Average Error: 58.5 → 0.3
Time: 21.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - 2 \cdot \varepsilon\right) - \frac{2}{5} \cdot {\varepsilon}^{5}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - 2 \cdot \varepsilon\right) - \frac{2}{5} \cdot {\varepsilon}^{5}
double f(double eps) {
        double r2373820 = 1.0;
        double r2373821 = eps;
        double r2373822 = r2373820 - r2373821;
        double r2373823 = r2373820 + r2373821;
        double r2373824 = r2373822 / r2373823;
        double r2373825 = log(r2373824);
        return r2373825;
}
↓
double f(double eps) {
        double r2373826 = eps;
        double r2373827 = r2373826 * r2373826;
        double r2373828 = r2373827 * r2373826;
        double r2373829 = -0.6666666666666666;
        double r2373830 = r2373828 * r2373829;
        double r2373831 = 2.0;
        double r2373832 = r2373831 * r2373826;
        double r2373833 = r2373830 - r2373832;
        double r2373834 = 0.4;
        double r2373835 = 5.0;
        double r2373836 = pow(r2373826, r2373835);
        double r2373837 = r2373834 * r2373836;
        double r2373838 = r2373833 - r2373837;
        return r2373838;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019141 
(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))))