174 lines
6.3 KiB
C
174 lines
6.3 KiB
C
/* Double-precision e^x function.
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Copyright (c) 2018 Arm Ltd. All rights reserved.
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SPDX-License-Identifier: BSD-3-Clause
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Redistribution and use in source and binary forms, with or without
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modification, are permitted provided that the following conditions
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are met:
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1. Redistributions of source code must retain the above copyright
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notice, this list of conditions and the following disclaimer.
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2. Redistributions in binary form must reproduce the above copyright
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notice, this list of conditions and the following disclaimer in the
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documentation and/or other materials provided with the distribution.
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3. The name of the company may not be used to endorse or promote
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products derived from this software without specific prior written
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permission.
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THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS'' AND ANY EXPRESS OR IMPLIED
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WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
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MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
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TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
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PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
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LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
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NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
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SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
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#include "fdlibm.h"
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#if !__OBSOLETE_MATH
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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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#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 (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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scale = asdouble (sbits);
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y = scale + scale * tmp;
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if (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;
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lo = scale - y + scale * tmp;
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hi = 1.0 + y;
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lo = 1.0 - hi + y + lo;
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y = eval_as_double (hi + lo) - 1.0;
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/* Avoid -0.0 with downward rounding. */
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if (WANT_ROUNDING && y == 0.0)
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y = 0.0;
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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 (y);
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}
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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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double
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exp (double x)
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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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/* Avoid spurious underflow for tiny x. */
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/* Note: 0 is common input. */
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return WANT_ROUNDING ? 1.0 + x : 1.0;
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if (abstop >= top12 (1024.0))
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{
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if (asuint64 (x) == asuint64 (-INFINITY))
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return 0.0;
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if (abstop >= top12 (INFINITY))
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return 1.0 + x;
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if (asuint64 (x) >> 63)
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return __math_uflow (0);
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else
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return __math_oflow (0);
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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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/* 2^(k/N) ~= scale * (1 + tail). */
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idx = 2 * (ki % N);
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top = ki << (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^-65 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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#endif
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