Average Error: 58.7 → 0.2
Time: 33.4s
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 r4164323 = 1.0;
        double r4164324 = eps;
        double r4164325 = r4164323 - r4164324;
        double r4164326 = r4164323 + r4164324;
        double r4164327 = r4164325 / r4164326;
        double r4164328 = log(r4164327);
        return r4164328;
}
↓
double f(double eps) {
        double r4164329 = eps;
        double r4164330 = 5.0;
        double r4164331 = pow(r4164329, r4164330);
        double r4164332 = -0.4;
        double r4164333 = r4164331 * r4164332;
        double r4164334 = 0.6666666666666666;
        double r4164335 = r4164334 * r4164329;
        double r4164336 = r4164335 * r4164329;
        double r4164337 = 2.0;
        double r4164338 = r4164336 + r4164337;
        double r4164339 = r4164329 * r4164338;
        double r4164340 = r4164333 - r4164339;
        return r4164340;
}

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 2019121 
(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))))