Average Error: 58.6 → 0.6
Time: 13.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \left(\varepsilon + \frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) + \log 1
double f(double eps) {
        double r3563622 = 1.0;
        double r3563623 = eps;
        double r3563624 = r3563622 - r3563623;
        double r3563625 = r3563622 + r3563623;
        double r3563626 = r3563624 / r3563625;
        double r3563627 = log(r3563626);
        return r3563627;
}
↓
double f(double eps) {
        double r3563628 = 2.0;
        double r3563629 = eps;
        double r3563630 = r3563629 * r3563629;
        double r3563631 = 1.0;
        double r3563632 = r3563629 / r3563631;
        double r3563633 = r3563632 * r3563632;
        double r3563634 = r3563629 + r3563633;
        double r3563635 = r3563630 - r3563634;
        double r3563636 = r3563628 * r3563635;
        double r3563637 = log(r3563631);
        double r3563638 = r3563636 + r3563637;
        return r3563638;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 
(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))))