Average Error: 58.6 → 0.2
Time: 18.2s
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 r2197778 = 1.0;
        double r2197779 = eps;
        double r2197780 = r2197778 - r2197779;
        double r2197781 = r2197778 + r2197779;
        double r2197782 = r2197780 / r2197781;
        double r2197783 = log(r2197782);
        return r2197783;
}
↓
double f(double eps) {
        double r2197784 = eps;
        double r2197785 = r2197784 * r2197784;
        double r2197786 = r2197785 * r2197784;
        double r2197787 = -0.6666666666666666;
        double r2197788 = r2197786 * r2197787;
        double r2197789 = 2.0;
        double r2197790 = r2197789 * r2197784;
        double r2197791 = r2197788 - r2197790;
        double r2197792 = 0.4;
        double r2197793 = 5.0;
        double r2197794 = pow(r2197784, r2197793);
        double r2197795 = r2197792 * r2197794;
        double r2197796 = r2197791 - r2197795;
        return r2197796;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019139 
(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))))