Average Error: 38.8 → 0.3
Time: 12.1s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000380361564467079915630165487527847:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.000000380361564467079915630165487527847:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\

\end{array}
double f(double x) {
        double r84681 = 1.0;
        double r84682 = x;
        double r84683 = r84681 + r84682;
        double r84684 = log(r84683);
        return r84684;
}
↓
double f(double x) {
        double r84685 = 1.0;
        double r84686 = x;
        double r84687 = r84685 + r84686;
        double r84688 = 1.0000003803615645;
        bool r84689 = r84687 <= r84688;
        double r84690 = 2.0;
        double r84691 = pow(r84686, r84690);
        double r84692 = pow(r84685, r84690);
        double r84693 = r84691 / r84692;
        double r84694 = -0.5;
        double r84695 = log(r84685);
        double r84696 = fma(r84685, r84686, r84695);
        double r84697 = fma(r84693, r84694, r84696);
        double r84698 = sqrt(r84687);
        double r84699 = log(r84698);
        double r84700 = r84699 + r84699;
        double r84701 = r84689 ? r84697 : r84700;
        return r84701;
}

Error

Bits error versus x

Target

Original38.8
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) < 1.0000003803615645

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 1.0000003803615645 < (+ 1.0 x)

    1. Initial program 0.2

      \[\log \left(1 + x\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.2

      \[\leadsto \log \color{blue}{\left(\sqrt{1 + x} \cdot \sqrt{1 + x}\right)}\]

    4. Applied log-prod0.2

      \[\leadsto \color{blue}{\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000380361564467079915630165487527847:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :herbie-target
  (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1)))

  (log (+ 1 x)))