ln(1 + x)

Average Error: 38.6 → 0.3

Time: 9.1s

Precision: 64

Math

\[\log \left(1 + x\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;x + 1 \le 1.000000000966332569518613127002026885748:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

C

double f(double x) {
        double r3205870 = 1.0;
        double r3205871 = x;
        double r3205872 = r3205870 + r3205871;
        double r3205873 = log(r3205872);
        return r3205873;
}

↓

double f(double x) {
        double r3205874 = x;
        double r3205875 = 1.0;
        double r3205876 = r3205874 + r3205875;
        double r3205877 = 1.0000000009663326;
        bool r3205878 = r3205876 <= r3205877;
        double r3205879 = r3205874 / r3205875;
        double r3205880 = r3205879 * r3205879;
        double r3205881 = -0.5;
        double r3205882 = log(r3205875);
        double r3205883 = fma(r3205875, r3205874, r3205882);
        double r3205884 = fma(r3205880, r3205881, r3205883);
        double r3205885 = log(r3205876);
        double r3205886 = r3205878 ? r3205884 : r3205885;
        return r3205886;
}

Error

Bits error versus x

Target

Original	38.6
Target	0.3
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (+ 1.0 x) < 1.0000000009663326

   1. Initial program 59.2

      \[\log \left(1 + x\right)\]

   2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(\log 1 + 1 \cdot x\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)}\]

   if 1.0000000009663326 < (+ 1.0 x)

   1. Initial program 0.3

      \[\log \left(1 + x\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \le 1.000000000966332569518613127002026885748:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))