Average Error: 58.6 → 0.0
Time: 17.5s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{log1p}\left(-\varepsilon\right) - \mathsf{log1p}\left(\varepsilon\right)
double f(double eps) {
        double r3152098 = 1.0;
        double r3152099 = eps;
        double r3152100 = r3152098 - r3152099;
        double r3152101 = r3152098 + r3152099;
        double r3152102 = r3152100 / r3152101;
        double r3152103 = log(r3152102);
        return r3152103;
}
↓
double f(double eps) {
        double r3152104 = eps;
        double r3152105 = -r3152104;
        double r3152106 = log1p(r3152105);
        double r3152107 = log1p(r3152104);
        double r3152108 = r3152106 - r3152107;
        return r3152108;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.6
Target0.2
Herbie0.0
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied log-div58.6

    \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) - \log \left(1 + \varepsilon\right)}\]

  4. Simplified50.5

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

  6. Applied log1p-expm1-u50.5

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))