Average Error: 61.0 → 0.0
Time: 21.1s
Precision: 64
\[-1 \lt x \land x \lt 1\]
\[\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}\]
↓
\[\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}\]
\frac{\log \left(1 - x\right)}{\log \left(1 + x\right)}
↓
\frac{\mathsf{log1p}\left(-x\right)}{\mathsf{log1p}\left(x\right)}
double f(double x) {
        double r3474637 = 1.0;
        double r3474638 = x;
        double r3474639 = r3474637 - r3474638;
        double r3474640 = log(r3474639);
        double r3474641 = r3474637 + r3474638;
        double r3474642 = log(r3474641);
        double r3474643 = r3474640 / r3474642;
        return r3474643;
}
↓
double f(double x) {
        double r3474644 = x;
        double r3474645 = -r3474644;
        double r3474646 = log1p(r3474645);
        double r3474647 = log1p(r3474644);
        double r3474648 = r3474646 / r3474647;
        return r3474648;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original61.0
Target0.3
Herbie0.0
\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + \frac{5}{12} \cdot {x}^{3}\right)\]

Derivation

  1. Initial program 61.0

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

  2. Simplified60.1

    \[\leadsto \color{blue}{\frac{\log \left(1 - x\right)}{\mathsf{log1p}\left(x\right)}}\]
  3. Using strategy rm

  4. Applied log1p-expm1-u60.1

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "qlog (example 3.10)"
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 5/12 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))