Average Error: 58.6 → 0.2
Time: 16.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left({\varepsilon}^{5} \cdot \frac{2}{5} + \left(\varepsilon \cdot 2 + \varepsilon \cdot \left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\left({\varepsilon}^{5} \cdot \frac{2}{5} + \left(\varepsilon \cdot 2 + \varepsilon \cdot \left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)
double f(double eps) {
        double r2698810 = 1.0;
        double r2698811 = eps;
        double r2698812 = r2698810 - r2698811;
        double r2698813 = r2698810 + r2698811;
        double r2698814 = r2698812 / r2698813;
        double r2698815 = log(r2698814);
        return r2698815;
}
↓
double f(double eps) {
        double r2698816 = eps;
        double r2698817 = 5.0;
        double r2698818 = pow(r2698816, r2698817);
        double r2698819 = 0.4;
        double r2698820 = r2698818 * r2698819;
        double r2698821 = 2.0;
        double r2698822 = r2698816 * r2698821;
        double r2698823 = 0.6666666666666666;
        double r2698824 = r2698816 * r2698816;
        double r2698825 = r2698823 * r2698824;
        double r2698826 = r2698816 * r2698825;
        double r2698827 = r2698822 + r2698826;
        double r2698828 = r2698820 + r2698827;
        double r2698829 = -r2698828;
        return r2698829;
}

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

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

  5. Applied distribute-rgt-in0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019134 
(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))))