Average Error: 58.6 → 0.2
Time: 14.8s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} + \left(\frac{-2}{3} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + -2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} + \left(\frac{-2}{3} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + -2 \cdot \varepsilon\right)
double f(double eps) {
        double r2901705 = 1.0;
        double r2901706 = eps;
        double r2901707 = r2901705 - r2901706;
        double r2901708 = r2901705 + r2901706;
        double r2901709 = r2901707 / r2901708;
        double r2901710 = log(r2901709);
        return r2901710;
}
↓
double f(double eps) {
        double r2901711 = eps;
        double r2901712 = 5.0;
        double r2901713 = pow(r2901711, r2901712);
        double r2901714 = -0.4;
        double r2901715 = r2901713 * r2901714;
        double r2901716 = -0.6666666666666666;
        double r2901717 = r2901711 * r2901711;
        double r2901718 = r2901711 * r2901717;
        double r2901719 = r2901716 * r2901718;
        double r2901720 = -2.0;
        double r2901721 = r2901720 * r2901711;
        double r2901722 = r2901719 + r2901721;
        double r2901723 = r2901715 + r2901722;
        return r2901723;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019129 
(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))))