Average Error: 58.5 → 0.2
Time: 13.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-2 \cdot \varepsilon + {\varepsilon}^{5} \cdot \frac{-2}{5}\right) - \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right) \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(-2 \cdot \varepsilon + {\varepsilon}^{5} \cdot \frac{-2}{5}\right) - \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{2}{3}\right) \cdot \varepsilon
double f(double eps) {
        double r2801463 = 1.0;
        double r2801464 = eps;
        double r2801465 = r2801463 - r2801464;
        double r2801466 = r2801463 + r2801464;
        double r2801467 = r2801465 / r2801466;
        double r2801468 = log(r2801467);
        return r2801468;
}
↓
double f(double eps) {
        double r2801469 = -2.0;
        double r2801470 = eps;
        double r2801471 = r2801469 * r2801470;
        double r2801472 = 5.0;
        double r2801473 = pow(r2801470, r2801472);
        double r2801474 = -0.4;
        double r2801475 = r2801473 * r2801474;
        double r2801476 = r2801471 + r2801475;
        double r2801477 = r2801470 * r2801470;
        double r2801478 = 0.6666666666666666;
        double r2801479 = r2801477 * r2801478;
        double r2801480 = r2801479 * r2801470;
        double r2801481 = r2801476 - r2801480;
        return r2801481;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  2. 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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(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))))