Average Error: 58.5 → 0.0
Time: 15.1s
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 r4156955 = 1.0;
        double r4156956 = eps;
        double r4156957 = r4156955 - r4156956;
        double r4156958 = r4156955 + r4156956;
        double r4156959 = r4156957 / r4156958;
        double r4156960 = log(r4156959);
        return r4156960;
}
↓
double f(double eps) {
        double r4156961 = eps;
        double r4156962 = -r4156961;
        double r4156963 = log1p(r4156962);
        double r4156964 = log1p(r4156961);
        double r4156965 = r4156963 - r4156964;
        return r4156965;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  3. Applied log-div58.5

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

  4. Simplified50.4

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

  6. Applied log1p-expm1-u50.4

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

  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 2019162 +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))))