Average Error: 38.8 → 0.3
Time: 52.9s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + 1 \le 1.000001508099978009269648282497655600309:\\ \;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{\left(\log \left(\sqrt[3]{\sqrt{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}}\right) + \log \left(\sqrt[3]{\sqrt{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}}\right)\right) + \log \left(\sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}} \cdot \sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}\right)}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;x + 1 \le 1.000001508099978009269648282497655600309:\\
\;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{\left(\log \left(\sqrt[3]{\sqrt{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}}\right) + \log \left(\sqrt[3]{\sqrt{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}}\right)\right) + \log \left(\sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}} \cdot \sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}\right)}{\frac{1}{x}}\\

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

\end{array}
double f(double x) {
        double r4928512 = 1.0;
        double r4928513 = x;
        double r4928514 = r4928512 + r4928513;
        double r4928515 = log(r4928514);
        return r4928515;
}
↓
double f(double x) {
        double r4928516 = x;
        double r4928517 = 1.0;
        double r4928518 = r4928516 + r4928517;
        double r4928519 = 1.000001508099978;
        bool r4928520 = r4928518 <= r4928519;
        double r4928521 = r4928517 * r4928516;
        double r4928522 = log(r4928517);
        double r4928523 = r4928521 + r4928522;
        double r4928524 = 0.5;
        double r4928525 = r4928517 / r4928516;
        double r4928526 = r4928524 / r4928525;
        double r4928527 = exp(r4928526);
        double r4928528 = sqrt(r4928527);
        double r4928529 = cbrt(r4928528);
        double r4928530 = log(r4928529);
        double r4928531 = r4928530 + r4928530;
        double r4928532 = cbrt(r4928527);
        double r4928533 = r4928532 * r4928532;
        double r4928534 = log(r4928533);
        double r4928535 = r4928531 + r4928534;
        double r4928536 = r4928535 / r4928525;
        double r4928537 = r4928523 - r4928536;
        double r4928538 = sqrt(r4928518);
        double r4928539 = log(r4928538);
        double r4928540 = r4928539 + r4928539;
        double r4928541 = r4928520 ? r4928537 : r4928540;
        return r4928541;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
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) < 1.000001508099978

    1. Initial program 59.1

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

      \[\leadsto \color{blue}{\left(1 \cdot x + \log 1\right) - \frac{\frac{\frac{1}{2}}{\frac{1}{x}}}{\frac{1}{x}}}\]
    4. Using strategy rm

    5. Applied add-log-exp0.4

      \[\leadsto \left(1 \cdot x + \log 1\right) - \frac{\color{blue}{\log \left(e^{\frac{\frac{1}{2}}{\frac{1}{x}}}\right)}}{\frac{1}{x}}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.4

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

    8. Applied log-prod0.4

      \[\leadsto \left(1 \cdot x + \log 1\right) - \frac{\color{blue}{\log \left(\sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}} \cdot \sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}\right) + \log \left(\sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}\right)}}{\frac{1}{x}}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt0.4

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

    11. Applied cbrt-prod0.4

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

    12. Applied log-prod0.4

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

    if 1.000001508099978 < (+ 1.0 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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \le 1.000001508099978009269648282497655600309:\\ \;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{\left(\log \left(\sqrt[3]{\sqrt{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}}\right) + \log \left(\sqrt[3]{\sqrt{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}}\right)\right) + \log \left(\sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}} \cdot \sqrt[3]{e^{\frac{\frac{1}{2}}{\frac{1}{x}}}}\right)}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + 1}\right) + \log \left(\sqrt{x + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "ln(1 + x)"

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))