Average Error: 38.8 → 0.3
Time: 10.5s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.000001719073508876789446730981580913067:\\ \;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(1 + x\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.000001719073508876789446730981580913067:\\
\;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\

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

\end{array}
double f(double x) {
        double r51765 = 1.0;
        double r51766 = x;
        double r51767 = r51765 + r51766;
        double r51768 = log(r51767);
        return r51768;
}
↓
double f(double x) {
        double r51769 = 1.0;
        double r51770 = x;
        double r51771 = r51769 + r51770;
        double r51772 = 1.0000017190735089;
        bool r51773 = r51771 <= r51772;
        double r51774 = r51769 * r51770;
        double r51775 = log(r51769);
        double r51776 = r51774 + r51775;
        double r51777 = 0.5;
        double r51778 = 2.0;
        double r51779 = pow(r51770, r51778);
        double r51780 = pow(r51769, r51778);
        double r51781 = r51779 / r51780;
        double r51782 = r51777 * r51781;
        double r51783 = r51776 - r51782;
        double r51784 = 1.0;
        double r51785 = r51784 / r51778;
        double r51786 = log(r51771);
        double r51787 = r51785 * r51786;
        double r51788 = sqrt(r51771);
        double r51789 = sqrt(r51788);
        double r51790 = log(r51789);
        double r51791 = r51790 + r51790;
        double r51792 = r51787 + r51791;
        double r51793 = r51773 ? r51783 : r51792;
        return r51793;
}

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) < 1.0000017190735089

    1. Initial program 59.0

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

    2. Taylor expanded around 0 0.4

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

    if 1.0000017190735089 < (+ 1.0 x)

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied log-prod0.2

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

    6. Applied pow10.2

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

    7. Applied sqrt-pow10.2

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

    8. Applied log-pow0.1

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

    10. Applied add-sqr-sqrt0.1

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

    11. Applied sqrt-prod0.2

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

    12. Applied log-prod0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.000001719073508876789446730981580913067:\\ \;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(1 + x\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt{\sqrt{1 + x}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

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

  (log (+ 1 x)))