Average Error: 39.1 → 0.3
Time: 18.5s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + 1 \le 1.000000010117869209125274210236966609955:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;x + 1 \le 1.000000010117869209125274210236966609955:\\
\;\;\;\;\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}
double f(double x) {
        double r3655838 = 1.0;
        double r3655839 = x;
        double r3655840 = r3655838 + r3655839;
        double r3655841 = log(r3655840);
        return r3655841;
}
↓
double f(double x) {
        double r3655842 = x;
        double r3655843 = 1.0;
        double r3655844 = r3655842 + r3655843;
        double r3655845 = 1.0000000101178692;
        bool r3655846 = r3655844 <= r3655845;
        double r3655847 = r3655842 / r3655843;
        double r3655848 = r3655847 * r3655847;
        double r3655849 = -0.5;
        double r3655850 = log(r3655843);
        double r3655851 = fma(r3655843, r3655842, r3655850);
        double r3655852 = fma(r3655848, r3655849, r3655851);
        double r3655853 = log(r3655844);
        double r3655854 = r3655846 ? r3655852 : r3655853;
        return r3655854;
}

Error

Bits error versus x

Target

Original39.1
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ 1.0 x) < 1.0000000101178692

    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. Simplified0.3

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

    if 1.0000000101178692 < (+ 1.0 x)

    1. Initial program 0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \le 1.000000010117869209125274210236966609955:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{1} \cdot \frac{x}{1}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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)))