Average Error: 58.4 → 0.2
Time: 34.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}
double f(double eps) {
        double r6280510 = 1.0;
        double r6280511 = eps;
        double r6280512 = r6280510 - r6280511;
        double r6280513 = r6280510 + r6280511;
        double r6280514 = r6280512 / r6280513;
        double r6280515 = log(r6280514);
        return r6280515;
}
↓
double f(double eps) {
        double r6280516 = eps;
        double r6280517 = 5.0;
        double r6280518 = pow(r6280516, r6280517);
        double r6280519 = -0.4;
        double r6280520 = r6280518 * r6280519;
        double r6280521 = 0.6666666666666666;
        double r6280522 = r6280521 * r6280516;
        double r6280523 = r6280522 * r6280516;
        double r6280524 = r6280523 * r6280523;
        double r6280525 = 4.0;
        double r6280526 = r6280524 - r6280525;
        double r6280527 = r6280526 * r6280516;
        double r6280528 = 2.0;
        double r6280529 = r6280523 - r6280528;
        double r6280530 = r6280527 / r6280529;
        double r6280531 = r6280520 - r6280530;
        return r6280531;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.4
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

  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. Using strategy rm

  5. Applied flip-+0.3

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

  6. Applied associate-*l/0.2

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

  7. Final simplification0.2

    \[\leadsto {\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) - 4\right) \cdot \varepsilon}{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon - 2}\]

Reproduce

herbie shell --seed 2019125 
(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))))