Average Error: 58.5 → 0.2
Time: 16.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{-2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right) - 2\right) \cdot \varepsilon - \frac{2}{5} \cdot {\varepsilon}^{5}\]
\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 {\varepsilon}^{5}
double f(double eps) {
        double r3456792 = 1.0;
        double r3456793 = eps;
        double r3456794 = r3456792 - r3456793;
        double r3456795 = r3456792 + r3456793;
        double r3456796 = r3456794 / r3456795;
        double r3456797 = log(r3456796);
        return r3456797;
}
↓
double f(double eps) {
        double r3456798 = -0.6666666666666666;
        double r3456799 = eps;
        double r3456800 = r3456799 * r3456799;
        double r3456801 = r3456798 * r3456800;
        double r3456802 = 2.0;
        double r3456803 = r3456801 - r3456802;
        double r3456804 = r3456803 * r3456799;
        double r3456805 = 0.4;
        double r3456806 = 5.0;
        double r3456807 = pow(r3456799, r3456806);
        double r3456808 = r3456805 * r3456807;
        double r3456809 = r3456804 - r3456808;
        return r3456809;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  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

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

  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

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 
(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))))