Average Error: 58.7 → 0.2
Time: 8.5s
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 r654647 = 1.0;
        double r654648 = eps;
        double r654649 = r654647 - r654648;
        double r654650 = r654647 + r654648;
        double r654651 = r654649 / r654650;
        double r654652 = log(r654651);
        return r654652;
}
↓
double f(double eps) {
        double r654653 = eps;
        double r654654 = r654653 * r654653;
        double r654655 = r654654 * r654653;
        double r654656 = -0.6666666666666666;
        double r654657 = r654655 * r654656;
        double r654658 = 2.0;
        double r654659 = r654653 * r654658;
        double r654660 = 5.0;
        double r654661 = pow(r654653, r654660);
        double r654662 = 0.4;
        double r654663 = r654661 * r654662;
        double r654664 = r654659 + r654663;
        double r654665 = r654657 - r654664;
        return r654665;
}

Error

Bits error versus eps

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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 2019156 
(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))))