Average Error: 39.6 → 0.3
Time: 15.1s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000000152443675460744998417794704437256:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.000000152443675460744998417794704437256:\\
\;\;\;\;\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 r80857 = 1.0;
        double r80858 = x;
        double r80859 = r80857 + r80858;
        double r80860 = log(r80859);
        return r80860;
}
↓
double f(double x) {
        double r80861 = 1.0;
        double r80862 = x;
        double r80863 = r80861 + r80862;
        double r80864 = 1.0000001524436755;
        bool r80865 = r80863 <= r80864;
        double r80866 = 2.0;
        double r80867 = pow(r80862, r80866);
        double r80868 = pow(r80861, r80866);
        double r80869 = r80867 / r80868;
        double r80870 = -0.5;
        double r80871 = log(r80861);
        double r80872 = fma(r80861, r80862, r80871);
        double r80873 = fma(r80869, r80870, r80872);
        double r80874 = sqrt(r80863);
        double r80875 = log(r80874);
        double r80876 = r80875 + r80875;
        double r80877 = r80865 ? r80873 : r80876;
        return r80877;
}

Error

Bits error versus x

Target

Original39.6
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) < 1.0000001524436755

    1. Initial program 59.1

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

    3. Applied add-cbrt-cube59.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \log \left(1 + x\right)}}\]

    4. Simplified59.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\log \left(1 + x\right)\right)}^{3}}}\]

    5. Taylor expanded around 0 0.4

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

    6. Simplified0.4

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

    if 1.0000001524436755 < (+ 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000000152443675460744998417794704437256:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{2}}{{1}^{2}}, \frac{-1}{2}, \mathsf{fma}\left(1, x, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))