Average Error: 61.3 → 0.4
Time: 8.4s
Precision: binary64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{1}{\frac{\log 1 + x \cdot \left(1 + \frac{\frac{-1}{2}}{1} \cdot \frac{x}{1}\right)}{\log 1 + x \cdot \left(\frac{\frac{-1}{2}}{1} \cdot \frac{x}{1} - 1\right)}}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{1}{\frac{\log 1 + x \cdot \left(1 + \frac{\frac{-1}{2}}{1} \cdot \frac{x}{1}\right)}{\log 1 + x \cdot \left(\frac{\frac{-1}{2}}{1} \cdot \frac{x}{1} - 1\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) (1.0 / ((double) (((double) (((double) log(1.0)) + ((double) (x * ((double) (1.0 + ((double) (((double) (-0.5 / 1.0)) * ((double) (x / 1.0)))))))))) / ((double) (((double) log(1.0)) + ((double) (x * ((double) (((double) (((double) (-0.5 / 1.0)) * ((double) (x / 1.0)))) - 1.0))))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.3
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 61.3

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

  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 0.4

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

  5. Simplified0.4

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

  7. Applied clear-num0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020185 
(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))))