Average Error: 39.2 → 0.2
Time: 16.1s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + 1 \le 1.000052777771681:\\ \;\;\;\;x + \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \left(\log \left(\sqrt{\sqrt{x + 1}}\right) + \log \left(x + 1\right) \cdot \frac{1}{4}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;x + 1 \le 1.000052777771681:\\
\;\;\;\;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) + \left(\log \left(\sqrt{\sqrt{x + 1}}\right) + \log \left(x + 1\right) \cdot \frac{1}{4}\right)\\

\end{array}
double f(double x) {
        double r3745833 = 1.0;
        double r3745834 = x;
        double r3745835 = r3745833 + r3745834;
        double r3745836 = log(r3745835);
        return r3745836;
}
↓
double f(double x) {
        double r3745837 = x;
        double r3745838 = 1.0;
        double r3745839 = r3745837 + r3745838;
        double r3745840 = 1.000052777771681;
        bool r3745841 = r3745839 <= r3745840;
        double r3745842 = 0.3333333333333333;
        double r3745843 = r3745842 * r3745837;
        double r3745844 = 0.5;
        double r3745845 = r3745843 - r3745844;
        double r3745846 = r3745837 * r3745837;
        double r3745847 = r3745845 * r3745846;
        double r3745848 = r3745837 + r3745847;
        double r3745849 = sqrt(r3745839);
        double r3745850 = log(r3745849);
        double r3745851 = sqrt(r3745849);
        double r3745852 = log(r3745851);
        double r3745853 = log(r3745839);
        double r3745854 = 0.25;
        double r3745855 = r3745853 * r3745854;
        double r3745856 = r3745852 + r3745855;
        double r3745857 = r3745850 + r3745856;
        double r3745858 = r3745841 ? r3745848 : r3745857;
        return r3745858;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.2
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) < 1.000052777771681

    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

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

    if 1.000052777771681 < (+ 1 x)

    1. Initial program 0.1

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

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied log-prod0.1

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

    9. Applied pow1/20.1

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

    10. Applied sqrt-pow10.1

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

    11. Applied log-pow0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \le 1.000052777771681:\\ \;\;\;\;x + \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \left(\log \left(\sqrt{\sqrt{x + 1}}\right) + \log \left(x + 1\right) \cdot \frac{1}{4}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "ln(1 + x)"

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

  (log (+ 1 x)))