Average Error: 61.4 → 0.4
Time: 8.0s
Precision: binary64
\[-1 < x \land x < 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{x \cdot \left(1 - \frac{-0.5}{1} \cdot \frac{x}{1}\right) - \log 1}{\left(-\log 1\right) - x \cdot \left(1 + \frac{-0.5}{1} \cdot \frac{x}{1}\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{x \cdot \left(1 - \frac{-0.5}{1} \cdot \frac{x}{1}\right) - \log 1}{\left(-\log 1\right) - x \cdot \left(1 + \frac{-0.5}{1} \cdot \frac{x}{1}\right)}
double code(double x) {
	return (((double) log(((double) (1.0 - x)))) / ((double) log(((double) (1.0 + x)))));
}
↓
double code(double x) {
	return (((double) (((double) (x * ((double) (1.0 - ((double) ((-0.5 / 1.0) * (x / 1.0))))))) - ((double) log(1.0)))) / ((double) (((double) -(((double) log(1.0)))) - ((double) (x * ((double) (1.0 + ((double) ((-0.5 / 1.0) * (x / 1.0))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.4
Target0.3
Herbie0.4
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666667 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.4

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

  2. Taylor expanded around 0 60.5

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{\left(1 \cdot x + \log 1\right) - 0.5 \cdot \frac{{x}^{2}}{{1}^{2}}}}\]

  3. Simplified60.5

    \[\leadsto \frac{\log \left(1 - x\right)}{\color{blue}{1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot -0.5\right)}}\]

  4. Taylor expanded around 0 0.4

    \[\leadsto \frac{\color{blue}{\log 1 - \left(1 \cdot x + 0.5 \cdot \frac{{x}^{2}}{{1}^{2}}\right)}}{1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot -0.5\right)}\]

  5. Simplified0.4

    \[\leadsto \frac{\color{blue}{\log 1 + \left(x \cdot \frac{x \cdot \frac{-0.5}{1}}{1} - 1 \cdot x\right)}}{1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot -0.5\right)}\]
  6. Using strategy rm

  7. Applied frac-2neg0.4

    \[\leadsto \color{blue}{\frac{-\left(\log 1 + \left(x \cdot \frac{x \cdot \frac{-0.5}{1}}{1} - 1 \cdot x\right)\right)}{-\left(1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot -0.5\right)\right)}}\]

  8. Simplified0.4

    \[\leadsto \frac{\color{blue}{x \cdot \left(1 - \frac{-0.5}{1} \cdot \frac{x}{1}\right) - \log 1}}{-\left(1 \cdot x + \left(\log 1 + \left(\frac{x}{1} \cdot \frac{x}{1}\right) \cdot -0.5\right)\right)}\]

  9. Simplified0.4

    \[\leadsto \frac{x \cdot \left(1 - \frac{-0.5}{1} \cdot \frac{x}{1}\right) - \log 1}{\color{blue}{\left(-\log 1\right) - x \cdot \left(1 + \frac{-0.5}{1} \cdot \frac{x}{1}\right)}}\]

  10. Final simplification0.4

    \[\leadsto \frac{x \cdot \left(1 - \frac{-0.5}{1} \cdot \frac{x}{1}\right) - \log 1}{\left(-\log 1\right) - x \cdot \left(1 + \frac{-0.5}{1} \cdot \frac{x}{1}\right)}\]

Reproduce

herbie shell --seed 2020199 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1.0 x) (< x 1.0))

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

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