135 lines
4.4 KiB
C
135 lines
4.4 KiB
C
/*
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* Double-precision e^x function.
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*
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* Copyright (c) 2018, Arm Limited.
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* SPDX-License-Identifier: MIT
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*/
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#include <math.h>
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#include <stdint.h>
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#include "libm.h"
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#include "exp_data.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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/* 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 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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/* 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 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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scale = asdouble(sbits);
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y = scale + scale * tmp;
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if (y < 1.0) {
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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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fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
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}
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y = 0x1p-1022 * y;
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return eval_as_double(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 top12(double x)
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{
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return asuint64(x) >> 52;
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}
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double 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 kd, z, r, r2, scale, tail, tmp;
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abstop = top12(x) & 0x7ff;
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if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
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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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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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tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
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if (predict_false(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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