Average Error: 58.5 → 0.2
Time: 25.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}
double f(double eps) {
        double r5951348 = 1.0;
        double r5951349 = eps;
        double r5951350 = r5951348 - r5951349;
        double r5951351 = r5951348 + r5951349;
        double r5951352 = r5951350 / r5951351;
        double r5951353 = log(r5951352);
        return r5951353;
}
↓
double f(double eps) {
        double r5951354 = eps;
        double r5951355 = 5.0;
        double r5951356 = pow(r5951354, r5951355);
        double r5951357 = -0.4;
        double r5951358 = r5951356 * r5951357;
        double r5951359 = 0.6666666666666666;
        double r5951360 = r5951359 * r5951354;
        double r5951361 = r5951360 * r5951354;
        double r5951362 = r5951361 * r5951361;
        double r5951363 = 4.0;
        double r5951364 = r5951362 - r5951363;
        double r5951365 = r5951364 * r5951354;
        double r5951366 = 2.0;
        double r5951367 = r5951361 - r5951366;
        double r5951368 = r5951365 / r5951367;
        double r5951369 = r5951358 - r5951368;
        return r5951369;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  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. Simplified0.3

    \[\leadsto \color{blue}{\frac{-2}{5} \cdot {\varepsilon}^{5} - \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon + 2\right) \cdot \varepsilon}\]
  4. Using strategy rm

  5. Applied flip-+0.3

    \[\leadsto \frac{-2}{5} \cdot {\varepsilon}^{5} - \color{blue}{\frac{\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 2 \cdot 2}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}} \cdot \varepsilon\]

  6. Applied associate-*l/0.2

    \[\leadsto \frac{-2}{5} \cdot {\varepsilon}^{5} - \color{blue}{\frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 2 \cdot 2\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}}\]

  7. Final simplification0.2

    \[\leadsto {\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}\]

Reproduce

herbie shell --seed 2019107 
(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))))