/* * Double-precision log2(x) function. * * Copyright (c) 2018, Arm Limited. * SPDX-License-Identifier: MIT */ #include <math.h> #include <stdint.h> #include "libm.h" #include "log2_data.h" #define T __log2_data.tab #define T2 __log2_data.tab2 #define B __log2_data.poly1 #define A __log2_data.poly #define InvLn2hi __log2_data.invln2hi #define InvLn2lo __log2_data.invln2lo #define N (1 << LOG2_TABLE_BITS) #define OFF 0x3fe6000000000000 /* Top 16 bits of a double. */ static inline uint32_t top16(double x) { return asuint64(x) >> 48; } double log2(double x) { double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p; uint64_t ix, iz, tmp; uint32_t top; int k, i; ix = asuint64(x); top = top16(x); #define LO asuint64(1.0 - 0x1.5b51p-5) #define HI asuint64(1.0 + 0x1.6ab2p-5) if (predict_false(ix - LO < HI - LO)) { /* Handle close to 1.0 inputs separately. */ /* Fix sign of zero with downward rounding when x==1. */ if (WANT_ROUNDING && predict_false(ix == asuint64(1.0))) return 0; r = x - 1.0; #if __FP_FAST_FMA hi = r * InvLn2hi; lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi); #else double_t rhi, rlo; rhi = asdouble(asuint64(r) & -1ULL << 32); rlo = r - rhi; hi = rhi * InvLn2hi; lo = rlo * InvLn2hi + r * InvLn2lo; #endif r2 = r * r; /* rounding error: 0x1p-62. */ r4 = r2 * r2; /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */ p = r2 * (B[0] + r * B[1]); y = hi + p; lo += hi - y + p; lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9]))); y += lo; return eval_as_double(y); } if (predict_false(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 - LOG2_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); kd = (double_t)k; /* log2(x) = log2(z/c) + log2(c) + k. */ /* r ~= z/c - 1, |r| < 1/(2*N). */ #if __FP_FAST_FMA /* rounding error: 0x1p-55/N. */ r = __builtin_fma(z, invc, -1.0); t1 = r * InvLn2hi; t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1); #else double_t rhi, rlo; /* rounding error: 0x1p-55/N + 0x1p-65. */ r = (z - T2[i].chi - T2[i].clo) * invc; rhi = asdouble(asuint64(r) & -1ULL << 32); rlo = r - rhi; t1 = rhi * InvLn2hi; t2 = rlo * InvLn2hi + r * InvLn2lo; #endif /* hi + lo = r/ln2 + log2(c) + k. */ t3 = kd + logc; hi = t3 + t1; lo = t3 - hi + t1 + t2; /* log2(r+1) = r/ln2 + r^2*poly(r). */ /* Evaluation is optimized assuming superscalar pipelined execution. */ r2 = r * r; /* rounding error: 0x1p-54/N^2. */ r4 = r2 * r2; /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma). ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */ p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]); y = lo + r2 * p + hi; return eval_as_double(y); }