Average Error: 39.6 → 0.3
Time: 10.1s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le \frac{35184377452467}{35184372088832}:\\ \;\;\;\;\log 1 - \left(\frac{\frac{{x}^{2}}{{1}^{2}}}{2} - 1 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le \frac{35184377452467}{35184372088832}:\\
\;\;\;\;\log 1 - \left(\frac{\frac{{x}^{2}}{{1}^{2}}}{2} - 1 \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r88701 = 1.0;
        double r88702 = x;
        double r88703 = r88701 + r88702;
        double r88704 = log(r88703);
        return r88704;
}
↓
double f(double x) {
        double r88705 = 1.0;
        double r88706 = x;
        double r88707 = r88705 + r88706;
        double r88708 = 35184377452467.0;
        double r88709 = 35184372088832.0;
        double r88710 = r88708 / r88709;
        bool r88711 = r88707 <= r88710;
        double r88712 = log(r88705);
        double r88713 = 2.0;
        double r88714 = pow(r88706, r88713);
        double r88715 = pow(r88705, r88713);
        double r88716 = r88714 / r88715;
        double r88717 = r88716 / r88713;
        double r88718 = r88705 * r88706;
        double r88719 = r88717 - r88718;
        double r88720 = r88712 - r88719;
        double r88721 = sqrt(r88707);
        double r88722 = log(r88721);
        double r88723 = r88722 + r88722;
        double r88724 = r88711 ? r88720 : r88723;
        return r88724;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le \frac{35184377452467}{35184372088832}:\\ \;\;\;\;\log 1 - \left(\frac{\frac{{x}^{2}}{{1}^{2}}}{2} - 1 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \log \left(\sqrt{1 + x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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)))