| /* 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 */ |
| |
| #[inline] |
| #[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 |
| } |