Average Error: 61.5 → 0.4
Time: 39.9s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}} - \log \left(e^{\frac{1 \cdot x}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - \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}}} - \log \left(e^{\frac{1 \cdot x}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\right)
double code(double x) {
	return (log((1.0 - x)) / log((1.0 + x)));
}
↓
double code(double x) {
	return (((log(1.0) - (0.5 * (pow(x, 2.0) / pow(1.0, 2.0)))) / (((1.0 * x) + log(1.0)) - (0.5 * (pow(x, 2.0) / pow(1.0, 2.0))))) - log(exp(((1.0 * x) / (((1.0 * x) + log(1.0)) - (0.5 * (pow(x, 2.0) / pow(1.0, 2.0))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.5
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 61.5

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

  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 +-commutative0.4

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

  6. Applied associate--r+0.4

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

  7. Applied div-sub0.4

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

  9. Applied add-log-exp0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020066 
(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))))