Average Error: 58.4 → 0.0
Time: 10.8s
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 r181124 = 1.0;
        double r181125 = eps;
        double r181126 = r181124 - r181125;
        double r181127 = r181124 + r181125;
        double r181128 = r181126 / r181127;
        double r181129 = log(r181128);
        return r181129;
}
↓
double f(double eps) {
        double r181130 = eps;
        double r181131 = -r181130;
        double r181132 = log1p(r181131);
        double r181133 = log1p(r181130);
        double r181134 = r181132 - r181133;
        return r181134;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.2
Herbie0.0
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  3. Applied log-div58.4

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

  4. Simplified50.3

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

  6. Applied log1p-expm1-u50.3

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

  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 2019154 +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))))