-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2016 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. */
-/* */
-/***************************************************************************/
+/* Double-precision x^y function.
+ Copyright (C) 2018-2021 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
+ <https://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.h>
+#include <stdint.h>
+#include <math-barriers.h>
+#include <math-narrow-eval.h>
+#include <math-svid-compat.h>
+#include <libm-alias-finite.h>
+#include <libm-alias-double.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, 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);
+/* 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, 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)
- 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;
+ /* 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, &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);
- }
+ /* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */
+ invc = T[i].invc;
+ logc = T[i].logc;
+ logctail = T[i].logctail;
- if (isinf (retval))
- retval = huge * huge;
- else if (retval == 0)
- retval = tiny * tiny;
- else
- math_check_force_underflow_nonneg (retval);
- return retval;
- }
+ /* 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
- if (x == 0)
+ /* k*Ln2 + log(c) + r. */
+ t1 = kd * Ln2hi + logc;
+ t2 = t1 + r;
+ lo1 = kd * Ln2lo + logctail;
+ lo2 = t1 - t2 + r;
+
+ /* 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]
+
+/* 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;
- 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;
- if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0)) /* NaN */
- return x == 1.0 ? 1.0 : 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;
+ return __math_oflow (sign_bias);
}
- /* 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;
+ /* Large x is special cased below. */
+ abstop = 0;
}
- 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)
+ /* 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;
+}
-/* Compute x^y using more accurate but more slow log routine. */
-static double
-SECTION
-power1 (double x, double y)
+/* 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)
{
- 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);
+ 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;
}
-/* 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)
+/* Returns 1 if input is the bit representation of 0, infinity or nan. */
+static inline int
+zeroinfnan (uint64_t i)
{
- 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
+ return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
+}
- 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];
- }
+#ifndef SECTION
+# define SECTION
+#endif
- 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;
- }
+double
+SECTION
+__pow (double x, double y)
+{
+ uint32_t sign_bias = 0;
+ uint64_t ix, iy;
+ uint32_t topx, topy;
- 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 */
+ ix = asuint64 (x);
+ iy = asuint64 (y);
+ topx = top12 (x);
+ topy = top12 (y);
+ if (__glibc_unlikely (topx - 0x001 >= 0x7ff - 0x001
+ || (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be))
{
- 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)))
+ {
+ 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)))
{
- 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
+ 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)
{
- 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;
+ /* 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;
}
- } /* 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 */
-}
+ 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;
+ }
+ }
-/* 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 */
+ 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
-# ifdef LITTLE_ENDI
- mynumber /**/ two52 = {{0x00000000, 0x43300000}}; /* 2**52 */
-# endif
+ 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
-
- 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;
+ return exp_inline (ehi, elo, sign_bias);
}
+#ifndef __pow
+strong_alias (__pow, __ieee754_pow)
+libm_alias_finite (__ieee754_pow, __pow)
+# if LIBM_SVID_COMPAT
+versioned_symbol (libm, __pow, pow, GLIBC_2_29);
+libm_alias_double_other (__pow, pow)
+# else
+libm_alias_double (__pow, pow)
+# endif
+#endif
projects / thirdparty / glibc.git / blobdiff