Average Error: 61.3 → 0.4
Time: 8.3s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - 1 \cdot x}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}} - \frac{\frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}{\left(1 \cdot x + \log 1\right) - \frac{x \cdot \frac{1}{2}}{\frac{{1}^{2}}{{x}^{\left(\frac{2}{2}\right)}}}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - 1 \cdot x}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}} - \frac{\frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}{\left(1 \cdot x + \log 1\right) - \frac{x \cdot \frac{1}{2}}{\frac{{1}^{2}}{{x}^{\left(\frac{2}{2}\right)}}}}
double code(double x) {
	return (log((1.0 - x)) / log((1.0 + x)));
}
↓
double code(double x) {
	return (((log(1.0) - (1.0 * x)) / (((1.0 * x) + log(1.0)) - (0.5 * (pow(x, 2.0) / pow(1.0, 2.0))))) - ((0.5 * (pow(x, 2.0) / pow(1.0, 2.0))) / (((1.0 * x) + log(1.0)) - ((x * 0.5) / (pow(1.0, 2.0) / pow(x, (2.0 / 2.0)))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.3
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.3

    \[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]

  2. Taylor expanded around 0 60.5

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

  3. Taylor expanded around 0 0.4

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

  5. Applied associate--r+0.4

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

  6. Applied div-sub0.4

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

  8. Applied sqr-pow0.4

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

  9. Applied associate-/l*0.4

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

  10. Applied associate-*r/0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020071 
(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))))