Average Error: 58.6 → 0.2
Time: 43.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} - \varepsilon \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon + 2\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} - \varepsilon \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon + 2\right)
double f(double eps) {
        double r3799596 = 1.0;
        double r3799597 = eps;
        double r3799598 = r3799596 - r3799597;
        double r3799599 = r3799596 + r3799597;
        double r3799600 = r3799598 / r3799599;
        double r3799601 = log(r3799600);
        return r3799601;
}
↓
double f(double eps) {
        double r3799602 = eps;
        double r3799603 = 5.0;
        double r3799604 = pow(r3799602, r3799603);
        double r3799605 = -0.4;
        double r3799606 = r3799604 * r3799605;
        double r3799607 = 0.6666666666666666;
        double r3799608 = r3799607 * r3799602;
        double r3799609 = r3799608 * r3799602;
        double r3799610 = 2.0;
        double r3799611 = r3799609 + r3799610;
        double r3799612 = r3799602 * r3799611;
        double r3799613 = r3799606 - r3799612;
        return r3799613;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  2. 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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019124 
(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))))