Average Error: 61.2 → 0.5
Time: 9.4s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\sqrt[3]{{\left(\frac{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}} - 1\right)}^{3}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\sqrt[3]{{\left(\frac{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}} - 1\right)}^{3}}
double code(double x) {
	return (log((1.0 - x)) / log((1.0 + x)));
}
↓
double code(double x) {
	return cbrt(pow((((log(1.0) - ((1.0 * pow(x, 2.0)) + (0.5 * (pow(x, 4.0) / pow(1.0, 2.0))))) / (((1.0 * x) + log(1.0)) - (0.5 * (pow(x, 2.0) / pow(1.0, 2.0))))) - 1.0), 3.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.2
Target0.3
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.2

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

  3. Applied flip--60.8

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

  4. Applied log-div61.0

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

  5. Taylor expanded around 0 60.7

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

  7. Applied add-cbrt-cube60.7

    \[\leadsto \frac{\left(\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)\right) - \log \left(1 + x\right)}{\color{blue}{\sqrt[3]{\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \log \left(1 + x\right)}}}\]

  8. Applied add-cbrt-cube60.7

    \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)\right) - \log \left(1 + x\right)\right) \cdot \left(\left(\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)\right) - \log \left(1 + x\right)\right)\right) \cdot \left(\left(\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)\right) - \log \left(1 + x\right)\right)}}}{\sqrt[3]{\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \log \left(1 + x\right)}}\]

  9. Applied cbrt-undiv60.7

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)\right) - \log \left(1 + x\right)\right) \cdot \left(\left(\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)\right) - \log \left(1 + x\right)\right)\right) \cdot \left(\left(\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)\right) - \log \left(1 + x\right)\right)}{\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \log \left(1 + x\right)}}}\]

  10. Simplified60.7

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}{\log \left(1 + x\right)} - 1\right)}^{3}}}\]

  11. Taylor expanded around 0 0.5

    \[\leadsto \sqrt[3]{{\left(\frac{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}{\color{blue}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}} - 1\right)}^{3}}\]

  12. Final simplification0.5

    \[\leadsto \sqrt[3]{{\left(\frac{\log 1 - \left(1 \cdot {x}^{2} + \frac{1}{2} \cdot \frac{{x}^{4}}{{1}^{2}}\right)}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}} - 1\right)}^{3}}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 0.4166666666666667 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))