Average Error: 39.6 → 0.3
Time: 4.1s
Precision: 64
\[\log \left(1 + x\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;1 + x \le 1.0000000379893819:\\ \;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt[3]{{\left(\sqrt{\sqrt{1 + x}}\right)}^{3}}\right)\right)\\ \end{array}\]
\log \left(1 + x\right)
↓
\begin{array}{l}
\mathbf{if}\;1 + x \le 1.0000000379893819:\\
\;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{1 + x}\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt[3]{{\left(\sqrt{\sqrt{1 + x}}\right)}^{3}}\right)\right)\\

\end{array}
double code(double x) {
	return log((1.0 + x));
}
↓
double code(double x) {
	double temp;
	if (((1.0 + x) <= 1.0000000379893819)) {
		temp = (((1.0 * x) + log(1.0)) - (0.5 * (pow(x, 2.0) / pow(1.0, 2.0))));
	} else {
		temp = (log(sqrt((1.0 + x))) + (log(sqrt(sqrt((1.0 + x)))) + log(cbrt(pow(sqrt(sqrt((1.0 + x))), 3.0)))));
	}
	return temp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.6
Target0.1
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) < 1.0000000379893819

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.2

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

    if 1.0000000379893819 < (+ 1.0 x)

    1. Initial program 0.3

      \[\log \left(1 + x\right)\]
    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 add-sqr-sqrt0.3

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

    7. Applied sqrt-prod0.3

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

    8. Applied log-prod0.3

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

    10. Applied add-cbrt-cube0.3

      \[\leadsto \log \left(\sqrt{1 + x}\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \color{blue}{\left(\sqrt[3]{\left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}\right) \cdot \sqrt{\sqrt{1 + x}}}\right)}\right)\]

    11. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 + x \le 1.0000000379893819:\\ \;\;\;\;\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1 + x}\right) + \left(\log \left(\sqrt{\sqrt{1 + x}}\right) + \log \left(\sqrt[3]{{\left(\sqrt{\sqrt{1 + x}}\right)}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(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)))