adding exp/expf
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388b059643
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@ -84,6 +84,9 @@ float log2f(float);
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double pow(double, double);
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double pow(double, double);
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float powf(float, float);
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float powf(float, float);
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double exp(double);
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float expf(float);
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double scalbn(double, int);
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double scalbn(double, int);
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double sin(double);
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double sin(double);
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@ -0,0 +1,136 @@
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/*
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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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//WARN(orca): we don't have fp_barrier in wasm
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//fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
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fp_force_eval((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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@ -0,0 +1,182 @@
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/*
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* Shared data between exp, exp2 and pow.
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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 "exp_data.h"
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#define N (1 << EXP_TABLE_BITS)
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const struct exp_data __exp_data = {
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// N/ln2
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.invln2N = 0x1.71547652b82fep0 * N,
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// -ln2/N
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.negln2hiN = -0x1.62e42fefa0000p-8,
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.negln2loN = -0x1.cf79abc9e3b3ap-47,
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// Used for rounding when !TOINT_INTRINSICS
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#if EXP_USE_TOINT_NARROW
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.shift = 0x1800000000.8p0,
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#else
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.shift = 0x1.8p52,
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#endif
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// exp polynomial coefficients.
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.poly = {
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// abs error: 1.555*2^-66
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// ulp error: 0.509 (0.511 without fma)
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// if |x| < ln2/256+eps
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// abs error if |x| < ln2/256+0x1p-15: 1.09*2^-65
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// abs error if |x| < ln2/128: 1.7145*2^-56
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0x1.ffffffffffdbdp-2,
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0x1.555555555543cp-3,
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0x1.55555cf172b91p-5,
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0x1.1111167a4d017p-7,
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},
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.exp2_shift = 0x1.8p52 / N,
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// exp2 polynomial coefficients.
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.exp2_poly = {
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// abs error: 1.2195*2^-65
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// ulp error: 0.507 (0.511 without fma)
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// if |x| < 1/256
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// abs error if |x| < 1/128: 1.9941*2^-56
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0x1.62e42fefa39efp-1,
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0x1.ebfbdff82c424p-3,
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0x1.c6b08d70cf4b5p-5,
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0x1.3b2abd24650ccp-7,
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0x1.5d7e09b4e3a84p-10,
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},
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// 2^(k/N) ~= H[k]*(1 + T[k]) for int k in [0,N)
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// tab[2*k] = asuint64(T[k])
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// tab[2*k+1] = asuint64(H[k]) - (k << 52)/N
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.tab = {
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0x0, 0x3ff0000000000000,
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0x3c9b3b4f1a88bf6e, 0x3feff63da9fb3335,
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0xbc7160139cd8dc5d, 0x3fefec9a3e778061,
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0xbc905e7a108766d1, 0x3fefe315e86e7f85,
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0x3c8cd2523567f613, 0x3fefd9b0d3158574,
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0xbc8bce8023f98efa, 0x3fefd06b29ddf6de,
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0x3c60f74e61e6c861, 0x3fefc74518759bc8,
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0x3c90a3e45b33d399, 0x3fefbe3ecac6f383,
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0x3c979aa65d837b6d, 0x3fefb5586cf9890f,
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0x3c8eb51a92fdeffc, 0x3fefac922b7247f7,
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0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2,
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0xbc6a033489906e0b, 0x3fef9b66affed31b,
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0xbc9556522a2fbd0e, 0x3fef9301d0125b51,
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0xbc5080ef8c4eea55, 0x3fef8abdc06c31cc,
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0xbc91c923b9d5f416, 0x3fef829aaea92de0,
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0x3c80d3e3e95c55af, 0x3fef7a98c8a58e51,
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0xbc801b15eaa59348, 0x3fef72b83c7d517b,
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0xbc8f1ff055de323d, 0x3fef6af9388c8dea,
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0x3c8b898c3f1353bf, 0x3fef635beb6fcb75,
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0xbc96d99c7611eb26, 0x3fef5be084045cd4,
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0x3c9aecf73e3a2f60, 0x3fef54873168b9aa,
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0xbc8fe782cb86389d, 0x3fef4d5022fcd91d,
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0x3c8a6f4144a6c38d, 0x3fef463b88628cd6,
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0x3c807a05b0e4047d, 0x3fef3f49917ddc96,
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0x3c968efde3a8a894, 0x3fef387a6e756238,
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0x3c875e18f274487d, 0x3fef31ce4fb2a63f,
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0x3c80472b981fe7f2, 0x3fef2b4565e27cdd,
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0xbc96b87b3f71085e, 0x3fef24dfe1f56381,
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0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1,
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0xbc3d219b1a6fbffa, 0x3fef187fd0dad990,
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0x3c8b3782720c0ab4, 0x3fef1285a6e4030b,
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0x3c6e149289cecb8f, 0x3fef0cafa93e2f56,
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0x3c834d754db0abb6, 0x3fef06fe0a31b715,
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0x3c864201e2ac744c, 0x3fef0170fc4cd831,
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0x3c8fdd395dd3f84a, 0x3feefc08b26416ff,
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0xbc86a3803b8e5b04, 0x3feef6c55f929ff1,
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0xbc924aedcc4b5068, 0x3feef1a7373aa9cb,
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0xbc9907f81b512d8e, 0x3feeecae6d05d866,
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0xbc71d1e83e9436d2, 0x3feee7db34e59ff7,
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0xbc991919b3ce1b15, 0x3feee32dc313a8e5,
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0x3c859f48a72a4c6d, 0x3feedea64c123422,
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0xbc9312607a28698a, 0x3feeda4504ac801c,
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0xbc58a78f4817895b, 0x3feed60a21f72e2a,
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0xbc7c2c9b67499a1b, 0x3feed1f5d950a897,
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0x3c4363ed60c2ac11, 0x3feece086061892d,
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0x3c9666093b0664ef, 0x3feeca41ed1d0057,
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0x3c6ecce1daa10379, 0x3feec6a2b5c13cd0,
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0x3c93ff8e3f0f1230, 0x3feec32af0d7d3de,
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0x3c7690cebb7aafb0, 0x3feebfdad5362a27,
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0x3c931dbdeb54e077, 0x3feebcb299fddd0d,
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0xbc8f94340071a38e, 0x3feeb9b2769d2ca7,
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0xbc87deccdc93a349, 0x3feeb6daa2cf6642,
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0xbc78dec6bd0f385f, 0x3feeb42b569d4f82,
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0xbc861246ec7b5cf6, 0x3feeb1a4ca5d920f,
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0x3c93350518fdd78e, 0x3feeaf4736b527da,
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0x3c7b98b72f8a9b05, 0x3feead12d497c7fd,
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0x3c9063e1e21c5409, 0x3feeab07dd485429,
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0x3c34c7855019c6ea, 0x3feea9268a5946b7,
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0x3c9432e62b64c035, 0x3feea76f15ad2148,
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0xbc8ce44a6199769f, 0x3feea5e1b976dc09,
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0xbc8c33c53bef4da8, 0x3feea47eb03a5585,
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0xbc845378892be9ae, 0x3feea34634ccc320,
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0xbc93cedd78565858, 0x3feea23882552225,
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0x3c5710aa807e1964, 0x3feea155d44ca973,
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0xbc93b3efbf5e2228, 0x3feea09e667f3bcd,
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0xbc6a12ad8734b982, 0x3feea012750bdabf,
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0xbc6367efb86da9ee, 0x3fee9fb23c651a2f,
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0xbc80dc3d54e08851, 0x3fee9f7df9519484,
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0xbc781f647e5a3ecf, 0x3fee9f75e8ec5f74,
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0xbc86ee4ac08b7db0, 0x3fee9f9a48a58174,
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0xbc8619321e55e68a, 0x3fee9feb564267c9,
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0x3c909ccb5e09d4d3, 0x3feea0694fde5d3f,
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0xbc7b32dcb94da51d, 0x3feea11473eb0187,
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0x3c94ecfd5467c06b, 0x3feea1ed0130c132,
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0x3c65ebe1abd66c55, 0x3feea2f336cf4e62,
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0xbc88a1c52fb3cf42, 0x3feea427543e1a12,
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0xbc9369b6f13b3734, 0x3feea589994cce13,
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0xbc805e843a19ff1e, 0x3feea71a4623c7ad,
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0xbc94d450d872576e, 0x3feea8d99b4492ed,
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0x3c90ad675b0e8a00, 0x3feeaac7d98a6699,
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0x3c8db72fc1f0eab4, 0x3feeace5422aa0db,
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0xbc65b6609cc5e7ff, 0x3feeaf3216b5448c,
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0x3c7bf68359f35f44, 0x3feeb1ae99157736,
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|
0xbc93091fa71e3d83, 0x3feeb45b0b91ffc6,
|
||||||
|
0xbc5da9b88b6c1e29, 0x3feeb737b0cdc5e5,
|
||||||
|
0xbc6c23f97c90b959, 0x3feeba44cbc8520f,
|
||||||
|
0xbc92434322f4f9aa, 0x3feebd829fde4e50,
|
||||||
|
0xbc85ca6cd7668e4b, 0x3feec0f170ca07ba,
|
||||||
|
0x3c71affc2b91ce27, 0x3feec49182a3f090,
|
||||||
|
0x3c6dd235e10a73bb, 0x3feec86319e32323,
|
||||||
|
0xbc87c50422622263, 0x3feecc667b5de565,
|
||||||
|
0x3c8b1c86e3e231d5, 0x3feed09bec4a2d33,
|
||||||
|
0xbc91bbd1d3bcbb15, 0x3feed503b23e255d,
|
||||||
|
0x3c90cc319cee31d2, 0x3feed99e1330b358,
|
||||||
|
0x3c8469846e735ab3, 0x3feede6b5579fdbf,
|
||||||
|
0xbc82dfcd978e9db4, 0x3feee36bbfd3f37a,
|
||||||
|
0x3c8c1a7792cb3387, 0x3feee89f995ad3ad,
|
||||||
|
0xbc907b8f4ad1d9fa, 0x3feeee07298db666,
|
||||||
|
0xbc55c3d956dcaeba, 0x3feef3a2b84f15fb,
|
||||||
|
0xbc90a40e3da6f640, 0x3feef9728de5593a,
|
||||||
|
0xbc68d6f438ad9334, 0x3feeff76f2fb5e47,
|
||||||
|
0xbc91eee26b588a35, 0x3fef05b030a1064a,
|
||||||
|
0x3c74ffd70a5fddcd, 0x3fef0c1e904bc1d2,
|
||||||
|
0xbc91bdfbfa9298ac, 0x3fef12c25bd71e09,
|
||||||
|
0x3c736eae30af0cb3, 0x3fef199bdd85529c,
|
||||||
|
0x3c8ee3325c9ffd94, 0x3fef20ab5fffd07a,
|
||||||
|
0x3c84e08fd10959ac, 0x3fef27f12e57d14b,
|
||||||
|
0x3c63cdaf384e1a67, 0x3fef2f6d9406e7b5,
|
||||||
|
0x3c676b2c6c921968, 0x3fef3720dcef9069,
|
||||||
|
0xbc808a1883ccb5d2, 0x3fef3f0b555dc3fa,
|
||||||
|
0xbc8fad5d3ffffa6f, 0x3fef472d4a07897c,
|
||||||
|
0xbc900dae3875a949, 0x3fef4f87080d89f2,
|
||||||
|
0x3c74a385a63d07a7, 0x3fef5818dcfba487,
|
||||||
|
0xbc82919e2040220f, 0x3fef60e316c98398,
|
||||||
|
0x3c8e5a50d5c192ac, 0x3fef69e603db3285,
|
||||||
|
0x3c843a59ac016b4b, 0x3fef7321f301b460,
|
||||||
|
0xbc82d52107b43e1f, 0x3fef7c97337b9b5f,
|
||||||
|
0xbc892ab93b470dc9, 0x3fef864614f5a129,
|
||||||
|
0x3c74b604603a88d3, 0x3fef902ee78b3ff6,
|
||||||
|
0x3c83c5ec519d7271, 0x3fef9a51fbc74c83,
|
||||||
|
0xbc8ff7128fd391f0, 0x3fefa4afa2a490da,
|
||||||
|
0xbc8dae98e223747d, 0x3fefaf482d8e67f1,
|
||||||
|
0x3c8ec3bc41aa2008, 0x3fefba1bee615a27,
|
||||||
|
0x3c842b94c3a9eb32, 0x3fefc52b376bba97,
|
||||||
|
0x3c8a64a931d185ee, 0x3fefd0765b6e4540,
|
||||||
|
0xbc8e37bae43be3ed, 0x3fefdbfdad9cbe14,
|
||||||
|
0x3c77893b4d91cd9d, 0x3fefe7c1819e90d8,
|
||||||
|
0x3c5305c14160cc89, 0x3feff3c22b8f71f1,
|
||||||
|
},
|
||||||
|
};
|
||||||
|
|
@ -0,0 +1,80 @@
|
||||||
|
/*
|
||||||
|
* Single-precision e^x function.
|
||||||
|
*
|
||||||
|
* Copyright (c) 2017-2018, Arm Limited.
|
||||||
|
* SPDX-License-Identifier: MIT
|
||||||
|
*/
|
||||||
|
|
||||||
|
#include <math.h>
|
||||||
|
#include <stdint.h>
|
||||||
|
#include "libm.h"
|
||||||
|
#include "exp2f_data.h"
|
||||||
|
|
||||||
|
/*
|
||||||
|
EXP2F_TABLE_BITS = 5
|
||||||
|
EXP2F_POLY_ORDER = 3
|
||||||
|
|
||||||
|
ULP error: 0.502 (nearest rounding.)
|
||||||
|
Relative error: 1.69 * 2^-34 in [-ln2/64, ln2/64] (before rounding.)
|
||||||
|
Wrong count: 170635 (all nearest rounding wrong results with fma.)
|
||||||
|
Non-nearest ULP error: 1 (rounded ULP error)
|
||||||
|
*/
|
||||||
|
|
||||||
|
#define N (1 << EXP2F_TABLE_BITS)
|
||||||
|
#define InvLn2N __exp2f_data.invln2_scaled
|
||||||
|
#define T __exp2f_data.tab
|
||||||
|
#define C __exp2f_data.poly_scaled
|
||||||
|
|
||||||
|
static inline uint32_t top12(float x)
|
||||||
|
{
|
||||||
|
return asuint(x) >> 20;
|
||||||
|
}
|
||||||
|
|
||||||
|
float expf(float x)
|
||||||
|
{
|
||||||
|
uint32_t abstop;
|
||||||
|
uint64_t ki, t;
|
||||||
|
double_t kd, xd, z, r, r2, y, s;
|
||||||
|
|
||||||
|
xd = (double_t)x;
|
||||||
|
abstop = top12(x) & 0x7ff;
|
||||||
|
if (predict_false(abstop >= top12(88.0f))) {
|
||||||
|
/* |x| >= 88 or x is nan. */
|
||||||
|
if (asuint(x) == asuint(-INFINITY))
|
||||||
|
return 0.0f;
|
||||||
|
if (abstop >= top12(INFINITY))
|
||||||
|
return x + x;
|
||||||
|
if (x > 0x1.62e42ep6f) /* x > log(0x1p128) ~= 88.72 */
|
||||||
|
return __math_oflowf(0);
|
||||||
|
if (x < -0x1.9fe368p6f) /* x < log(0x1p-150) ~= -103.97 */
|
||||||
|
return __math_uflowf(0);
|
||||||
|
}
|
||||||
|
|
||||||
|
/* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k. */
|
||||||
|
z = InvLn2N * xd;
|
||||||
|
|
||||||
|
/* Round and convert z to int, the result is in [-150*N, 128*N] and
|
||||||
|
ideally ties-to-even rule is used, otherwise the magnitude of r
|
||||||
|
can be bigger which gives larger approximation error. */
|
||||||
|
#if TOINT_INTRINSICS
|
||||||
|
kd = roundtoint(z);
|
||||||
|
ki = converttoint(z);
|
||||||
|
#else
|
||||||
|
# define SHIFT __exp2f_data.shift
|
||||||
|
kd = eval_as_double(z + SHIFT);
|
||||||
|
ki = asuint64(kd);
|
||||||
|
kd -= SHIFT;
|
||||||
|
#endif
|
||||||
|
r = z - kd;
|
||||||
|
|
||||||
|
/* exp(x) = 2^(k/N) * 2^(r/N) ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
|
||||||
|
t = T[ki % N];
|
||||||
|
t += ki << (52 - EXP2F_TABLE_BITS);
|
||||||
|
s = asdouble(t);
|
||||||
|
z = C[0] * r + C[1];
|
||||||
|
r2 = r * r;
|
||||||
|
y = C[2] * r + 1;
|
||||||
|
y = z * r2 + y;
|
||||||
|
y = y * s;
|
||||||
|
return eval_as_float(y);
|
||||||
|
}
|
||||||
Loading…
Reference in New Issue