Average Error: 58.7 → 0.2
Time: 18.9s
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 r1119504 = 1.0;
        double r1119505 = eps;
        double r1119506 = r1119504 - r1119505;
        double r1119507 = r1119504 + r1119505;
        double r1119508 = r1119506 / r1119507;
        double r1119509 = log(r1119508);
        return r1119509;
}
↓
double f(double eps) {
        double r1119510 = eps;
        double r1119511 = r1119510 * r1119510;
        double r1119512 = r1119511 * r1119510;
        double r1119513 = -0.6666666666666666;
        double r1119514 = r1119512 * r1119513;
        double r1119515 = 2.0;
        double r1119516 = r1119510 * r1119515;
        double r1119517 = 5.0;
        double r1119518 = pow(r1119510, r1119517);
        double r1119519 = 0.4;
        double r1119520 = r1119518 * r1119519;
        double r1119521 = r1119516 + r1119520;
        double r1119522 = r1119514 - r1119521;
        return r1119522;
}

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 2019144 
(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))))