Average Error: 58.6 → 0.6
Time: 10.8s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\varepsilon \cdot \left(\left(\varepsilon - \frac{\varepsilon}{{1}^{2}}\right) \cdot 2 - 2\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\varepsilon \cdot \left(\left(\varepsilon - \frac{\varepsilon}{{1}^{2}}\right) \cdot 2 - 2\right) + \log 1
double f(double eps) {
        double r69993 = 1.0;
        double r69994 = eps;
        double r69995 = r69993 - r69994;
        double r69996 = r69993 + r69994;
        double r69997 = r69995 / r69996;
        double r69998 = log(r69997);
        return r69998;
}
↓
double f(double eps) {
        double r69999 = eps;
        double r70000 = 1.0;
        double r70001 = 2.0;
        double r70002 = pow(r70000, r70001);
        double r70003 = r69999 / r70002;
        double r70004 = r69999 - r70003;
        double r70005 = 2.0;
        double r70006 = r70004 * r70005;
        double r70007 = r70006 - r70005;
        double r70008 = r69999 * r70007;
        double r70009 = log(r70000);
        double r70010 = r70008 + r70009;
        return r70010;
}

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

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 197574269 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))