Average Error: 61.2 → 0.6
Time: 9.9s
Precision: binary64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - \left(1 \cdot \left(x \cdot x\right) + \frac{1}{2} \cdot \frac{{x}^{4}}{1 \cdot 1}\right)}{1 \cdot x + \left(\log 1 - \frac{1}{2} \cdot \left(x \cdot \frac{x}{1 \cdot 1}\right)\right)} - 1\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - \left(1 \cdot \left(x \cdot x\right) + \frac{1}{2} \cdot \frac{{x}^{4}}{1 \cdot 1}\right)}{1 \cdot x + \left(\log 1 - \frac{1}{2} \cdot \left(x \cdot \frac{x}{1 \cdot 1}\right)\right)} - 1
double code(double x) {
	return ((double) (((double) log(((double) (1.0 - x)))) / ((double) log(((double) (1.0 + x))))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) (((double) log(1.0)) - ((double) (((double) (1.0 * ((double) (x * x)))) + ((double) (0.5 * ((double) (((double) pow(x, 4.0)) / ((double) (1.0 * 1.0)))))))))) / ((double) (((double) (1.0 * x)) + ((double) (((double) log(1.0)) - ((double) (0.5 * ((double) (x * ((double) (x / ((double) (1.0 * 1.0)))))))))))))) - 1.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.2
Target0.4
Herbie0.6
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

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. Applied div-sub61.0

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

  6. Simplified61.0

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

  7. Taylor expanded around 0 1.1

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

  8. Simplified1.1

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

  9. Taylor expanded around 0 0.6

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

  10. Simplified0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1.0 x) (< x 1.0))

  :herbie-target
  (neg (+ (+ (+ 1.0 x) (/ (* x x) 2.0)) (* 0.4166666666666667 (pow x 3.0))))

  (/ (log (- 1.0 x)) (log (+ 1.0 x))))