Average Error: 61.5 → 0.4
Time: 17.7min
Precision: binary64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\log 1 - x \cdot \left(\frac{\frac{1}{2}}{{1}^{2}} \cdot x + 1\right)}{\log 1 + \left(x \cdot 1 + \frac{\frac{-1}{2}}{{1}^{2}} \cdot \left(x \cdot x\right)\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\log 1 - x \cdot \left(\frac{\frac{1}{2}}{{1}^{2}} \cdot x + 1\right)}{\log 1 + \left(x \cdot 1 + \frac{\frac{-1}{2}}{{1}^{2}} \cdot \left(x \cdot x\right)\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) log(1.0)) - ((double) (x * ((double) (((double) ((0.5 / ((double) pow(1.0, 2.0))) * x)) + 1.0)))))) / ((double) (((double) log(1.0)) + ((double) (((double) (x * 1.0)) + ((double) ((-0.5 / ((double) pow(1.0, 2.0))) * ((double) (x * x)))))))));
}

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 60.6

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

  3. Simplified60.6

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

  4. Taylor expanded around 0 0.4

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

  5. Simplified0.4

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

  7. Applied sub-neg0.4

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

  8. Applied distribute-lft-in0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020181 
(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))))