Average Error: 61.3 → 0.5
Time: 8.2s
Precision: binary64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\left(\log 1 - 1 \cdot x\right) + x \cdot \frac{x \cdot \frac{\frac{-1}{2}}{1}}{1}}{1 \cdot x + \left(\log 1 + \frac{-1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\left(\log 1 - 1 \cdot x\right) + x \cdot \frac{x \cdot \frac{\frac{-1}{2}}{1}}{1}}{1 \cdot x + \left(\log 1 + \frac{-1}{2} \cdot \left(\frac{x}{1} \cdot \frac{x}{1}\right)\right)}
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) (1.0 * x)))) + ((double) (x * ((double) (((double) (x * ((double) (-0.5 / 1.0)))) / 1.0)))))) / ((double) (((double) (1.0 * x)) + ((double) (((double) log(1.0)) + ((double) (-0.5 * ((double) (((double) (x / 1.0)) * ((double) (x / 1.0))))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.3
Target0.4
Herbie0.5
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.416666666666666685 \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.3

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

  2. Taylor expanded around 0 60.4

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

  3. Simplified60.4

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020184 
(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))))