libm/math/
log.rs

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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
63const LN2_HI: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
64const LN2_LO: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
65const LG1: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
66const LG2: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
67const LG3: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
68const LG4: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
69const LG5: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
70const LG6: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
71const LG7: f64 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
72
73/// The natural logarithm of `x` (f64).
74#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
75pub fn log(mut x: f64) -> f64 {
76    let x1p54 = f64::from_bits(0x4350000000000000); // 0x1p54 === 2 ^ 54
77
78    let mut ui = x.to_bits();
79    let mut hx: u32 = (ui >> 32) as u32;
80    let mut k: i32 = 0;
81
82    if (hx < 0x00100000) || ((hx >> 31) != 0) {
83        /* x < 2**-126  */
84        if ui << 1 == 0 {
85            return -1. / (x * x); /* log(+-0)=-inf */
86        }
87        if hx >> 31 != 0 {
88            return (x - x) / 0.0; /* log(-#) = NaN */
89        }
90        /* subnormal number, scale x up */
91        k -= 54;
92        x *= x1p54;
93        ui = x.to_bits();
94        hx = (ui >> 32) as u32;
95    } else if hx >= 0x7ff00000 {
96        return x;
97    } else if hx == 0x3ff00000 && ui << 32 == 0 {
98        return 0.;
99    }
100
101    /* reduce x into [sqrt(2)/2, sqrt(2)] */
102    hx += 0x3ff00000 - 0x3fe6a09e;
103    k += ((hx >> 20) as i32) - 0x3ff;
104    hx = (hx & 0x000fffff) + 0x3fe6a09e;
105    ui = ((hx as u64) << 32) | (ui & 0xffffffff);
106    x = f64::from_bits(ui);
107
108    let f: f64 = x - 1.0;
109    let hfsq: f64 = 0.5 * f * f;
110    let s: f64 = f / (2.0 + f);
111    let z: f64 = s * s;
112    let w: f64 = z * z;
113    let t1: f64 = w * (LG2 + w * (LG4 + w * LG6));
114    let t2: f64 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7)));
115    let r: f64 = t2 + t1;
116    let dk: f64 = k as f64;
117    s * (hfsq + r) + dk * LN2_LO - hfsq + f + dk * LN2_HI
118}
libm/math/log.rs

libm/math/
log.rs
