Average Error: 58.8 → 0.6
Time: 13.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{1 \cdot 1}\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{1 \cdot 1}\right)\right) + \log 1
double f(double eps) {
        double r2543688 = 1.0;
        double r2543689 = eps;
        double r2543690 = r2543688 - r2543689;
        double r2543691 = r2543688 + r2543689;
        double r2543692 = r2543690 / r2543691;
        double r2543693 = log(r2543692);
        return r2543693;
}
↓
double f(double eps) {
        double r2543694 = 2.0;
        double r2543695 = eps;
        double r2543696 = r2543695 * r2543695;
        double r2543697 = 1.0;
        double r2543698 = r2543697 * r2543697;
        double r2543699 = r2543696 / r2543698;
        double r2543700 = r2543695 + r2543699;
        double r2543701 = r2543696 - r2543700;
        double r2543702 = r2543694 * r2543701;
        double r2543703 = log(r2543697);
        double r2543704 = r2543702 + r2543703;
        return r2543704;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.2
Herbie0.6
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.8

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

  2. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\left(\log 1 + 2 \cdot {\varepsilon}^{2}\right) - \left(2 \cdot \frac{{\varepsilon}^{2}}{{1}^{2}} + 2 \cdot \varepsilon\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{1 \cdot 1}\right)\right)}\]

  4. Final simplification0.6

    \[\leadsto 2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{1 \cdot 1}\right)\right) + \log 1\]

Reproduce

herbie shell --seed 2019171 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))