Average Error: 58.6 → 0.2
Time: 7.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon
double f(double eps) {
        double r107359 = 1.0;
        double r107360 = eps;
        double r107361 = r107359 - r107360;
        double r107362 = r107359 + r107360;
        double r107363 = r107361 / r107362;
        double r107364 = log(r107363);
        return r107364;
}
↓
double f(double eps) {
        double r107365 = eps;
        double r107366 = 1.0;
        double r107367 = r107365 / r107366;
        double r107368 = 3.0;
        double r107369 = pow(r107367, r107368);
        double r107370 = -0.6666666666666666;
        double r107371 = r107369 * r107370;
        double r107372 = 0.4;
        double r107373 = 5.0;
        double r107374 = pow(r107365, r107373);
        double r107375 = pow(r107366, r107373);
        double r107376 = r107374 / r107375;
        double r107377 = r107372 * r107376;
        double r107378 = r107371 - r107377;
        double r107379 = 2.0;
        double r107380 = r107379 * r107365;
        double r107381 = r107378 - r107380;
        return r107381;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied log-div58.6

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

  4. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  5. Simplified0.2

    \[\leadsto \color{blue}{\left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon}\]

  6. Final simplification0.2

    \[\leadsto \left({\left(\frac{\varepsilon}{1}\right)}^{3} \cdot \frac{-2}{3} - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon\]

Reproduce

herbie shell --seed 2020045 
(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))))