Average Error: 58.6 → 0.2
Time: 19.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{8}{27}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + 8\right) \cdot \varepsilon}{\left(4 - \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right) \cdot 2\right) + \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right)}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{8}{27}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + 8\right) \cdot \varepsilon}{\left(4 - \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right) \cdot 2\right) + \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right)}
double f(double eps) {
        double r5290553 = 1.0;
        double r5290554 = eps;
        double r5290555 = r5290553 - r5290554;
        double r5290556 = r5290553 + r5290554;
        double r5290557 = r5290555 / r5290556;
        double r5290558 = log(r5290557);
        return r5290558;
}
↓
double f(double eps) {
        double r5290559 = eps;
        double r5290560 = 5.0;
        double r5290561 = pow(r5290559, r5290560);
        double r5290562 = -0.4;
        double r5290563 = r5290561 * r5290562;
        double r5290564 = r5290559 * r5290559;
        double r5290565 = 0.2962962962962963;
        double r5290566 = r5290564 * r5290565;
        double r5290567 = r5290564 * r5290564;
        double r5290568 = r5290566 * r5290567;
        double r5290569 = 8.0;
        double r5290570 = r5290568 + r5290569;
        double r5290571 = r5290570 * r5290559;
        double r5290572 = 4.0;
        double r5290573 = 0.6666666666666666;
        double r5290574 = r5290559 * r5290573;
        double r5290575 = r5290559 * r5290574;
        double r5290576 = 2.0;
        double r5290577 = r5290575 * r5290576;
        double r5290578 = r5290572 - r5290577;
        double r5290579 = r5290575 * r5290575;
        double r5290580 = r5290578 + r5290579;
        double r5290581 = r5290571 / r5290580;
        double r5290582 = r5290563 - r5290581;
        return r5290582;
}

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

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

  5. Applied flip3-+0.2

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

  6. Applied associate-*l/0.2

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

  7. Simplified0.2

    \[\leadsto \frac{-2}{5} \cdot {\varepsilon}^{5} - \frac{\color{blue}{\left(8 + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\frac{8}{27} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \varepsilon}}{\left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) + \left(2 \cdot 2 - \left(\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot 2\right)}\]

  8. Final simplification0.2

    \[\leadsto {\varepsilon}^{5} \cdot \frac{-2}{5} - \frac{\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{8}{27}\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + 8\right) \cdot \varepsilon}{\left(4 - \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right) \cdot 2\right) + \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \frac{2}{3}\right)\right)}\]

Reproduce

herbie shell --seed 2019104 
(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))))