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164 lines
4.6 KiB
164 lines
4.6 KiB
/*
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* Double-precision log(x) function.
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*
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* Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
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* See https://llvm.org/LICENSE.txt for license information.
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* SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
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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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#define T __log_data.tab
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#define T2 __log_data.tab2
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#define B __log_data.poly1
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#define A __log_data.poly
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#define Ln2hi __log_data.ln2hi
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#define Ln2lo __log_data.ln2lo
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#define N (1 << LOG_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
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top16 (double x)
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{
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return asuint64 (x) >> 48;
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}
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double
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log (double x)
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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 w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
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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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#if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11
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# define LO asuint64 (1.0 - 0x1p-5)
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# define HI asuint64 (1.0 + 0x1.1p-5)
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#elif LOG_POLY1_ORDER == 12
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# define LO asuint64 (1.0 - 0x1p-4)
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# define HI asuint64 (1.0 + 0x1.09p-4)
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#endif
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if (unlikely (ix - LO < HI - LO))
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{
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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 && unlikely (ix == asuint64 (1.0)))
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return 0;
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r = x - 1.0;
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r2 = r * r;
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r3 = r * r2;
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#if LOG_POLY1_ORDER == 10
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/* Worst-case error is around 0.516 ULP. */
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y = r3 * (B[1] + r * B[2] + r2 * B[3]
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+ r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8])));
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w = B[0] * r2; /* B[0] == -0.5. */
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hi = r + w;
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y += r - hi + w;
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y += hi;
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#elif LOG_POLY1_ORDER == 11
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/* Worst-case error is around 0.516 ULP. */
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y = r3 * (B[1] + r * B[2]
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+ r2 * (B[3] + r * B[4] + r2 * B[5]
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+ r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9])));
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w = B[0] * r2; /* B[0] == -0.5. */
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hi = r + w;
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y += r - hi + w;
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y += hi;
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#elif LOG_POLY1_ORDER == 12
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y = r3 * (B[1] + r * B[2] + r2 * B[3]
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+ r3 * (B[4] + r * B[5] + r2 * B[6]
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+ r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
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# if N <= 64
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/* Worst-case error is around 0.532 ULP. */
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w = B[0] * r2; /* B[0] == -0.5. */
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hi = r + w;
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y += r - hi + w;
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y += hi;
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# else
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/* Worst-case error is around 0.507 ULP. */
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w = r * 0x1p27;
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double_t rhi = r + w - w;
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double_t rlo = r - rhi;
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w = rhi * rhi * B[0]; /* B[0] == -0.5. */
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hi = r + w;
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lo = r - hi + w;
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lo += B[0] * rlo * (rhi + r);
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y += lo;
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y += hi;
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# endif
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#endif
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return eval_as_double (y);
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}
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if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
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{
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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 - 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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invc = T[i].invc;
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logc = T[i].logc;
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z = asdouble (iz);
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/* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
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/* r ~= z/c - 1, |r| < 1/(2*N). */
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#if HAVE_FAST_FMA
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/* rounding error: 0x1p-55/N. */
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r = fma (z, invc, -1.0);
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#else
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/* rounding error: 0x1p-55/N + 0x1p-66. */
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r = (z - T2[i].chi - T2[i].clo) * invc;
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#endif
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kd = (double_t) k;
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/* hi + lo = r + log(c) + k*Ln2. */
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w = kd * Ln2hi + logc;
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hi = w + r;
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lo = w - hi + r + kd * Ln2lo;
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/* log(x) = lo + (log1p(r) - r) + hi. */
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r2 = r * r; /* rounding error: 0x1p-54/N^2. */
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/* Worst case error if |y| > 0x1p-5:
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0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
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Worst case error if |y| > 0x1p-4:
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0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
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#if LOG_POLY_ORDER == 6
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y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
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#elif LOG_POLY_ORDER == 7
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y = lo
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+ r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])
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+ r2 * r2 * (A[4] + r * A[5]))
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+ hi;
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#endif
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return eval_as_double (y);
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}
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#if USE_GLIBC_ABI
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strong_alias (log, __log_finite)
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hidden_alias (log, __ieee754_log)
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# if LDBL_MANT_DIG == 53
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long double logl (long double x) { return log (x); }
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# endif
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#endif
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