vendor/libm/src/math/log1p.rs

   1 /* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   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
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* double log1p(double x)
  13  * Return the natural logarithm of 1+x.
  14  *
  15  * Method :
  16  *   1. Argument Reduction: find k and f such that
  17  *                      1+x = 2^k * (1+f),
  18  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  19  *
  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))
  26  *
  27  *   2. Approximation of log(1+f): See log.c
  28  *
  29  *   3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
  30  *
  31  * Special cases:
  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.
  35  *
  36  * Accuracy:
  37  *      according to an error analysis, the error is always less than
  38  *      1 ulp (unit in the last place).
  39  *
  40  * Constants:
  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.
  45  *
  46  * Note: Assuming log() return accurate answer, the following
  47  *       algorithm can be used to compute log1p(x) to within a few ULP:
  48  *
  49  *              u = 1+x;
  50  *              if(u==1.0) return x ; else
  51  *                         return log(u)*(x/(u-1.0));
  52  *
  53  *       See HP-15C Advanced Functions Handbook, p.193.
  54  */
  55
  56 use core::f64;
  57
  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 */
  67
  68 #[inline]
  69 #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
  70 pub fn log1p(x: f64) -> f64 {
  71     let mut ui: u64 = x.to_bits();
  72     let hfsq: f64;
  73     let mut f: f64 = 0.;
  74     let mut c: f64 = 0.;
  75     let s: f64;
  76     let z: f64;
  77     let r: f64;
  78     let w: f64;
  79     let t1: f64;
  80     let t2: f64;
  81     let dk: f64;
  82     let hx: u32;
  83     let mut hu: u32;
  84     let mut k: i32;
  85
  86     hx = (ui >> 32) as u32;
  87     k = 1;
  88     if hx < 0x3fda827a || (hx >> 31) > 0 {
  89         /* 1+x < sqrt(2)+ */
  90         if hx >= 0xbff00000 {
  91             /* x <= -1.0 */
  92             if x == -1. {
  93                 return x / 0.0; /* log1p(-1) = -inf */
  94             }
  95             return (x - x) / 0.0; /* log1p(x<-1) = NaN */
  96         }
  97         if hx << 1 < 0x3ca00000 << 1 {
  98             /* |x| < 2**-53 */
  99             /* underflow if subnormal */
 100             if (hx & 0x7ff00000) == 0 {
 101                 force_eval!(x as f32);
 102             }
 103             return x;
 104         }
 105         if hx <= 0xbfd2bec4 {
 106             /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
 107             k = 0;
 108             c = 0.;
 109             f = x;
 110         }
 111     } else if hx >= 0x7ff00000 {
 112         return x;
 113     }
 114     if k > 0 {
 115         ui = (1. + x).to_bits();
 116         hu = (ui >> 32) as u32;
 117         hu += 0x3ff00000 - 0x3fe6a09e;
 118         k = (hu >> 20) as i32 - 0x3ff;
 119         /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
 120         if k < 54 {
 121             c = if k >= 2 {
 122                 1. - (f64::from_bits(ui) - x)
 123             } else {
 124                 x - (f64::from_bits(ui) - 1.)
 125             };
 126             c /= f64::from_bits(ui);
 127         } else {
 128             c = 0.;
 129         }
 130         /* reduce u into [sqrt(2)/2, sqrt(2)] */
 131         hu = (hu & 0x000fffff) + 0x3fe6a09e;
 132         ui = (hu as u64) << 32 | (ui & 0xffffffff);
 133         f = f64::from_bits(ui) - 1.;
 134     }
 135     hfsq = 0.5 * f * f;
 136     s = f / (2.0 + f);
 137     z = s * s;
 138     w = z * z;
 139     t1 = w * (LG2 + w * (LG4 + w * LG6));
 140     t2 = z * (LG1 + w * (LG3 + w * (LG5 + w * LG7)));
 141     r = t2 + t1;
 142     dk = k as f64;
 143     s * (hfsq + r) + (dk * LN2_LO + c) - hfsq + f + dk * LN2_HI
 144 }