Average Error: 58.7 → 0.2
Time: 11.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{-2}{3} - \left(2 \cdot \varepsilon + {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{-2}{3} - \left(2 \cdot \varepsilon + {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r990807 = 1.0;
        double r990808 = eps;
        double r990809 = r990807 - r990808;
        double r990810 = r990807 + r990808;
        double r990811 = r990809 / r990810;
        double r990812 = log(r990811);
        return r990812;
}
↓
double f(double eps) {
        double r990813 = eps;
        double r990814 = r990813 * r990813;
        double r990815 = r990813 * r990814;
        double r990816 = -0.6666666666666666;
        double r990817 = r990815 * r990816;
        double r990818 = 2.0;
        double r990819 = r990818 * r990813;
        double r990820 = 5.0;
        double r990821 = pow(r990813, r990820);
        double r990822 = 0.4;
        double r990823 = r990821 * r990822;
        double r990824 = r990819 + r990823;
        double r990825 = r990817 - r990824;
        return r990825;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 
(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))))