1/*
2 * Double-precision log2(x) function.
3 *
4 * Copyright (c) 2018-2019, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8#include <float.h>
9#include <math.h>
10#include <stdint.h>
11#include "math_config.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
24top16 (double x)
25{
26  return asuint64 (x) >> 48;
27}
28
29double
30log2 (double x)
31{
32  /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
33  double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
34  uint64_t ix, iz, tmp;
35  uint32_t top;
36  int k, i;
37
38  ix = asuint64 (x);
39  top = top16 (x);
40
41#if LOG2_POLY1_ORDER == 11
42# define LO asuint64 (1.0 - 0x1.5b51p-5)
43# define HI asuint64 (1.0 + 0x1.6ab2p-5)
44#endif
45  if (unlikely (ix - LO < HI - LO))
46    {
47      /* Handle close to 1.0 inputs separately.  */
48      /* Fix sign of zero with downward rounding when x==1.  */
49      if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
50	return 0;
51      r = x - 1.0;
52#if HAVE_FAST_FMA
53      hi = r * InvLn2hi;
54      lo = r * InvLn2lo + fma (r, InvLn2hi, -hi);
55#else
56      double_t rhi, rlo;
57      rhi = asdouble (asuint64 (r) & -1ULL << 32);
58      rlo = r - rhi;
59      hi = rhi * InvLn2hi;
60      lo = rlo * InvLn2hi + r * InvLn2lo;
61#endif
62      r2 = r * r; /* rounding error: 0x1p-62.  */
63      r4 = r2 * r2;
64#if LOG2_POLY1_ORDER == 11
65      /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma).  */
66      p = r2 * (B[0] + r * B[1]);
67      y = hi + p;
68      lo += hi - y + p;
69      lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5])
70		  + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
71      y += lo;
72#endif
73      return eval_as_double (y);
74    }
75  if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
76    {
77      /* x < 0x1p-1022 or inf or nan.  */
78      if (ix * 2 == 0)
79	return __math_divzero (1);
80      if (ix == asuint64 (INFINITY)) /* log(inf) == inf.  */
81	return x;
82      if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
83	return __math_invalid (x);
84      /* x is subnormal, normalize it.  */
85      ix = asuint64 (x * 0x1p52);
86      ix -= 52ULL << 52;
87    }
88
89  /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
90     The range is split into N subintervals.
91     The ith subinterval contains z and c is near its center.  */
92  tmp = ix - OFF;
93  i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
94  k = (int64_t) tmp >> 52; /* arithmetic shift */
95  iz = ix - (tmp & 0xfffULL << 52);
96  invc = T[i].invc;
97  logc = T[i].logc;
98  z = asdouble (iz);
99  kd = (double_t) k;
100
101  /* log2(x) = log2(z/c) + log2(c) + k.  */
102  /* r ~= z/c - 1, |r| < 1/(2*N).  */
103#if HAVE_FAST_FMA
104  /* rounding error: 0x1p-55/N.  */
105  r = fma (z, invc, -1.0);
106  t1 = r * InvLn2hi;
107  t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1);
108#else
109  double_t rhi, rlo;
110  /* rounding error: 0x1p-55/N + 0x1p-65.  */
111  r = (z - T2[i].chi - T2[i].clo) * invc;
112  rhi = asdouble (asuint64 (r) & -1ULL << 32);
113  rlo = r - rhi;
114  t1 = rhi * InvLn2hi;
115  t2 = rlo * InvLn2hi + r * InvLn2lo;
116#endif
117
118  /* hi + lo = r/ln2 + log2(c) + k.  */
119  t3 = kd + logc;
120  hi = t3 + t1;
121  lo = t3 - hi + t1 + t2;
122
123  /* log2(r+1) = r/ln2 + r^2*poly(r).  */
124  /* Evaluation is optimized assuming superscalar pipelined execution.  */
125  r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
126  r4 = r2 * r2;
127#if LOG2_POLY_ORDER == 7
128  /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
129     ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma).  */
130  p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
131  y = lo + r2 * p + hi;
132#endif
133  return eval_as_double (y);
134}
135#if USE_GLIBC_ABI
136strong_alias (log2, __log2_finite)
137hidden_alias (log2, __ieee754_log2)
138# if LDBL_MANT_DIG == 53
139long double log2l (long double x) { return log2 (x); }
140# endif
141#endif
142