| /* |
| * Double-precision log(x) function. |
| * |
| * Copyright (c) 2018-2023, Arm Limited. |
| * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception |
| */ |
| |
| #include <float.h> |
| #include <math.h> |
| #include <stdint.h> |
| #include "math_config.h" |
| |
| #define T __log_data.tab |
| #define T2 __log_data.tab2 |
| #define B __log_data.poly1 |
| #define A __log_data.poly |
| #define Ln2hi __log_data.ln2hi |
| #define Ln2lo __log_data.ln2lo |
| #define N (1 << LOG_TABLE_BITS) |
| #define OFF 0x3fe6000000000000 |
| |
| /* Top 16 bits of a double. */ |
| static inline uint32_t |
| top16 (double x) |
| { |
| return asuint64 (x) >> 48; |
| } |
| |
| double |
| optr_aor_log_f64 (double x) |
| { |
| /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ |
| double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo; |
| uint64_t ix, iz, tmp; |
| uint32_t top; |
| int k, i; |
| |
| ix = asuint64 (x); |
| top = top16 (x); |
| |
| #if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11 |
| #define LO asuint64 (1.0 - 0x1p-5) |
| #define HI asuint64 (1.0 + 0x1.1p-5) |
| #elif LOG_POLY1_ORDER == 12 |
| #define LO asuint64 (1.0 - 0x1p-4) |
| #define HI asuint64 (1.0 + 0x1.09p-4) |
| #endif |
| if (unlikely (ix - LO < HI - LO)) |
| { |
| /* Handle close to 1.0 inputs separately. */ |
| /* Fix sign of zero with downward rounding when x==1. */ |
| if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0))) |
| return 0; |
| r = x - 1.0; |
| r2 = r * r; |
| r3 = r * r2; |
| #if LOG_POLY1_ORDER == 10 |
| /* Worst-case error is around 0.516 ULP. */ |
| y = r3 |
| * (B[1] + r * B[2] + r2 * B[3] |
| + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8]))); |
| w = B[0] * r2; /* B[0] == -0.5. */ |
| hi = r + w; |
| y += r - hi + w; |
| y += hi; |
| #elif LOG_POLY1_ORDER == 11 |
| /* Worst-case error is around 0.516 ULP. */ |
| y = r3 |
| * (B[1] + r * B[2] |
| + r2 |
| * (B[3] + r * B[4] + r2 * B[5] |
| + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9]))); |
| w = B[0] * r2; /* B[0] == -0.5. */ |
| hi = r + w; |
| y += r - hi + w; |
| y += hi; |
| #elif LOG_POLY1_ORDER == 12 |
| y = r3 |
| * (B[1] + r * B[2] + r2 * B[3] |
| + r3 |
| * (B[4] + r * B[5] + r2 * B[6] |
| + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10]))); |
| #if N <= 64 |
| /* Worst-case error is around 0.532 ULP. */ |
| w = B[0] * r2; /* B[0] == -0.5. */ |
| hi = r + w; |
| y += r - hi + w; |
| y += hi; |
| #else |
| /* Worst-case error is around 0.507 ULP. */ |
| w = r * 0x1p27; |
| double_t rhi = r + w - w; |
| double_t rlo = r - rhi; |
| w = rhi * rhi * B[0]; /* B[0] == -0.5. */ |
| hi = r + w; |
| lo = r - hi + w; |
| lo += B[0] * rlo * (rhi + r); |
| y += lo; |
| y += hi; |
| #endif |
| #endif |
| return eval_as_double (y); |
| } |
| if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010)) |
| { |
| /* x < 0x1p-1022 or inf or nan. */ |
| if (ix * 2 == 0) |
| return __math_divzero (1); |
| if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */ |
| return x; |
| if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) |
| return __math_invalid (x); |
| /* x is subnormal, normalize it. */ |
| ix = asuint64 (x * 0x1p52); |
| ix -= 52ULL << 52; |
| } |
| |
| /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. |
| The range is split into N subintervals. |
| The ith subinterval contains z and c is near its center. */ |
| tmp = ix - OFF; |
| i = (tmp >> (52 - LOG_TABLE_BITS)) % N; |
| k = (int64_t) tmp >> 52; /* arithmetic shift */ |
| iz = ix - (tmp & 0xfffULL << 52); |
| invc = T[i].invc; |
| logc = T[i].logc; |
| z = asdouble (iz); |
| |
| /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */ |
| /* r ~= z/c - 1, |r| < 1/(2*N). */ |
| #if HAVE_FAST_FMA |
| /* rounding error: 0x1p-55/N. */ |
| r = fma (z, invc, -1.0); |
| #else |
| /* rounding error: 0x1p-55/N + 0x1p-66. */ |
| r = (z - T2[i].chi - T2[i].clo) * invc; |
| #endif |
| kd = (double_t) k; |
| |
| /* hi + lo = r + log(c) + k*Ln2. */ |
| w = kd * Ln2hi + logc; |
| hi = w + r; |
| lo = w - hi + r + kd * Ln2lo; |
| |
| /* log(x) = lo + (log1p(r) - r) + hi. */ |
| r2 = r * r; /* rounding error: 0x1p-54/N^2. */ |
| /* Worst case error if |y| > 0x1p-5: |
| 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma) |
| Worst case error if |y| > 0x1p-4: |
| 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */ |
| #if LOG_POLY_ORDER == 6 |
| y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi; |
| #elif LOG_POLY_ORDER == 7 |
| y = lo |
| + r2 |
| * (A[0] + r * A[1] + r2 * (A[2] + r * A[3]) |
| + r2 * r2 * (A[4] + r * A[5])) |
| + hi; |
| #endif |
| return eval_as_double (y); |
| } |