Average Error: 39.1 → 0.2
Time: 14.0s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.00016988036905788884:\\ \;\;\;\;x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le 0.00016988036905788884:\\
\;\;\;\;x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x\\

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

\end{array}
double f(double x) {
        double r1653676 = 1.0;
        double r1653677 = x;
        double r1653678 = r1653676 + r1653677;
        double r1653679 = log(r1653678);
        return r1653679;
}
↓
double f(double x) {
        double r1653680 = x;
        double r1653681 = 0.00016988036905788884;
        bool r1653682 = r1653680 <= r1653681;
        double r1653683 = -0.5;
        double r1653684 = 0.3333333333333333;
        double r1653685 = r1653684 * r1653680;
        double r1653686 = r1653683 + r1653685;
        double r1653687 = r1653680 * r1653686;
        double r1653688 = r1653687 * r1653680;
        double r1653689 = r1653680 + r1653688;
        double r1653690 = 1.0;
        double r1653691 = r1653680 + r1653690;
        double r1653692 = sqrt(r1653691);
        double r1653693 = log(r1653692);
        double r1653694 = r1653693 + r1653693;
        double r1653695 = r1653682 ? r1653689 : r1653694;
        return r1653695;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < 0.00016988036905788884

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot {x}^{3}\right) - \frac{1}{2} \cdot {x}^{2}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{x + x \cdot \left(x \cdot \left(\frac{1}{3} \cdot x + \frac{-1}{2}\right)\right)}\]

    if 0.00016988036905788884 < x

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied log-prod0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.00016988036905788884:\\ \;\;\;\;x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))