Average Error: 58.7 → 0.2
Time: 36.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} - \varepsilon \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon + 2\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} - \varepsilon \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon + 2\right)
double f(double eps) {
        double r6199622 = 1.0;
        double r6199623 = eps;
        double r6199624 = r6199622 - r6199623;
        double r6199625 = r6199622 + r6199623;
        double r6199626 = r6199624 / r6199625;
        double r6199627 = log(r6199626);
        return r6199627;
}
↓
double f(double eps) {
        double r6199628 = eps;
        double r6199629 = 5.0;
        double r6199630 = pow(r6199628, r6199629);
        double r6199631 = -0.4;
        double r6199632 = r6199630 * r6199631;
        double r6199633 = 0.6666666666666666;
        double r6199634 = r6199633 * r6199628;
        double r6199635 = r6199634 * r6199628;
        double r6199636 = 2.0;
        double r6199637 = r6199635 + r6199636;
        double r6199638 = r6199628 * r6199637;
        double r6199639 = r6199632 - r6199638;
        return r6199639;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019112 
(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))))