Average Error: 58.4 → 0.4
Time: 11.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log \left(e^{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3}}\right) - \left(2 \cdot \varepsilon + {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\log \left(e^{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3}}\right) - \left(2 \cdot \varepsilon + {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r951355 = 1.0;
        double r951356 = eps;
        double r951357 = r951355 - r951356;
        double r951358 = r951355 + r951356;
        double r951359 = r951357 / r951358;
        double r951360 = log(r951359);
        return r951360;
}
↓
double f(double eps) {
        double r951361 = eps;
        double r951362 = r951361 * r951361;
        double r951363 = r951362 * r951361;
        double r951364 = -0.6666666666666666;
        double r951365 = r951363 * r951364;
        double r951366 = exp(r951365);
        double r951367 = log(r951366);
        double r951368 = 2.0;
        double r951369 = r951368 * r951361;
        double r951370 = 5.0;
        double r951371 = pow(r951361, r951370);
        double r951372 = 0.4;
        double r951373 = r951371 * r951372;
        double r951374 = r951369 + r951373;
        double r951375 = r951367 - r951374;
        return r951375;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot {\varepsilon}^{3} + \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\right)}\]

  3. Simplified0.2

    \[\leadsto \color{blue}{\frac{-2}{3} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) - \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)}\]
  4. Using strategy rm

  5. Applied add-log-exp0.4

    \[\leadsto \color{blue}{\log \left(e^{\frac{-2}{3} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)}\right)} - \left(\frac{2}{5} \cdot {\varepsilon}^{5} + 2 \cdot \varepsilon\right)\]

  6. Final simplification0.4

    \[\leadsto \log \left(e^{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3}}\right) - \left(2 \cdot \varepsilon + {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]

Reproduce

herbie shell --seed 2019154 
(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))))