Average Error: 58.7 → 0.2
Time: 11.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - \left(\varepsilon \cdot 2 + {\varepsilon}^{5} \cdot \frac{2}{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - \left(\varepsilon \cdot 2 + {\varepsilon}^{5} \cdot \frac{2}{5}\right)
double f(double eps) {
        double r1790662 = 1.0;
        double r1790663 = eps;
        double r1790664 = r1790662 - r1790663;
        double r1790665 = r1790662 + r1790663;
        double r1790666 = r1790664 / r1790665;
        double r1790667 = log(r1790666);
        return r1790667;
}
↓
double f(double eps) {
        double r1790668 = eps;
        double r1790669 = r1790668 * r1790668;
        double r1790670 = r1790669 * r1790668;
        double r1790671 = -0.6666666666666666;
        double r1790672 = r1790670 * r1790671;
        double r1790673 = 2.0;
        double r1790674 = r1790668 * r1790673;
        double r1790675 = 5.0;
        double r1790676 = pow(r1790668, r1790675);
        double r1790677 = 0.4;
        double r1790678 = r1790676 * r1790677;
        double r1790679 = r1790674 + r1790678;
        double r1790680 = r1790672 - r1790679;
        return r1790680;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 
(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))))