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1 /* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
10 * ====================================================
12 /* double log1p(double x)
13 * Return the natural logarithm of 1+x.
16 * 1. Argument Reduction: find k and f such that
18 * where sqrt(2)/2 < 1+f < sqrt(2) .
20 * Note. If k=0, then f=x is exact. However, if k!=0, then f
21 * may not be representable exactly. In that case, a correction
22 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
23 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
24 * and add back the correction term c/u.
25 * (Note: when x > 2**53, one can simply return log(x))
27 * 2. Approximation of log(1+f): See log.c
29 * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
32 * log1p(x) is NaN with signal if x < -1 (including -INF) ;
33 * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
34 * log1p(NaN) is that NaN with no signal.
37 * according to an error analysis, the error is always less than
38 * 1 ulp (unit in the last place).
41 * The hexadecimal values are the intended ones for the following
42 * constants. The decimal values may be used, provided that the
43 * compiler will convert from decimal to binary accurately enough
44 * to produce the hexadecimal values shown.
46 * Note: Assuming log() return accurate answer, the following
47 * algorithm can be used to compute log1p(x) to within a few ULP:
50 * if(u==1.0) return x ; else
51 * return log(u)*(x/(u-1.0));
53 * See HP-15C Advanced Functions Handbook, p.193.
58 const LN2_HI
: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
59 const LN2_LO
: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
60 const LG1
: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
61 const LG2
: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
62 const LG3
: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
63 const LG4
: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
64 const LG5
: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
65 const LG6
: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
66 const LG7
: f64 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
68 #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
69 pub fn log1p(x
: f64) -> f64 {
70 let mut ui
: u64 = x
.to_bits();
85 hx
= (ui
>> 32) as u32;
87 if hx
< 0x3fda827a || (hx
>> 31) > 0 {
92 return x
/ 0.0; /* log1p(-1) = -inf */
94 return (x
- x
) / 0.0; /* log1p(x<-1) = NaN */
96 if hx
<< 1 < 0x3ca00000 << 1 {
98 /* underflow if subnormal */
99 if (hx
& 0x7ff00000) == 0 {
100 force_eval
!(x
as f32);
104 if hx
<= 0xbfd2bec4 {
105 /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
110 } else if hx
>= 0x7ff00000 {
114 ui
= (1. + x
).to_bits();
115 hu
= (ui
>> 32) as u32;
116 hu
+= 0x3ff00000 - 0x3fe6a09e;
117 k
= (hu
>> 20) as i32 - 0x3ff;
118 /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
121 1. - (f64::from_bits(ui
) - x
)
123 x
- (f64::from_bits(ui
) - 1.)
125 c
/= f64::from_bits(ui
);
129 /* reduce u into [sqrt(2)/2, sqrt(2)] */
130 hu
= (hu
& 0x000fffff) + 0x3fe6a09e;
131 ui
= (hu
as u64) << 32 | (ui
& 0xffffffff);
132 f
= f64::from_bits(ui
) - 1.;
138 t1
= w
* (LG2
+ w
* (LG4
+ w
* LG6
));
139 t2
= z
* (LG1
+ w
* (LG3
+ w
* (LG5
+ w
* LG7
)));
142 s
* (hfsq
+ r
) + (dk
* LN2_LO
+ c
) - hfsq
+ f
+ dk
* LN2_HI