Average Error: 39.3 → 0.3
Time: 11.4s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.00000072941427853:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(x, 1, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(1 + x\right) \cdot \frac{1}{2}\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.00000072941427853:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(x, 1, \log 1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(1 + x\right) \cdot \frac{1}{2}\\

\end{array}
double f(double x) {
        double r34834 = 1.0;
        double r34835 = x;
        double r34836 = r34834 + r34835;
        double r34837 = log(r34836);
        return r34837;
}
↓
double f(double x) {
        double r34838 = 1.0;
        double r34839 = x;
        double r34840 = r34838 + r34839;
        double r34841 = 1.0000007294142785;
        bool r34842 = r34840 <= r34841;
        double r34843 = -0.5;
        double r34844 = 2.0;
        double r34845 = pow(r34839, r34844);
        double r34846 = pow(r34838, r34844);
        double r34847 = r34845 / r34846;
        double r34848 = log(r34838);
        double r34849 = fma(r34839, r34838, r34848);
        double r34850 = fma(r34843, r34847, r34849);
        double r34851 = sqrt(r34840);
        double r34852 = log(r34851);
        double r34853 = log(r34840);
        double r34854 = 0.5;
        double r34855 = r34853 * r34854;
        double r34856 = r34852 + r34855;
        double r34857 = r34842 ? r34850 : r34856;
        return r34857;
}

Error

Bits error versus x

Target

Original39.3
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) < 1.0000007294142785

    1. Initial program 59.1

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

    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

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

    if 1.0000007294142785 < (+ 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

      \[\leadsto \color{blue}{\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)}\]
    5. Using strategy rm

    6. Applied pow10.2

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

    7. Applied sqrt-pow10.2

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

    8. Applied log-pow0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.00000072941427853:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{2}, \frac{{x}^{2}}{{1}^{2}}, \mathsf{fma}\left(x, 1, \log 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(1 + x\right) \cdot \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +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)))