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374 lines
12 KiB
C
374 lines
12 KiB
C
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/* Double-precision x^y function.
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Copyright (C) 2024 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library 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 GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<https://www.gnu.org/licenses/>. */
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#include "math_config.h"
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/* Scalar version of pow used for fallbacks in vector implementations. */
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/* Data is defined in v_pow_log_data.c. */
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#define N_LOG (1 << V_POW_LOG_TABLE_BITS)
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#define Off 0x3fe6955500000000
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#define As __v_pow_log_data.poly
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/* Data is defined in v_pow_exp_data.c. */
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#define N_EXP (1 << V_POW_EXP_TABLE_BITS)
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#define SignBias (0x800 << V_POW_EXP_TABLE_BITS)
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#define SmallExp 0x3c9 /* top12(0x1p-54). */
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#define BigExp 0x408 /* top12(512.0). */
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#define ThresExp 0x03f /* BigExp - SmallExp. */
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#define InvLn2N __v_pow_exp_data.n_over_ln2
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#define Ln2HiN __v_pow_exp_data.ln2_over_n_hi
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#define Ln2LoN __v_pow_exp_data.ln2_over_n_lo
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#define SBits __v_pow_exp_data.sbits
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#define Cs __v_pow_exp_data.poly
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/* Constants associated with pow. */
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#define SmallPowX 0x001 /* top12(0x1p-126). */
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#define BigPowX 0x7ff /* top12(INFINITY). */
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#define ThresPowX 0x7fe /* BigPowX - SmallPowX. */
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#define SmallPowY 0x3be /* top12(0x1.e7b6p-65). */
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#define BigPowY 0x43e /* top12(0x1.749p62). */
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#define ThresPowY 0x080 /* BigPowY - SmallPowY. */
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/* Top 12 bits of a double (sign and exponent bits). */
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static inline uint32_t
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top12 (double x)
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{
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return asuint64 (x) >> 52;
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}
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/* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about
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additional 15 bits precision. IX is the bit representation of x, but
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normalized in the subnormal range using the sign bit for the exponent. */
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static inline double
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log_inline (uint64_t ix, double *tail)
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{
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/* x = 2^k z; where z is in range [Off,2*Off) and exact.
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The range is split into N subintervals.
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The ith subinterval contains z and c is near its center. */
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uint64_t tmp = ix - Off;
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int i = (tmp >> (52 - V_POW_LOG_TABLE_BITS)) & (N_LOG - 1);
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int k = (int64_t) tmp >> 52; /* arithmetic shift. */
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uint64_t iz = ix - (tmp & 0xfffULL << 52);
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double z = asdouble (iz);
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double kd = (double) k;
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/* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */
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double invc = __v_pow_log_data.invc[i];
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double logc = __v_pow_log_data.logc[i];
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double logctail = __v_pow_log_data.logctail[i];
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/* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and
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|z/c - 1| < 1/N, so r = z/c - 1 is exactly representible. */
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double r = fma (z, invc, -1.0);
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/* k*Ln2 + log(c) + r. */
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double t1 = kd * __v_pow_log_data.ln2_hi + logc;
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double t2 = t1 + r;
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double lo1 = kd * __v_pow_log_data.ln2_lo + logctail;
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double lo2 = t1 - t2 + r;
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/* Evaluation is optimized assuming superscalar pipelined execution. */
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double ar = As[0] * r;
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double ar2 = r * ar;
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double ar3 = r * ar2;
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/* k*Ln2 + log(c) + r + A[0]*r*r. */
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double hi = t2 + ar2;
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double lo3 = fma (ar, r, -ar2);
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double lo4 = t2 - hi + ar2;
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/* p = log1p(r) - r - A[0]*r*r. */
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double p = (ar3
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* (As[1] + r * As[2]
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+ ar2 * (As[3] + r * As[4] + ar2 * (As[5] + r * As[6]))));
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double lo = lo1 + lo2 + lo3 + lo4 + p;
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double y = hi + lo;
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*tail = hi - y + lo;
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return y;
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}
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/* Handle cases that may overflow or underflow when computing the result that
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is scale*(1+TMP) without intermediate rounding. The bit representation of
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scale is in SBITS, however it has a computed exponent that may have
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overflown into the sign bit so that needs to be adjusted before using it as
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a double. (int32_t)KI is the k used in the argument reduction and exponent
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adjustment of scale, positive k here means the result may overflow and
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negative k means the result may underflow. */
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static inline double
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special_case (double tmp, uint64_t sbits, uint64_t ki)
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{
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double scale, y;
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if ((ki & 0x80000000) == 0)
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{
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/* k > 0, the exponent of scale might have overflowed by <= 460. */
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sbits -= 1009ull << 52;
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scale = asdouble (sbits);
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y = 0x1p1009 * (scale + scale * tmp);
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return y;
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}
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/* k < 0, need special care in the subnormal range. */
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sbits += 1022ull << 52;
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/* Note: sbits is signed scale. */
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scale = asdouble (sbits);
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y = scale + scale * tmp;
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#if WANT_SIMD_EXCEPT
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if (fabs (y) < 1.0)
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{
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/* Round y to the right precision before scaling it into the subnormal
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range to avoid double rounding that can cause 0.5+E/2 ulp error where
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E is the worst-case ulp error outside the subnormal range. So this
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is only useful if the goal is better than 1 ulp worst-case error. */
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double hi, lo, one = 1.0;
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if (y < 0.0)
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one = -1.0;
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lo = scale - y + scale * tmp;
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hi = one + y;
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lo = one - hi + y + lo;
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y = (hi + lo) - one;
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/* Fix the sign of 0. */
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if (y == 0.0)
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y = asdouble (sbits & 0x8000000000000000);
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/* The underflow exception needs to be signaled explicitly. */
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force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);
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}
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#endif
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y = 0x1p-1022 * y;
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return y;
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}
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/* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
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The sign_bias argument is SignBias or 0 and sets the sign to -1 or 1. */
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static inline double
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exp_inline (double x, double xtail, uint32_t sign_bias)
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{
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uint32_t abstop = top12 (x) & 0x7ff;
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if (__glibc_unlikely (abstop - SmallExp >= ThresExp))
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{
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if (abstop - SmallExp >= 0x80000000)
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{
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/* Avoid spurious underflow for tiny x. */
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/* Note: 0 is common input. */
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return sign_bias ? -1.0 : 1.0;
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}
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if (abstop >= top12 (1024.0))
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{
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/* Note: inf and nan are already handled. */
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/* Skip errno handling. */
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#if WANT_SIMD_EXCEPT
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return asuint64 (x) >> 63 ? __math_uflow (sign_bias)
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: __math_oflow (sign_bias);
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#else
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double res_uoflow = asuint64 (x) >> 63 ? 0.0 : INFINITY;
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return sign_bias ? -res_uoflow : res_uoflow;
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#endif
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}
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/* Large x is special cased below. */
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abstop = 0;
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}
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/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
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/* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
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double z = InvLn2N * x;
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double kd = round (z);
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uint64_t ki = lround (z);
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double r = x - kd * Ln2HiN - kd * Ln2LoN;
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/* The code assumes 2^-200 < |xtail| < 2^-8/N. */
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r += xtail;
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/* 2^(k/N) ~= scale. */
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uint64_t idx = ki & (N_EXP - 1);
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uint64_t top = (ki + sign_bias) << (52 - V_POW_EXP_TABLE_BITS);
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/* This is only a valid scale when -1023*N < k < 1024*N. */
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uint64_t sbits = SBits[idx] + top;
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/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (exp(r) - 1). */
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/* Evaluation is optimized assuming superscalar pipelined execution. */
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double r2 = r * r;
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double tmp = r + r2 * Cs[0] + r * r2 * (Cs[1] + r * Cs[2]);
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if (__glibc_unlikely (abstop == 0))
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return special_case (tmp, sbits, ki);
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double scale = asdouble (sbits);
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/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
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is no spurious underflow here even without fma. */
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return scale + scale * tmp;
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}
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/* Computes exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
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A version of exp_inline that is not inlined and for which sign_bias is
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equal to 0. */
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static double NOINLINE
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exp_nosignbias (double x, double xtail)
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{
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uint32_t abstop = top12 (x) & 0x7ff;
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if (__glibc_unlikely (abstop - SmallExp >= ThresExp))
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{
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/* Avoid spurious underflow for tiny x. */
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if (abstop - SmallExp >= 0x80000000)
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return 1.0;
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/* Note: inf and nan are already handled. */
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if (abstop >= top12 (1024.0))
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#if WANT_SIMD_EXCEPT
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return asuint64 (x) >> 63 ? __math_uflow (0) : __math_oflow (0);
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#else
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return asuint64 (x) >> 63 ? 0.0 : INFINITY;
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#endif
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/* Large x is special cased below. */
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abstop = 0;
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}
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/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
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/* x = ln2/N*k + r, with k integer and r in [-ln2/2N, ln2/2N]. */
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double z = InvLn2N * x;
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double kd = round (z);
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uint64_t ki = lround (z);
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double r = x - kd * Ln2HiN - kd * Ln2LoN;
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/* The code assumes 2^-200 < |xtail| < 2^-8/N. */
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r += xtail;
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/* 2^(k/N) ~= scale. */
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uint64_t idx = ki & (N_EXP - 1);
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uint64_t top = ki << (52 - V_POW_EXP_TABLE_BITS);
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/* This is only a valid scale when -1023*N < k < 1024*N. */
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uint64_t sbits = SBits[idx] + top;
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/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
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double r2 = r * r;
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double tmp = r + r2 * Cs[0] + r * r2 * (Cs[1] + r * Cs[2]);
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if (__glibc_unlikely (abstop == 0))
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return special_case (tmp, sbits, ki);
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double scale = asdouble (sbits);
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/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
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is no spurious underflow here even without fma. */
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return scale + scale * tmp;
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}
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/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
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the bit representation of a non-zero finite floating-point value. */
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static inline int
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checkint (uint64_t iy)
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{
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int e = iy >> 52 & 0x7ff;
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if (e < 0x3ff)
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return 0;
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if (e > 0x3ff + 52)
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return 2;
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if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
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return 0;
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if (iy & (1ULL << (0x3ff + 52 - e)))
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return 1;
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return 2;
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}
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/* Returns 1 if input is the bit representation of 0, infinity or nan. */
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static inline int
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zeroinfnan (uint64_t i)
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{
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return 2 * i - 1 >= 2 * asuint64 (INFINITY) - 1;
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}
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static double NOINLINE
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pow_scalar_special_case (double x, double y)
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{
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uint32_t sign_bias = 0;
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uint64_t ix, iy;
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uint32_t topx, topy;
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ix = asuint64 (x);
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iy = asuint64 (y);
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topx = top12 (x);
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topy = top12 (y);
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if (__glibc_unlikely (topx - SmallPowX >= ThresPowX
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|| (topy & 0x7ff) - SmallPowY >= ThresPowY))
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{
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/* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
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and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */
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/* Special cases: (x < 0x1p-126 or inf or nan) or
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(|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */
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if (__glibc_unlikely (zeroinfnan (iy)))
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{
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if (2 * iy == 0)
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return issignaling_inline (x) ? x + y : 1.0;
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if (ix == asuint64 (1.0))
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return issignaling_inline (y) ? x + y : 1.0;
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if (2 * ix > 2 * asuint64 (INFINITY)
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|| 2 * iy > 2 * asuint64 (INFINITY))
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return x + y;
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if (2 * ix == 2 * asuint64 (1.0))
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return 1.0;
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if ((2 * ix < 2 * asuint64 (1.0)) == !(iy >> 63))
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return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
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return y * y;
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}
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if (__glibc_unlikely (zeroinfnan (ix)))
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{
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double x2 = x * x;
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if (ix >> 63 && checkint (iy) == 1)
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{
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x2 = -x2;
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sign_bias = 1;
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}
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#if WANT_SIMD_EXCEPT
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if (2 * ix == 0 && iy >> 63)
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return __math_divzero (sign_bias);
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#endif
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return iy >> 63 ? 1 / x2 : x2;
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}
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/* Here x and y are non-zero finite. */
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if (ix >> 63)
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{
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/* Finite x < 0. */
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int yint = checkint (iy);
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if (yint == 0)
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#if WANT_SIMD_EXCEPT
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return __math_invalid (x);
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#else
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return __builtin_nan ("");
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#endif
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if (yint == 1)
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sign_bias = SignBias;
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ix &= 0x7fffffffffffffff;
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topx &= 0x7ff;
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}
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if ((topy & 0x7ff) - SmallPowY >= ThresPowY)
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{
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/* Note: sign_bias == 0 here because y is not odd. */
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if (ix == asuint64 (1.0))
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return 1.0;
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/* |y| < 2^-65, x^y ~= 1 + y*log(x). */
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if ((topy & 0x7ff) < SmallPowY)
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return 1.0;
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#if WANT_SIMD_EXCEPT
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return (ix > asuint64 (1.0)) == (topy < 0x800) ? __math_oflow (0)
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: __math_uflow (0);
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#else
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return (ix > asuint64 (1.0)) == (topy < 0x800) ? INFINITY : 0;
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#endif
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}
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if (topx == 0)
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{
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/* Normalize subnormal x so exponent becomes negative. */
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ix = asuint64 (x * 0x1p52);
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ix &= 0x7fffffffffffffff;
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ix -= 52ULL << 52;
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}
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}
|
||
|
|
||
|
double lo;
|
||
|
double hi = log_inline (ix, &lo);
|
||
|
double ehi = y * hi;
|
||
|
double elo = y * lo + fma (y, hi, -ehi);
|
||
|
return exp_inline (ehi, elo, sign_bias);
|
||
|
}
|