Average Error: 58.4 → 0.3
Time: 16.5s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-2 \cdot \varepsilon + \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \frac{-2}{5} - \varepsilon \cdot \left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-2 \cdot \varepsilon + \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \frac{-2}{5} - \varepsilon \cdot \left(\frac{2}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)
double f(double eps) {
        double r3044669 = 1.0;
        double r3044670 = eps;
        double r3044671 = r3044669 - r3044670;
        double r3044672 = r3044669 + r3044670;
        double r3044673 = r3044671 / r3044672;
        double r3044674 = log(r3044673);
        return r3044674;
}
↓
double f(double eps) {
        double r3044675 = -2.0;
        double r3044676 = eps;
        double r3044677 = r3044675 * r3044676;
        double r3044678 = r3044676 * r3044676;
        double r3044679 = r3044678 * r3044676;
        double r3044680 = r3044678 * r3044679;
        double r3044681 = -0.4;
        double r3044682 = r3044680 * r3044681;
        double r3044683 = 0.6666666666666666;
        double r3044684 = r3044683 * r3044678;
        double r3044685 = r3044676 * r3044684;
        double r3044686 = r3044682 - r3044685;
        double r3044687 = r3044677 + r3044686;
        return r3044687;
}

Error

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Taylor expanded around 0 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019163 
(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))))