Average Error: 58.7 → 0.2
Time: 28.6s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - \left(\varepsilon \cdot 2 + {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - \left(\varepsilon \cdot 2 + {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r3106814 = 1.0;
        double r3106815 = eps;
        double r3106816 = r3106814 - r3106815;
        double r3106817 = r3106814 + r3106815;
        double r3106818 = r3106816 / r3106817;
        double r3106819 = log(r3106818);
        return r3106819;
}
↓
double f(double eps) {
        double r3106820 = eps;
        double r3106821 = r3106820 * r3106820;
        double r3106822 = r3106821 * r3106820;
        double r3106823 = -0.6666666666666666;
        double r3106824 = r3106822 * r3106823;
        double r3106825 = 2.0;
        double r3106826 = r3106820 * r3106825;
        double r3106827 = 5.0;
        double r3106828 = pow(r3106820, r3106827);
        double r3106829 = 0.4;
        double r3106830 = r3106828 * r3106829;
        double r3106831 = r3106826 + r3106830;
        double r3106832 = r3106824 - r3106831;
        return r3106832;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019142 
(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))))