Average Error: 58.6 → 0.2
Time: 20.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2\right) \cdot \varepsilon - \frac{2}{5} \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) + -2\right) \cdot \varepsilon - \frac{2}{5} \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)
double f(double eps) {
        double r2866247 = 1.0;
        double r2866248 = eps;
        double r2866249 = r2866247 - r2866248;
        double r2866250 = r2866247 + r2866248;
        double r2866251 = r2866249 / r2866250;
        double r2866252 = log(r2866251);
        return r2866252;
}
↓
double f(double eps) {
        double r2866253 = -0.6666666666666666;
        double r2866254 = eps;
        double r2866255 = r2866254 * r2866254;
        double r2866256 = r2866253 * r2866255;
        double r2866257 = -2.0;
        double r2866258 = r2866256 + r2866257;
        double r2866259 = r2866258 * r2866254;
        double r2866260 = 0.4;
        double r2866261 = r2866255 * r2866255;
        double r2866262 = r2866261 * r2866254;
        double r2866263 = r2866260 * r2866262;
        double r2866264 = r2866259 - r2866263;
        return r2866264;
}

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. 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

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

  4. Taylor expanded around 0 0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019138 
(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))))