Average Error: 39.2 → 0.2
Time: 9.5s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 9.703167262933319 \cdot 10^{-05}:\\ \;\;\;\;x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le 9.703167262933319 \cdot 10^{-05}:\\
\;\;\;\;x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x\\

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

\end{array}
double f(double x) {
        double r2771328 = 1.0;
        double r2771329 = x;
        double r2771330 = r2771328 + r2771329;
        double r2771331 = log(r2771330);
        return r2771331;
}
↓
double f(double x) {
        double r2771332 = x;
        double r2771333 = 9.703167262933319e-05;
        bool r2771334 = r2771332 <= r2771333;
        double r2771335 = -0.5;
        double r2771336 = 0.3333333333333333;
        double r2771337 = r2771336 * r2771332;
        double r2771338 = r2771335 + r2771337;
        double r2771339 = r2771332 * r2771338;
        double r2771340 = r2771339 * r2771332;
        double r2771341 = r2771332 + r2771340;
        double r2771342 = 1.0;
        double r2771343 = r2771332 + r2771342;
        double r2771344 = log(r2771343);
        double r2771345 = r2771334 ? r2771341 : r2771344;
        return r2771345;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.2
Target0.3
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 < 9.703167262933319e-05

    1. Initial program 59.0

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

    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

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

    if 9.703167262933319e-05 < x

    1. Initial program 0.1

      \[\log \left(1 + x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 9.703167262933319 \cdot 10^{-05}:\\ \;\;\;\;x + \left(x \cdot \left(\frac{-1}{2} + \frac{1}{3} \cdot x\right)\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))