Average Error: 39.6 → 0.3
Time: 14.5s
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 r44816 = 1.0;
        double r44817 = x;
        double r44818 = r44816 + r44817;
        double r44819 = log(r44818);
        return r44819;
}
↓
double f(double x) {
        double r44820 = 1.0;
        double r44821 = x;
        double r44822 = r44820 + r44821;
        double r44823 = 1.0000001524436755;
        bool r44824 = r44822 <= r44823;
        double r44825 = 2.0;
        double r44826 = pow(r44821, r44825);
        double r44827 = pow(r44820, r44825);
        double r44828 = r44826 / r44827;
        double r44829 = -0.5;
        double r44830 = log(r44820);
        double r44831 = fma(r44820, r44821, r44830);
        double r44832 = fma(r44828, r44829, r44831);
        double r44833 = sqrt(r44822);
        double r44834 = log(r44833);
        double r44835 = r44834 + r44834;
        double r44836 = r44824 ? r44832 : r44835;
        return r44836;
}

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)))