Average Error: 58.7 → 0.2
Time: 58.6s
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 r9029772 = 1.0;
        double r9029773 = eps;
        double r9029774 = r9029772 - r9029773;
        double r9029775 = r9029772 + r9029773;
        double r9029776 = r9029774 / r9029775;
        double r9029777 = log(r9029776);
        return r9029777;
}
↓
double f(double eps) {
        double r9029778 = eps;
        double r9029779 = 5.0;
        double r9029780 = pow(r9029778, r9029779);
        double r9029781 = -0.4;
        double r9029782 = r9029780 * r9029781;
        double r9029783 = 0.6666666666666666;
        double r9029784 = r9029783 * r9029778;
        double r9029785 = r9029784 * r9029778;
        double r9029786 = 2.0;
        double r9029787 = r9029785 + r9029786;
        double r9029788 = r9029778 * r9029787;
        double r9029789 = r9029782 - r9029788;
        return r9029789;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  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 2019128 
(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))))