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381 lines
12 KiB
381 lines
12 KiB
/*
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* Double-precision x^y function.
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*
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* Copyright (c) 2018-2020, Arm Limited.
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* SPDX-License-Identifier: MIT
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*/
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#include <float.h>
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#include <math.h>
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#include <stdint.h>
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#include "math_config.h"
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/*
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Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
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relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
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ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma)
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*/
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#define T __pow_log_data.tab
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#define A __pow_log_data.poly
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#define Ln2hi __pow_log_data.ln2hi
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#define Ln2lo __pow_log_data.ln2lo
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#define N (1 << POW_LOG_TABLE_BITS)
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#define OFF 0x3fe6955500000000
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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_t
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log_inline (uint64_t ix, double_t *tail)
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{
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/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
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double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p;
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uint64_t iz, tmp;
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int k, i;
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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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tmp = ix - OFF;
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i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N;
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k = (int64_t) tmp >> 52; /* arithmetic shift */
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iz = ix - (tmp & 0xfffULL << 52);
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z = asdouble (iz);
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kd = (double_t) k;
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/* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */
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invc = T[i].invc;
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logc = T[i].logc;
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logctail = T[i].logctail;
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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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#if HAVE_FAST_FMA
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r = fma (z, invc, -1.0);
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#else
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/* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */
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double_t zhi = asdouble ((iz + (1ULL << 31)) & (-1ULL << 32));
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double_t zlo = z - zhi;
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double_t rhi = zhi * invc - 1.0;
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double_t rlo = zlo * invc;
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r = rhi + rlo;
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#endif
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/* k*Ln2 + log(c) + r. */
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t1 = kd * Ln2hi + logc;
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t2 = t1 + r;
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lo1 = kd * Ln2lo + logctail;
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lo2 = t1 - t2 + r;
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/* Evaluation is optimized assuming superscalar pipelined execution. */
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double_t ar, ar2, ar3, lo3, lo4;
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ar = A[0] * r; /* A[0] = -0.5. */
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ar2 = r * ar;
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ar3 = r * ar2;
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/* k*Ln2 + log(c) + r + A[0]*r*r. */
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#if HAVE_FAST_FMA
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hi = t2 + ar2;
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lo3 = fma (ar, r, -ar2);
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lo4 = t2 - hi + ar2;
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#else
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double_t arhi = A[0] * rhi;
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double_t arhi2 = rhi * arhi;
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hi = t2 + arhi2;
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lo3 = rlo * (ar + arhi);
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lo4 = t2 - hi + arhi2;
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#endif
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/* p = log1p(r) - r - A[0]*r*r. */
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#if POW_LOG_POLY_ORDER == 8
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p = (ar3
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* (A[1] + r * A[2] + ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6]))));
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#endif
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lo = lo1 + lo2 + lo3 + lo4 + p;
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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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#undef N
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#undef T
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#define N (1 << EXP_TABLE_BITS)
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#define InvLn2N __exp_data.invln2N
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#define NegLn2hiN __exp_data.negln2hiN
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#define NegLn2loN __exp_data.negln2loN
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#define Shift __exp_data.shift
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#define T __exp_data.tab
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#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
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#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
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#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
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#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
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#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
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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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specialcase (double_t tmp, uint64_t sbits, uint64_t ki)
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{
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double_t 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 check_oflow (eval_as_double (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 (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_t 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 = eval_as_double (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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y = 0x1p-1022 * y;
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return check_uflow (eval_as_double (y));
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}
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#define SIGN_BIAS (0x800 << EXP_TABLE_BITS)
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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 SIGN_BIAS or 0 and sets the sign to -1 or 1. */
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static inline double
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exp_inline (double_t x, double_t xtail, uint32_t sign_bias)
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{
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uint32_t abstop;
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uint64_t ki, idx, top, sbits;
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/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
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double_t kd, z, r, r2, scale, tail, tmp;
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abstop = top12 (x) & 0x7ff;
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if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))
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{
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if (abstop - top12 (0x1p-54) >= 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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double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
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return sign_bias ? -one : one;
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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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if (asuint64 (x) >> 63)
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return __math_uflow (sign_bias);
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else
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return __math_oflow (sign_bias);
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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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z = InvLn2N * x;
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#if TOINT_INTRINSICS
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kd = roundtoint (z);
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ki = converttoint (z);
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#elif EXP_USE_TOINT_NARROW
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/* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
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kd = eval_as_double (z + Shift);
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ki = asuint64 (kd) >> 16;
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kd = (double_t) (int32_t) ki;
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#else
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/* z - kd is in [-1, 1] in non-nearest rounding modes. */
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kd = eval_as_double (z + Shift);
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ki = asuint64 (kd);
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kd -= Shift;
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#endif
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r = x + kd * NegLn2hiN + kd * NegLn2loN;
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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 * (1 + tail). */
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idx = 2 * (ki % N);
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top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
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tail = asdouble (T[idx]);
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/* This is only a valid scale when -1023*N < k < 1024*N. */
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sbits = T[idx + 1] + top;
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/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
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/* Evaluation is optimized assuming superscalar pipelined execution. */
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r2 = r * r;
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/* Without fma the worst case error is 0.25/N ulp larger. */
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/* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
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#if EXP_POLY_ORDER == 4
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tmp = tail + r + r2 * C2 + r * r2 * (C3 + r * C4);
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#elif EXP_POLY_ORDER == 5
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tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
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#elif EXP_POLY_ORDER == 6
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tmp = tail + r + r2 * (0.5 + r * C3) + r2 * r2 * (C4 + r * C5 + r2 * C6);
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#endif
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if (unlikely (abstop == 0))
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return specialcase (tmp, sbits, ki);
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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 eval_as_double (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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double
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pow (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 (unlikely (topx - 0x001 >= 0x7ff - 0x001
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|| (topy & 0x7ff) - 0x3be >= 0x43e - 0x3be))
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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 (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 (unlikely (zeroinfnan (ix)))
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{
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double_t 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_ERRNO && 2 * ix == 0 && iy >> 63)
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return __math_divzero (sign_bias);
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/* Without the barrier some versions of clang hoist the 1/x2 and
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thus division by zero exception can be signaled spuriously. */
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return iy >> 63 ? opt_barrier_double (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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return __math_invalid (x);
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if (yint == 1)
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sign_bias = SIGN_BIAS;
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ix &= 0x7fffffffffffffff;
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topx &= 0x7ff;
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}
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if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)
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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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if ((topy & 0x7ff) < 0x3be)
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{
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/* |y| < 2^-65, x^y ~= 1 + y*log(x). */
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if (WANT_ROUNDING)
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return ix > asuint64 (1.0) ? 1.0 + y : 1.0 - y;
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else
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return 1.0;
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}
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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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}
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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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/* Without the barrier some versions of clang evalutate the mul
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unconditionally causing spurious overflow exceptions. */
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ix = asuint64 (opt_barrier_double (x) * 0x1p52);
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ix &= 0x7fffffffffffffff;
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ix -= 52ULL << 52;
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}
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}
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double_t lo;
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double_t hi = log_inline (ix, &lo);
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double_t ehi, elo;
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#if HAVE_FAST_FMA
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ehi = y * hi;
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elo = y * lo + fma (y, hi, -ehi);
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#else
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double_t yhi = asdouble (iy & -1ULL << 27);
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double_t ylo = y - yhi;
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double_t lhi = asdouble (asuint64 (hi) & -1ULL << 27);
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double_t llo = hi - lhi + lo;
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ehi = yhi * lhi;
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elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */
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#endif
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return exp_inline (ehi, elo, sign_bias);
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}
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#if USE_GLIBC_ABI
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strong_alias (pow, __pow_finite)
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hidden_alias (pow, __ieee754_pow)
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# if LDBL_MANT_DIG == 53
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long double powl (long double x, long double y) { return pow (x, y); }
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# endif
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#endif
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