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123 lines
3.4 KiB
C
123 lines
3.4 KiB
C
/*
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* Double-precision log2(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 "log2_data.h"
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#define T __log2_data.tab
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#define T2 __log2_data.tab2
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#define B __log2_data.poly1
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#define A __log2_data.poly
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#define InvLn2hi __log2_data.invln2hi
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#define InvLn2lo __log2_data.invln2lo
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#define N (1 << LOG2_TABLE_BITS)
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#define OFF 0x3fe6000000000000
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/* Top 16 bits of a double. */
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static inline uint32_t top16(double x)
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{
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return asuint64(x) >> 48;
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}
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double log2(double x)
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{
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double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
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uint64_t ix, iz, tmp;
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uint32_t top;
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int k, i;
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ix = asuint64(x);
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top = top16(x);
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#define LO asuint64(1.0 - 0x1.5b51p-5)
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#define HI asuint64(1.0 + 0x1.6ab2p-5)
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if (predict_false(ix - LO < HI - LO)) {
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/* Handle close to 1.0 inputs separately. */
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/* Fix sign of zero with downward rounding when x==1. */
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if (WANT_ROUNDING && predict_false(ix == asuint64(1.0)))
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return 0;
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r = x - 1.0;
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#if __FP_FAST_FMA
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hi = r * InvLn2hi;
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lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi);
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#else
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double_t rhi, rlo;
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rhi = asdouble(asuint64(r) & -1ULL << 32);
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rlo = r - rhi;
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hi = rhi * InvLn2hi;
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lo = rlo * InvLn2hi + r * InvLn2lo;
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#endif
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r2 = r * r; /* rounding error: 0x1p-62. */
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r4 = r2 * r2;
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/* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
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p = r2 * (B[0] + r * B[1]);
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y = hi + p;
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lo += hi - y + p;
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lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) +
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r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
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y += lo;
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return eval_as_double(y);
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}
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if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
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/* x < 0x1p-1022 or inf or nan. */
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if (ix * 2 == 0)
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return __math_divzero(1);
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if (ix == asuint64(INFINITY)) /* log(inf) == inf. */
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return x;
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if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
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return __math_invalid(x);
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/* x is subnormal, normalize it. */
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ix = asuint64(x * 0x1p52);
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ix -= 52ULL << 52;
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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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tmp = ix - OFF;
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i = (tmp >> (52 - LOG2_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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invc = T[i].invc;
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logc = T[i].logc;
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z = asdouble(iz);
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kd = (double_t)k;
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/* log2(x) = log2(z/c) + log2(c) + k. */
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/* r ~= z/c - 1, |r| < 1/(2*N). */
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#if __FP_FAST_FMA
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/* rounding error: 0x1p-55/N. */
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r = __builtin_fma(z, invc, -1.0);
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t1 = r * InvLn2hi;
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t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1);
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#else
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double_t rhi, rlo;
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/* rounding error: 0x1p-55/N + 0x1p-65. */
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r = (z - T2[i].chi - T2[i].clo) * invc;
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rhi = asdouble(asuint64(r) & -1ULL << 32);
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rlo = r - rhi;
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t1 = rhi * InvLn2hi;
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t2 = rlo * InvLn2hi + r * InvLn2lo;
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#endif
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/* hi + lo = r/ln2 + log2(c) + k. */
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t3 = kd + logc;
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hi = t3 + t1;
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lo = t3 - hi + t1 + t2;
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/* log2(r+1) = r/ln2 + r^2*poly(r). */
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/* Evaluation is optimized assuming superscalar pipelined execution. */
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r2 = r * r; /* rounding error: 0x1p-54/N^2. */
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r4 = r2 * r2;
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/* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
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~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
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p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
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y = lo + r2 * p + hi;
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return eval_as_double(y);
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}
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