Average Error: 61.2 → 0.0
Time: 9.7s
Precision: binary64
\[-1 < x \land x < 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)} \]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(-x\right) \cdot \frac{1}{\mathsf{log1p}\left(x\right)}\right)\right) \]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(-x\right) \cdot \frac{1}{\mathsf{log1p}\left(x\right)}\right)\right)

(FPCore (x) :precision binary64 (/ (log (- 1.0 x)) (log (+ 1.0 x))))
↓
(FPCore (x)
 :precision binary64
 (log1p (expm1 (* (log1p (- x)) (/ 1.0 (log1p x))))))
double code(double x) {
	return log(1.0 - x) / log(1.0 + x);
}
↓
double code(double x) {
	return log1p(expm1(log1p(-x) * (1.0 / log1p(x))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.2
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 61.2

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

  2. Simplified0.0

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

  3. Applied log1p-expm1-u_binary640.0

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}\right)\right)} \]

  4. Applied div-inv_binary640.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(-x\right) \cdot \frac{1}{\mathsf{log1p}\left(x\right)}}\right)\right) \]

  5. Final simplification0.0

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(-x\right) \cdot \frac{1}{\mathsf{log1p}\left(x\right)}\right)\right) \]

Reproduce

herbie shell --seed 2022024 
(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))))