1/*
  2 * Double-precision log(x) function.
  3 *
  4 * Copyright (c) 2018, Arm Limited.
  5 * SPDX-License-Identifier: MIT
  6 */
  7
  8#include <math.h>
  9#include <stdint.h>
 10#include "libm.h"
 11#include "log_data.h"
 12
 13#define T __log_data.tab
 14#define T2 __log_data.tab2
 15#define B __log_data.poly1
 16#define A __log_data.poly
 17#define Ln2hi __log_data.ln2hi
 18#define Ln2lo __log_data.ln2lo
 19#define N (1 << LOG_TABLE_BITS)
 20#define OFF 0x3fe6000000000000
 21
 22/* Top 16 bits of a double.  */
 23static inline uint32_t top16(double x)
 24{
 25	return asuint64(x) >> 48;
 26}
 27
 28double log(double x)
 29{
 30	double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
 31	uint64_t ix, iz, tmp;
 32	uint32_t top;
 33	int k, i;
 34
 35	ix = asuint64(x);
 36	top = top16(x);
 37#define LO asuint64(1.0 - 0x1p-4)
 38#define HI asuint64(1.0 + 0x1.09p-4)
 39	if (predict_false(ix - LO < HI - LO)) {
 40		/* Handle close to 1.0 inputs separately.  */
 41		/* Fix sign of zero with downward rounding when x==1.  */
 42		if (WANT_ROUNDING && predict_false(ix == asuint64(1.0)))
 43			return 0;
 44		r = x - 1.0;
 45		r2 = r * r;
 46		r3 = r * r2;
 47		y = r3 *
 48		    (B[1] + r * B[2] + r2 * B[3] +
 49		     r3 * (B[4] + r * B[5] + r2 * B[6] +
 50			   r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
 51		/* Worst-case error is around 0.507 ULP.  */
 52		w = r * 0x1p27;
 53		double_t rhi = r + w - w;
 54		double_t rlo = r - rhi;
 55		w = rhi * rhi * B[0]; /* B[0] == -0.5.  */
 56		hi = r + w;
 57		lo = r - hi + w;
 58		lo += B[0] * rlo * (rhi + r);
 59		y += lo;
 60		y += hi;
 61		return eval_as_double(y);
 62	}
 63	if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
 64		/* x < 0x1p-1022 or inf or nan.  */
 65		if (ix * 2 == 0)
 66			return __math_divzero(1);
 67		if (ix == asuint64(INFINITY)) /* log(inf) == inf.  */
 68			return x;
 69		if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
 70			return __math_invalid(x);
 71		/* x is subnormal, normalize it.  */
 72		ix = asuint64(x * 0x1p52);
 73		ix -= 52ULL << 52;
 74	}
 75
 76	/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
 77	   The range is split into N subintervals.
 78	   The ith subinterval contains z and c is near its center.  */
 79	tmp = ix - OFF;
 80	i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
 81	k = (int64_t)tmp >> 52; /* arithmetic shift */
 82	iz = ix - (tmp & 0xfffULL << 52);
 83	invc = T[i].invc;
 84	logc = T[i].logc;
 85	z = asdouble(iz);
 86
 87	/* log(x) = log1p(z/c-1) + log(c) + k*Ln2.  */
 88	/* r ~= z/c - 1, |r| < 1/(2*N).  */
 89#if __FP_FAST_FMA
 90	/* rounding error: 0x1p-55/N.  */
 91	r = __builtin_fma(z, invc, -1.0);
 92#else
 93	/* rounding error: 0x1p-55/N + 0x1p-66.  */
 94	r = (z - T2[i].chi - T2[i].clo) * invc;
 95#endif
 96	kd = (double_t)k;
 97
 98	/* hi + lo = r + log(c) + k*Ln2.  */
 99	w = kd * Ln2hi + logc;
100	hi = w + r;
101	lo = w - hi + r + kd * Ln2lo;
102
103	/* log(x) = lo + (log1p(r) - r) + hi.  */
104	r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
105	/* Worst case error if |y| > 0x1p-5:
106	   0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
107	   Worst case error if |y| > 0x1p-4:
108	   0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma).  */
109	y = lo + r2 * A[0] +
110	    r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
111	return eval_as_double(y);
112}