Average Error: 61.3 → 0.0
Time: 7.8s
Precision: binary64
\[-1 < x \land x < 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
↓
\[{\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1} \]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
{\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1}

(FPCore (x) :precision binary64 (/ (log (- 1.0 x)) (log (+ 1.0 x))))
↓
(FPCore (x) :precision binary64 (pow (/ (log1p x) (log1p (- x))) -1.0))
double code(double x) {
	return log(1.0 - x) / log(1.0 + x);
}
↓
double code(double x) {
	return pow((log1p(x) / log1p(-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
Target0.3
Herbie0.0
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666667 \cdot {x}^{3}\right) \]

Derivation

  1. Initial program 61.3

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}} \]
  3. Using strategy rm

  4. Applied clear-num_binary640.0

    \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}}} \]

  5. Applied inv-pow_binary640.0

    \[\leadsto \color{blue}{{\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1}} \]

  6. Final simplification0.0

    \[\leadsto {\left(\frac{\mathsf{log1p}\left(x\right)}{\mathsf{log1p}\left(-x\right)}\right)}^{-1} \]

Reproduce

herbie shell --seed 2021211 
(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))))