Average Error: 39.5 → 0.2
Time: 17.7s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + 1 \le 1.0000004263740856:\\ \;\;\;\;x + \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(x + 1\right) \cdot \frac{1}{2}\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;x + 1 \le 1.0000004263740856:\\
\;\;\;\;x + \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(x \cdot x\right)\\

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

\end{array}
double f(double x) {
        double r4586657 = 1.0;
        double r4586658 = x;
        double r4586659 = r4586657 + r4586658;
        double r4586660 = log(r4586659);
        return r4586660;
}
↓
double f(double x) {
        double r4586661 = x;
        double r4586662 = 1.0;
        double r4586663 = r4586661 + r4586662;
        double r4586664 = 1.0000004263740856;
        bool r4586665 = r4586663 <= r4586664;
        double r4586666 = 0.3333333333333333;
        double r4586667 = r4586666 * r4586661;
        double r4586668 = 0.5;
        double r4586669 = r4586667 - r4586668;
        double r4586670 = r4586661 * r4586661;
        double r4586671 = r4586669 * r4586670;
        double r4586672 = r4586661 + r4586671;
        double r4586673 = sqrt(r4586663);
        double r4586674 = log(r4586673);
        double r4586675 = log(r4586663);
        double r4586676 = r4586675 * r4586668;
        double r4586677 = r4586674 + r4586676;
        double r4586678 = r4586665 ? r4586672 : r4586677;
        return r4586678;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.5
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 (+ 1 x) < 1.0000004263740856

    1. Initial program 59.1

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

    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

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

    if 1.0000004263740856 < (+ 1 x)

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.3

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

    4. Applied log-prod0.3

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

    6. Applied pow1/20.3

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

    7. 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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \le 1.0000004263740856:\\ \;\;\;\;x + \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(x + 1\right) \cdot \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))