Average Error: 58.6 → 0.2
Time: 15.6s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - {\varepsilon}^{5} \cdot \frac{2}{5}\right) - 2 \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \frac{-2}{3} - {\varepsilon}^{5} \cdot \frac{2}{5}\right) - 2 \cdot \varepsilon
double f(double eps) {
        double r1327618 = 1.0;
        double r1327619 = eps;
        double r1327620 = r1327618 - r1327619;
        double r1327621 = r1327618 + r1327619;
        double r1327622 = r1327620 / r1327621;
        double r1327623 = log(r1327622);
        return r1327623;
}
↓
double f(double eps) {
        double r1327624 = eps;
        double r1327625 = r1327624 * r1327624;
        double r1327626 = r1327625 * r1327624;
        double r1327627 = -0.6666666666666666;
        double r1327628 = r1327626 * r1327627;
        double r1327629 = 5.0;
        double r1327630 = pow(r1327624, r1327629);
        double r1327631 = 0.4;
        double r1327632 = r1327630 * r1327631;
        double r1327633 = r1327628 - r1327632;
        double r1327634 = 2.0;
        double r1327635 = r1327634 * r1327624;
        double r1327636 = r1327633 - r1327635;
        return r1327636;
}

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

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

  5. Applied associate--r+0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 
(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))))