2 * Double-precision log(x) function.
4 * Copyright (c) 2018-2019, Arm Limited.
5 * SPDX-License-Identifier: MIT
11 #include "math_config.h"
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
22 /* Top 16 bits of a double. */
23 static inline uint32_t
26 return asuint64 (x) >> 48;
32 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
33 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
41 #if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11
42 # define LO asuint64 (1.0 - 0x1p-5)
43 # define HI asuint64 (1.0 + 0x1.1p-5)
44 #elif LOG_POLY1_ORDER == 12
45 # define LO asuint64 (1.0 - 0x1p-4)
46 # define HI asuint64 (1.0 + 0x1.09p-4)
48 if (unlikely (ix - LO < HI - LO))
50 /* Handle close to 1.0 inputs separately. */
51 /* Fix sign of zero with downward rounding when x==1. */
52 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
57 #if LOG_POLY1_ORDER == 10
58 /* Worst-case error is around 0.516 ULP. */
59 y = r3 * (B[1] + r * B[2] + r2 * B[3]
60 + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8])));
61 w = B[0] * r2; /* B[0] == -0.5. */
65 #elif LOG_POLY1_ORDER == 11
66 /* Worst-case error is around 0.516 ULP. */
67 y = r3 * (B[1] + r * B[2]
68 + r2 * (B[3] + r * B[4] + r2 * B[5]
69 + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9])));
70 w = B[0] * r2; /* B[0] == -0.5. */
74 #elif LOG_POLY1_ORDER == 12
75 y = r3 * (B[1] + r * B[2] + r2 * B[3]
76 + r3 * (B[4] + r * B[5] + r2 * B[6]
77 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
79 /* Worst-case error is around 0.532 ULP. */
80 w = B[0] * r2; /* B[0] == -0.5. */
85 /* Worst-case error is around 0.507 ULP. */
87 double_t rhi = r + w - w;
88 double_t rlo = r - rhi;
89 w = rhi * rhi * B[0]; /* B[0] == -0.5. */
92 lo += B[0] * rlo * (rhi + r);
97 return eval_as_double (y);
99 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
101 /* x < 0x1p-1022 or inf or nan. */
103 return __math_divzero (1);
104 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
106 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
107 return __math_invalid (x);
108 /* x is subnormal, normalize it. */
109 ix = asuint64 (x * 0x1p52);
113 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
114 The range is split into N subintervals.
115 The ith subinterval contains z and c is near its center. */
117 i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
118 k = (int64_t) tmp >> 52; /* arithmetic shift */
119 iz = ix - (tmp & 0xfffULL << 52);
124 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
125 /* r ~= z/c - 1, |r| < 1/(2*N). */
127 /* rounding error: 0x1p-55/N. */
128 r = fma (z, invc, -1.0);
130 /* rounding error: 0x1p-55/N + 0x1p-66. */
131 r = (z - T2[i].chi - T2[i].clo) * invc;
135 /* hi + lo = r + log(c) + k*Ln2. */
136 w = kd * Ln2hi + logc;
138 lo = w - hi + r + kd * Ln2lo;
140 /* log(x) = lo + (log1p(r) - r) + hi. */
141 r2 = r * r; /* rounding error: 0x1p-54/N^2. */
142 /* Worst case error if |y| > 0x1p-5:
143 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
144 Worst case error if |y| > 0x1p-4:
145 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
146 #if LOG_POLY_ORDER == 6
147 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
148 #elif LOG_POLY_ORDER == 7
150 + r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])
151 + r2 * r2 * (A[4] + r * A[5]))
154 return eval_as_double (y);
157 strong_alias (log, __log_finite)
158 hidden_alias (log, __ieee754_log)
159 # if LDBL_MANT_DIG == 53
160 long double logl (long double x) { return log (x); }