1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
|
package math
@(require_results)
ln_f64 :: proc "contextless" (x: f64) -> f64 {
// The original C code, the long comment, and the constants
// below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
// and came with this notice.
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// __ieee754_log(x)
// Return the logarithm of x
//
// Method :
// 1. Argument Reduction: find k and f such that
// x = 2**k * (1+f),
// where sqrt(2)/2 < 1+f < sqrt(2) .
//
// 2. Approximation of log(1+f).
// Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
// = 2s + 2/3 s**3 + 2/5 s**5 + .....,
// = 2s + s*R
// We use a special Reme algorithm on [0,0.1716] to generate
// a polynomial of degree 14 to approximate R. The maximum error
// of this polynomial approximation is bounded by 2**-58.45. In
// other words,
// 2 4 6 8 10 12 14
// R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s +L6*s +L7*s
// (the values of L1 to L7 are listed in the program) and
// | 2 14 | -58.45
// | L1*s +...+L7*s - R(z) | <= 2
// | |
// Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
// In order to guarantee error in log below 1ulp, we compute log by
// log(1+f) = f - s*(f - R) (if f is not too large)
// log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
//
// 3. Finally, log(x) = k*Ln2 + log(1+f).
// = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
// Here Ln2 is split into two floating point number:
// Ln2_hi + Ln2_lo,
// where n*Ln2_hi is always exact for |n| < 2000.
//
// Special cases:
// log(x) is NaN with signal if x < 0 (including -INF) ;
// log(+INF) is +INF; log(0) is -INF with signal;
// log(NaN) is that NaN with no signal.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// Constants:
// The hexadecimal values are the intended ones for the following
// constants. The decimal values may be used, provided that the
// compiler will convert from decimal to binary accurately enough
// to produce the hexadecimal values shown.
LN2_HI :: 0h3fe62e42_fee00000 // 6.93147180369123816490e-01
LN2_LO :: 0h3dea39ef_35793c76 // 1.90821492927058770002e-10
L1 :: 0h3fe55555_55555593 // 6.666666666666735130e-01
L2 :: 0h3fd99999_9997fa04 // 3.999999999940941908e-01
L3 :: 0h3fd24924_94229359 // 2.857142874366239149e-01
L4 :: 0h3fcc71c5_1d8e78af // 2.222219843214978396e-01
L5 :: 0h3fc74664_96cb03de // 1.818357216161805012e-01
L6 :: 0h3fc39a09_d078c69f // 1.531383769920937332e-01
L7 :: 0h3fc2f112_df3e5244 // 1.479819860511658591e-01
switch {
case is_nan(x) || is_inf(x, 1):
return x
case x < 0:
return nan_f64()
case x == 0:
return inf_f64(-1)
}
// reduce
f1, ki := frexp(x)
if f1 < SQRT_TWO/2 {
f1 *= 2
ki -= 1
}
f := f1 - 1
k := f64(ki)
// compute
s := f / (2 + f)
s2 := s * s
s4 := s2 * s2
t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
t2 := s4 * (L2 + s4*(L4+s4*L6))
R := t1 + t2
hfsq := 0.5 * f * f
return k*LN2_HI - ((hfsq - (s*(hfsq+R) + k*LN2_LO)) - f)
}
@(require_results) ln_f16 :: proc "contextless" (x: f16) -> f16 { return #force_inline f16(ln_f64(f64(x))) }
@(require_results) ln_f32 :: proc "contextless" (x: f32) -> f32 { return #force_inline f32(ln_f64(f64(x))) }
@(require_results) ln_f16le :: proc "contextless" (x: f16le) -> f16le { return #force_inline f16le(ln_f64(f64(x))) }
@(require_results) ln_f16be :: proc "contextless" (x: f16be) -> f16be { return #force_inline f16be(ln_f64(f64(x))) }
@(require_results) ln_f32le :: proc "contextless" (x: f32le) -> f32le { return #force_inline f32le(ln_f64(f64(x))) }
@(require_results) ln_f32be :: proc "contextless" (x: f32be) -> f32be { return #force_inline f32be(ln_f64(f64(x))) }
@(require_results) ln_f64le :: proc "contextless" (x: f64le) -> f64le { return #force_inline f64le(ln_f64(f64(x))) }
@(require_results) ln_f64be :: proc "contextless" (x: f64be) -> f64be { return #force_inline f64be(ln_f64(f64(x))) }
ln :: proc{
ln_f16, ln_f16le, ln_f16be,
ln_f32, ln_f32le, ln_f32be,
ln_f64, ln_f64le, ln_f64be,
}
|
