131914882SAlex Richardson /* 231914882SAlex Richardson * Double-precision log2(x) function. 331914882SAlex Richardson * 4*f3087befSAndrew Turner * Copyright (c) 2018-2024, Arm Limited. 5072a4ba8SAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 631914882SAlex Richardson */ 731914882SAlex Richardson 831914882SAlex Richardson #include <float.h> 931914882SAlex Richardson #include <math.h> 1031914882SAlex Richardson #include <stdint.h> 1131914882SAlex Richardson #include "math_config.h" 12*f3087befSAndrew Turner #include "test_defs.h" 13*f3087befSAndrew Turner #include "test_sig.h" 1431914882SAlex Richardson 1531914882SAlex Richardson #define T __log2_data.tab 1631914882SAlex Richardson #define T2 __log2_data.tab2 1731914882SAlex Richardson #define B __log2_data.poly1 1831914882SAlex Richardson #define A __log2_data.poly 1931914882SAlex Richardson #define InvLn2hi __log2_data.invln2hi 2031914882SAlex Richardson #define InvLn2lo __log2_data.invln2lo 2131914882SAlex Richardson #define N (1 << LOG2_TABLE_BITS) 2231914882SAlex Richardson #define OFF 0x3fe6000000000000 2331914882SAlex Richardson 2431914882SAlex Richardson /* Top 16 bits of a double. */ 2531914882SAlex Richardson static inline uint32_t 2631914882SAlex Richardson top16 (double x) 2731914882SAlex Richardson { 2831914882SAlex Richardson return asuint64 (x) >> 48; 2931914882SAlex Richardson } 3031914882SAlex Richardson 3131914882SAlex Richardson double 3231914882SAlex Richardson log2 (double x) 3331914882SAlex Richardson { 3431914882SAlex Richardson /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 3531914882SAlex Richardson double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p; 3631914882SAlex Richardson uint64_t ix, iz, tmp; 3731914882SAlex Richardson uint32_t top; 3831914882SAlex Richardson int k, i; 3931914882SAlex Richardson 4031914882SAlex Richardson ix = asuint64 (x); 4131914882SAlex Richardson top = top16 (x); 4231914882SAlex Richardson 4331914882SAlex Richardson #if LOG2_POLY1_ORDER == 11 4431914882SAlex Richardson # define LO asuint64 (1.0 - 0x1.5b51p-5) 4531914882SAlex Richardson # define HI asuint64 (1.0 + 0x1.6ab2p-5) 4631914882SAlex Richardson #endif 4731914882SAlex Richardson if (unlikely (ix - LO < HI - LO)) 4831914882SAlex Richardson { 4931914882SAlex Richardson /* Handle close to 1.0 inputs separately. */ 5031914882SAlex Richardson /* Fix sign of zero with downward rounding when x==1. */ 5131914882SAlex Richardson if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0))) 5231914882SAlex Richardson return 0; 5331914882SAlex Richardson r = x - 1.0; 5431914882SAlex Richardson #if HAVE_FAST_FMA 5531914882SAlex Richardson hi = r * InvLn2hi; 5631914882SAlex Richardson lo = r * InvLn2lo + fma (r, InvLn2hi, -hi); 5731914882SAlex Richardson #else 5831914882SAlex Richardson double_t rhi, rlo; 5931914882SAlex Richardson rhi = asdouble (asuint64 (r) & -1ULL << 32); 6031914882SAlex Richardson rlo = r - rhi; 6131914882SAlex Richardson hi = rhi * InvLn2hi; 6231914882SAlex Richardson lo = rlo * InvLn2hi + r * InvLn2lo; 6331914882SAlex Richardson #endif 6431914882SAlex Richardson r2 = r * r; /* rounding error: 0x1p-62. */ 6531914882SAlex Richardson r4 = r2 * r2; 6631914882SAlex Richardson #if LOG2_POLY1_ORDER == 11 6731914882SAlex Richardson /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */ 6831914882SAlex Richardson p = r2 * (B[0] + r * B[1]); 6931914882SAlex Richardson y = hi + p; 7031914882SAlex Richardson lo += hi - y + p; 7131914882SAlex Richardson lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) 7231914882SAlex Richardson + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9]))); 7331914882SAlex Richardson y += lo; 7431914882SAlex Richardson #endif 7531914882SAlex Richardson return eval_as_double (y); 7631914882SAlex Richardson } 7731914882SAlex Richardson if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010)) 7831914882SAlex Richardson { 7931914882SAlex Richardson /* x < 0x1p-1022 or inf or nan. */ 8031914882SAlex Richardson if (ix * 2 == 0) 8131914882SAlex Richardson return __math_divzero (1); 8231914882SAlex Richardson if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */ 8331914882SAlex Richardson return x; 8431914882SAlex Richardson if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 8531914882SAlex Richardson return __math_invalid (x); 8631914882SAlex Richardson /* x is subnormal, normalize it. */ 8731914882SAlex Richardson ix = asuint64 (x * 0x1p52); 8831914882SAlex Richardson ix -= 52ULL << 52; 8931914882SAlex Richardson } 9031914882SAlex Richardson 9131914882SAlex Richardson /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 9231914882SAlex Richardson The range is split into N subintervals. 9331914882SAlex Richardson The ith subinterval contains z and c is near its center. */ 9431914882SAlex Richardson tmp = ix - OFF; 9531914882SAlex Richardson i = (tmp >> (52 - LOG2_TABLE_BITS)) % N; 9631914882SAlex Richardson k = (int64_t) tmp >> 52; /* arithmetic shift */ 9731914882SAlex Richardson iz = ix - (tmp & 0xfffULL << 52); 9831914882SAlex Richardson invc = T[i].invc; 9931914882SAlex Richardson logc = T[i].logc; 10031914882SAlex Richardson z = asdouble (iz); 10131914882SAlex Richardson kd = (double_t) k; 10231914882SAlex Richardson 10331914882SAlex Richardson /* log2(x) = log2(z/c) + log2(c) + k. */ 10431914882SAlex Richardson /* r ~= z/c - 1, |r| < 1/(2*N). */ 10531914882SAlex Richardson #if HAVE_FAST_FMA 10631914882SAlex Richardson /* rounding error: 0x1p-55/N. */ 10731914882SAlex Richardson r = fma (z, invc, -1.0); 10831914882SAlex Richardson t1 = r * InvLn2hi; 10931914882SAlex Richardson t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1); 11031914882SAlex Richardson #else 11131914882SAlex Richardson double_t rhi, rlo; 11231914882SAlex Richardson /* rounding error: 0x1p-55/N + 0x1p-65. */ 11331914882SAlex Richardson r = (z - T2[i].chi - T2[i].clo) * invc; 11431914882SAlex Richardson rhi = asdouble (asuint64 (r) & -1ULL << 32); 11531914882SAlex Richardson rlo = r - rhi; 11631914882SAlex Richardson t1 = rhi * InvLn2hi; 11731914882SAlex Richardson t2 = rlo * InvLn2hi + r * InvLn2lo; 11831914882SAlex Richardson #endif 11931914882SAlex Richardson 12031914882SAlex Richardson /* hi + lo = r/ln2 + log2(c) + k. */ 12131914882SAlex Richardson t3 = kd + logc; 12231914882SAlex Richardson hi = t3 + t1; 12331914882SAlex Richardson lo = t3 - hi + t1 + t2; 12431914882SAlex Richardson 12531914882SAlex Richardson /* log2(r+1) = r/ln2 + r^2*poly(r). */ 12631914882SAlex Richardson /* Evaluation is optimized assuming superscalar pipelined execution. */ 12731914882SAlex Richardson r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 12831914882SAlex Richardson r4 = r2 * r2; 12931914882SAlex Richardson #if LOG2_POLY_ORDER == 7 13031914882SAlex Richardson /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma). 13131914882SAlex Richardson ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */ 13231914882SAlex Richardson p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]); 13331914882SAlex Richardson y = lo + r2 * p + hi; 13431914882SAlex Richardson #endif 13531914882SAlex Richardson return eval_as_double (y); 13631914882SAlex Richardson } 13731914882SAlex Richardson #if USE_GLIBC_ABI 13831914882SAlex Richardson strong_alias (log2, __log2_finite) 13931914882SAlex Richardson hidden_alias (log2, __ieee754_log2) 14031914882SAlex Richardson # if LDBL_MANT_DIG == 53 14131914882SAlex Richardson long double log2l (long double x) { return log2 (x); } 14231914882SAlex Richardson # endif 14331914882SAlex Richardson #endif 144*f3087befSAndrew Turner 145*f3087befSAndrew Turner TEST_SIG (S, D, 1, log2, 0.01, 11.1) 146*f3087befSAndrew Turner TEST_ULP (log2, 0.05) 147*f3087befSAndrew Turner TEST_ULP_NONNEAREST (log2, 0.5) 148*f3087befSAndrew Turner TEST_INTERVAL (log2, 0, 0xffff000000000000, 10000) 149*f3087befSAndrew Turner TEST_INTERVAL (log2, 0x1p-4, 0x1p4, 40000) 150*f3087befSAndrew Turner TEST_INTERVAL (log2, 0, inf, 40000) 151