better than 1 ulp worst-case error. */ + double_t hi, lo, one = 1.0; + if (y < 0.0) + one = -1.0; + lo = scale - y + scale * tmp; + hi = one + y; + lo = one - hi + y + lo; + y = math_narrow_eval (hi + lo) - one; + /* Fix the sign of 0. */ + if (y == 0.0) + y = asdouble (sbits & 0x8000000000000000); + /* The underflow exception needs to be signaled explicitly. */ + math_force_eval (math_opt_barrier (0x1p-1022) * 0x1p-1022); } + y = 0x1p-1022 * y; + return check_uflow (y); +} - qx = u.i[HIGH_HALF] & 0x7fffffff; /* no sign */ - qy = v.i[HIGH_HALF] & 0x7fffffff; /* no sign */ +#define SIGN_BIAS (0x800 << EXP_TABLE_BITS) - 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; +/* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|. + The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */ +static inline double +exp_inline (double x, double xtail, uint32_t sign_bias) +{ + uint32_t abstop; + uint64_t ki, idx, top, sbits; + /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ + double_t kd, z, r, r2, scale, tail, tmp; - /* if x<0 */ - if (u.i[HIGH_HALF] < 0) + abstop = top12 (x) & 0x7ff; + if (__glibc_unlikely (abstop - top12 (0x1p-54) + >= top12 (512.0) - top12 (0x1p-54))) { - k = checkint (y); - if (k == 0) + if (abstop - top12 (0x1p-54) >= 0x80000000) { - 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 */ + /* Avoid spurious underflow for tiny x. */ + /* Note: 0 is common input. */ + double_t one = WANT_ROUNDING ? 1.0 + x : 1.0; + return sign_bias ? -one : one; } - else if (qx == 0x7ff00000) + if (abstop >= top12 (1024.0)) { - if (k < 0) - return y < 0 ? nZERO.x : nINF.x; + /* Note: inf and nan are already handled. */ + if (asuint64 (x) >> 63) + return __math_uflow (sign_bias); 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; + return __math_oflow (sign_bias); } + /* Large x is special cased below. */ + abstop = 0; } - /* x>0 */ - if (qx == 0x7ff00000) /* x= 2^-0x3ff */ - return y > 0 ? x : 0; + /* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */ + /* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */ + z = InvLn2N * x; +#if TOINT_INTRINSICS + /* z - kd is in [-0.5, 0.5] in all rounding modes. */ + kd = roundtoint (z); + ki = converttoint (z); +#else + /* z - kd is in [-1, 1] in non-nearest rounding modes. */ + kd = math_narrow_eval (z + Shift); + ki = asuint64 (kd); + kd -= Shift; +#endif + r = x + kd * NegLn2hiN + kd * NegLn2loN; + /* The code assumes 2^-200 < |xtail| < 2^-8/N. */ + r += xtail; + /* 2^(k/N) ~= scale * (1 + tail). */ + idx = 2 * (ki % N); + top = (ki + sign_bias) << (52 - EXP_TABLE_BITS); + tail = asdouble (T[idx]); + /* This is only a valid scale when -1023*N < k < 1024*N. */ + sbits = T[idx + 1] + top; + /* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */ + /* Evaluation is optimized assuming superscalar pipelined execution. */ + r2 = r * r; + /* Without fma the worst case error is 0.25/N ulp larger. */ + /* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */ + tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5); + if (__glibc_unlikely (abstop == 0)) + return specialcase (tmp, sbits, ki); + scale = asdouble (sbits); + /* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there + is no spurious underflow here even without fma. */ + return scale + scale * tmp; +} - 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; - } +/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is + the bit representation of a non-zero finite floating-point value. */ +static inline int +checkint (uint64_t iy) +{ + int e = iy >> 52 & 0x7ff; + if (e < 0x3ff) + return 0; + if (e > 0x3ff + 52) + return 2; + if (iy & ((1ULL << (0x3ff + 52 - e)) - 1)) + return 0; + if (iy & (1ULL << (0x3ff + 52 - e))) + return 1; + return 2; +} - 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 */ +/* Returns 1 if input is the bit representation of 0, infinity or nan. */ +static inline int +zeroinfnan (uint64_t i) +{ + return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1; } -#ifndef __ieee754_pow -strong_alias (__ieee754_pow, __pow_finite) +#ifndef SECTION +# define SECTION #endif -/* Compute log(x) (x is left argument). The result is the returned double + the - parameter DELTA. */ -static double +double SECTION -log1 (double x, double *delta) +__ieee754_pow (double x, double y) { - 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]; - if (m < 0x00100000) /* Handle denormal x. */ - { - x = x * t52.x; - add = -52.0; - u.x = x; - m = u.i[HIGH_HALF]; - } + uint32_t sign_bias = 0; + uint64_t ix, iy; + uint32_t topx, topy; - if ((m & 0x000fffff) < 0x0006a09e) + ix = asuint64 (x); + iy = asuint64 (y); + topx = top12 (x); + topy = top12 (y); + if (__glibc_unlikely (topx - 0x001 >= 0x7ff - 0x001 + || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)) { - 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) /* Exponent of x is 0. */ - { - if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10 */ + /* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0 + and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */ + /* Special cases: (x < 0x1p-126 or inf or nan) or + (|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */ + if (__glibc_unlikely (zeroinfnan (iy))) { - 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; - *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 + if (2 * iy == 0) + return issignaling_inline (x) ? x + y : 1.0; + if (ix == asuint64 (1.0)) + return issignaling_inline (y) ? x + y : 1.0; + if (2 * ix > 2 * asuint64 (INFINITY) + || 2 * iy > 2 * asuint64 (INFINITY)) + return x + y; + if (2 * ix == 2 * asuint64 (1.0)) + return 1.0; + if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63)) + return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */ + return y * y; + } + if (__glibc_unlikely (zeroinfnan (ix))) { - 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; - *delta = (t1 - res) + t2; - /* Max relative error is 1.0e-24, so accurate to 79.7 bits. */ - return res; + double_t x2 = x * x; + if (ix >> 63 && checkint (iy) == 1) + { + x2 = -x2; + sign_bias = 1; + } + if (WANT_ERRNO && 2 * ix == 0 && iy >> 63) + return __math_divzero (sign_bias); + /* Without the barrier some versions of clang hoist the 1/x2 and + thus division by zero exception can be signaled spuriously. */ + return iy >> 63 ? math_opt_barrier (1 / x2) : x2; + } + /* Here x and y are non-zero finite. */ + if (ix >> 63) + { + /* Finite x < 0. */ + int yint = checkint (iy); + if (yint == 0) + return __math_invalid (x); + if (yint == 1) + sign_bias = SIGN_BIAS; + ix &= 0x7fffffffffffffff; + topx &= 0x7ff; + } + if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be) + { + /* Note: sign_bias == 0 here because y is not odd. */ + if (ix == asuint64 (1.0)) + return 1.0; + if ((topy & 0x7ff) < 0x3be) + { + /* |y| < 2^-65, x^y ~= 1 + y*log(x). */ + if (WANT_ROUNDING) + return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y; + else + return 1.0; + } + return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0) + : __math_uflow (0); + } + if (topx == 0) + { + /* Normalize subnormal x so exponent becomes negative. */ + ix = asuint64 (x * 0x1p52); + ix &= 0x7fffffffffffffff; + ix -= 52ULL << 52; } } - else /* Exponent of x != 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; - *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 -SECTION -checkint (double x) -{ - union - { - int4 i[2]; - double x; - } u; - int k; - unsigned int 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; + double_t lo; + double_t hi = log_inline (ix, &lo); + double_t ehi, elo; +#ifdef __FP_FAST_FMA + ehi = y * hi; + elo = y * lo + __builtin_fma (y, hi, -ehi); +#else + double_t yhi = asdouble (iy & -1ULL << 27); + double_t ylo = y - yhi; + double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27); + double_t llo = hi - lhi + lo; + ehi = yhi * lhi; + elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */ +#endif + return exp_inline (ehi, elo, sign_bias); } +#ifndef __ieee754_pow +strong_alias (__ieee754_pow, __pow_finite) +#endif |