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/* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
/*
* ====================================================
* 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.
* ====================================================
*/
/* double log1p(double x)
* Return the natural logarithm of 1+x.
*
* Method :
* 1. Argument Reduction: find k and f such that
* 1+x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* Note. If k=0, then f=x is exact. However, if k!=0, then f
* may not be representable exactly. In that case, a correction
* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
* and add back the correction term c/u.
* (Note: when x > 2**53, one can simply return log(x))
*
* 2. Approximation of log(1+f): See log.c
*
* 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
*
* Special cases:
* log1p(x) is NaN with signal if x < -1 (including -INF) ;
* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
* log1p(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.
*
* Note: Assuming log() return accurate answer, the following
* algorithm can be used to compute log1p(x) to within a few ULP:
*
* u = 1+x;
* if(u==1.0) return x ; else
* return log(u)*(x/(u-1.0));
*
* See HP-15C Advanced Functions Handbook, p.193.
*/
use core::f64;
const LN2_HI: f64 = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
const LN2_LO: f64 = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
const LG1: f64 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
const LG2: f64 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
const LG3: f64 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
const LG4: f64 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
const LG5: f64 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
const LG6: f64 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
const LG7: f64 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
/// The natural logarithm of 1+`x` (f64).
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn log1p(x: f64) -> f64 {
let mut ui: u64 = x.to_bits();
let hfsq: f64;
let mut f: f64 = 0.;
let mut c: f64 = 0.;
let s: f64;
let z: f64;
let r: f64;
let w: f64;
let t1: f64;
let t2: f64;
let dk: f64;
let hx: u32;
let mut hu: u32;
let mut k: i32;
hx = (ui >> 32) as u32;
k = 1;
if hx < 0x3fda827a || (hx >> 31) > 0 {
/* 1+x < sqrt(2)+ */
if hx >= 0xbff00000 {
/* x <= -1.0 */
if x == -1. {
return x / 0.0; /* log1p(-1) = -inf */
}
return (x - x) / 0.0; /* log1p(x<-1) = NaN */
}
if hx << 1 < 0x3ca00000 << 1 {
/* |x| < 2**-53 */
/* underflow if subnormal */
if (hx & 0x7ff00000) == 0 {
force_eval!(x as f32);
}
return x;
}
if hx <= 0xbfd2bec4 {
/* sqrt(2)/2- <= 1+x < sqrt(2)+ */
k = 0;
c = 0.;
f = x;
}
} else if hx >= 0x7ff00000 {
return x;
}
if k > 0 {
ui = (1. + x).to_bits();
hu = (ui >> 32) as u32;
hu += 0x3ff00000 - 0x3fe6a09e;
k = (hu >> 20) as i32 - 0x3ff;
/* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
if k < 54 {
c = if k >= 2 {
1. - (f64::from_bits(ui) - x)
} else {
x - (f64::from_bits(ui) - 1.)
};
c /= f64::from_bits(ui);
} else {
c = 0.;
}
/* reduce u into [sqrt(2)/2, sqrt(2)] */
hu = (hu & 0x000fffff) + 0x3fe6a09e;
ui = ((hu as u64) << 32) | (ui & 0xffffffff);
f = f64::from_bits(ui) - 1.;
}
hfsq = 0.5 * f * f;
s = f / (2.0 + f);
z = s * s;
w = z * z;
t1 = w * (LG2 + w * (LG4 + w * LG6));
t2 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7)));
r = t2 + t1;
dk = k as f64;
s * (hfsq + r) + (dk * LN2_LO + c) - hfsq + f + dk * LN2_HI
}