Average Error: 58.8 → 0.2
Time: 25.8s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \frac{-2}{5} + \left(\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right) - \varepsilon \cdot 2\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \frac{-2}{5} + \left(\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right) - \varepsilon \cdot 2
double f(double eps) {
        double r2185714 = 1.0;
        double r2185715 = eps;
        double r2185716 = r2185714 - r2185715;
        double r2185717 = r2185714 + r2185715;
        double r2185718 = r2185716 / r2185717;
        double r2185719 = log(r2185718);
        return r2185719;
}
↓
double f(double eps) {
        double r2185720 = eps;
        double r2185721 = r2185720 * r2185720;
        double r2185722 = r2185721 * r2185721;
        double r2185723 = r2185720 * r2185722;
        double r2185724 = -0.4;
        double r2185725 = r2185723 * r2185724;
        double r2185726 = -0.6666666666666666;
        double r2185727 = r2185726 * r2185721;
        double r2185728 = r2185727 * r2185720;
        double r2185729 = r2185725 + r2185728;
        double r2185730 = 2.0;
        double r2185731 = r2185720 * r2185730;
        double r2185732 = r2185729 - r2185731;
        return r2185732;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.8

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

  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

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

  5. Applied associate--r+0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019149 
(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))))