1 /* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunSoft, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12 /* log(x)
13 * Return the logarithm of x
14 *
15 * Method :
16 * 1. Argument Reduction: find k and f such that
17 * x = 2^k * (1+f),
18 * where sqrt(2)/2 < 1+f < sqrt(2) .
19 *
20 * 2. Approximation of log(1+f).
21 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
22 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
23 * = 2s + s*R
24 * We use a special Remez algorithm on [0,0.1716] to generate
25 * a polynomial of degree 14 to approximate R The maximum error
26 * of this polynomial approximation is bounded by 2**-58.45. In
27 * other words,
28 * 2 4 6 8 10 12 14
29 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
30 * (the values of Lg1 to Lg7 are listed in the program)
31 * and
32 * | 2 14 | -58.45
33 * | Lg1*s +...+Lg7*s - R(z) | <= 2
34 * | |
35 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
36 * In order to guarantee error in log below 1ulp, we compute log
37 * by
38 * log(1+f) = f - s*(f - R) (if f is not too large)
39 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
40 *
41 * 3. Finally, log(x) = k*ln2 + log(1+f).
42 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
43 * Here ln2 is split into two floating point number:
44 * ln2_hi + ln2_lo,
45 * where n*ln2_hi is always exact for |n| < 2000.
46 *
47 * Special cases:
48 * log(x) is NaN with signal if x < 0 (including -INF) ;
49 * log(+INF) is +INF; log(0) is -INF with signal;
50 * log(NaN) is that NaN with no signal.
51 *
52 * Accuracy:
53 * according to an error analysis, the error is always less than
54 * 1 ulp (unit in the last place).
55 *
56 * Constants:
57 * The hexadecimal values are the intended ones for the following
58 * constants. The decimal values may be used, provided that the
59 * compiler will convert from decimal to binary accurately enough
60 * to produce the hexadecimal values shown.
61 */
62
63 const LN2_HI: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
64 const LN2_LO: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
65 const LG1: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
66 const LG2: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
67 const LG3: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
68 const LG4: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
69 const LG5: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
70 const LG6: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
71 const LG7: f64 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
72
73 #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
log(mut x: f64) -> f6474 pub fn log(mut x: f64) -> f64 {
75 let x1p54 = f64::from_bits(0x4350000000000000); // 0x1p54 === 2 ^ 54
76
77 let mut ui = x.to_bits();
78 let mut hx: u32 = (ui >> 32) as u32;
79 let mut k: i32 = 0;
80
81 if (hx < 0x00100000) || ((hx >> 31) != 0) {
82 /* x < 2**-126 */
83 if ui << 1 == 0 {
84 return -1. / (x * x); /* log(+-0)=-inf */
85 }
86 if hx >> 31 != 0 {
87 return (x - x) / 0.0; /* log(-#) = NaN */
88 }
89 /* subnormal number, scale x up */
90 k -= 54;
91 x *= x1p54;
92 ui = x.to_bits();
93 hx = (ui >> 32) as u32;
94 } else if hx >= 0x7ff00000 {
95 return x;
96 } else if hx == 0x3ff00000 && ui << 32 == 0 {
97 return 0.;
98 }
99
100 /* reduce x into [sqrt(2)/2, sqrt(2)] */
101 hx += 0x3ff00000 - 0x3fe6a09e;
102 k += ((hx >> 20) as i32) - 0x3ff;
103 hx = (hx & 0x000fffff) + 0x3fe6a09e;
104 ui = ((hx as u64) << 32) | (ui & 0xffffffff);
105 x = f64::from_bits(ui);
106
107 let f: f64 = x - 1.0;
108 let hfsq: f64 = 0.5 * f * f;
109 let s: f64 = f / (2.0 + f);
110 let z: f64 = s * s;
111 let w: f64 = z * z;
112 let t1: f64 = w * (LG2 + w * (LG4 + w * LG6));
113 let t2: f64 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7)));
114 let r: f64 = t2 + t1;
115 let dk: f64 = k as f64;
116 s * (hfsq + r) + dk * LN2_LO - hfsq + f + dk * LN2_HI
117 }
118