1/*
  2 * Double-precision log2(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 "log2_data.h"
 12
 13#define T __log2_data.tab
 14#define T2 __log2_data.tab2
 15#define B __log2_data.poly1
 16#define A __log2_data.poly
 17#define InvLn2hi __log2_data.invln2hi
 18#define InvLn2lo __log2_data.invln2lo
 19#define N (1 << LOG2_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 log2(double x)
 29{
 30	double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
 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 - 0x1.5b51p-5)
 38#define HI asuint64(1.0 + 0x1.6ab2p-5)
 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#if __FP_FAST_FMA
 46		hi = r * InvLn2hi;
 47		lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi);
 48#else
 49		double_t rhi, rlo;
 50		rhi = asdouble(asuint64(r) & -1ULL << 32);
 51		rlo = r - rhi;
 52		hi = rhi * InvLn2hi;
 53		lo = rlo * InvLn2hi + r * InvLn2lo;
 54#endif
 55		r2 = r * r; /* rounding error: 0x1p-62.  */
 56		r4 = r2 * r2;
 57		/* Worst-case error is less than 0.54 ULP (0.55 ULP without fma).  */
 58		p = r2 * (B[0] + r * B[1]);
 59		y = hi + p;
 60		lo += hi - y + p;
 61		lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) +
 62			    r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
 63		y += lo;
 64		return eval_as_double(y);
 65	}
 66	if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
 67		/* x < 0x1p-1022 or inf or nan.  */
 68		if (ix * 2 == 0)
 69			return __math_divzero(1);
 70		if (ix == asuint64(INFINITY)) /* log(inf) == inf.  */
 71			return x;
 72		if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
 73			return __math_invalid(x);
 74		/* x is subnormal, normalize it.  */
 75		ix = asuint64(x * 0x1p52);
 76		ix -= 52ULL << 52;
 77	}
 78
 79	/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
 80	   The range is split into N subintervals.
 81	   The ith subinterval contains z and c is near its center.  */
 82	tmp = ix - OFF;
 83	i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
 84	k = (int64_t)tmp >> 52; /* arithmetic shift */
 85	iz = ix - (tmp & 0xfffULL << 52);
 86	invc = T[i].invc;
 87	logc = T[i].logc;
 88	z = asdouble(iz);
 89	kd = (double_t)k;
 90
 91	/* log2(x) = log2(z/c) + log2(c) + k.  */
 92	/* r ~= z/c - 1, |r| < 1/(2*N).  */
 93#if __FP_FAST_FMA
 94	/* rounding error: 0x1p-55/N.  */
 95	r = __builtin_fma(z, invc, -1.0);
 96	t1 = r * InvLn2hi;
 97	t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1);
 98#else
 99	double_t rhi, rlo;
100	/* rounding error: 0x1p-55/N + 0x1p-65.  */
101	r = (z - T2[i].chi - T2[i].clo) * invc;
102	rhi = asdouble(asuint64(r) & -1ULL << 32);
103	rlo = r - rhi;
104	t1 = rhi * InvLn2hi;
105	t2 = rlo * InvLn2hi + r * InvLn2lo;
106#endif
107
108	/* hi + lo = r/ln2 + log2(c) + k.  */
109	t3 = kd + logc;
110	hi = t3 + t1;
111	lo = t3 - hi + t1 + t2;
112
113	/* log2(r+1) = r/ln2 + r^2*poly(r).  */
114	/* Evaluation is optimized assuming superscalar pipelined execution.  */
115	r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
116	r4 = r2 * r2;
117	/* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
118	   ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma).  */
119	p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
120	y = lo + r2 * p + hi;
121	return eval_as_double(y);
122}