Average Error: 58.5 → 0.2
Time: 36.5s
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 r6337568 = 1.0;
        double r6337569 = eps;
        double r6337570 = r6337568 - r6337569;
        double r6337571 = r6337568 + r6337569;
        double r6337572 = r6337570 / r6337571;
        double r6337573 = log(r6337572);
        return r6337573;
}
↓
double f(double eps) {
        double r6337574 = eps;
        double r6337575 = 5.0;
        double r6337576 = pow(r6337574, r6337575);
        double r6337577 = -0.4;
        double r6337578 = r6337576 * r6337577;
        double r6337579 = 0.6666666666666666;
        double r6337580 = r6337579 * r6337574;
        double r6337581 = r6337580 * r6337574;
        double r6337582 = r6337581 * r6337581;
        double r6337583 = 4.0;
        double r6337584 = r6337582 - r6337583;
        double r6337585 = r6337584 * r6337574;
        double r6337586 = 2.0;
        double r6337587 = r6337581 - r6337586;
        double r6337588 = r6337585 / r6337587;
        double r6337589 = r6337578 - r6337588;
        return r6337589;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  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-+0.2

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

  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 2019120 
(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))))