Average Error: 58.6 → 0.2
Time: 5.1s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r69832 = 1.0;
        double r69833 = eps;
        double r69834 = r69832 - r69833;
        double r69835 = r69832 + r69833;
        double r69836 = r69834 / r69835;
        double r69837 = log(r69836);
        return r69837;
}
↓
double f(double eps) {
        double r69838 = 0.6666666666666666;
        double r69839 = eps;
        double r69840 = 3.0;
        double r69841 = pow(r69839, r69840);
        double r69842 = 1.0;
        double r69843 = pow(r69842, r69840);
        double r69844 = r69841 / r69843;
        double r69845 = r69838 * r69844;
        double r69846 = 0.4;
        double r69847 = 5.0;
        double r69848 = pow(r69839, r69847);
        double r69849 = pow(r69842, r69847);
        double r69850 = r69848 / r69849;
        double r69851 = r69846 * r69850;
        double r69852 = r69845 + r69851;
        double r69853 = 2.0;
        double r69854 = r69853 * r69839;
        double r69855 = r69852 + r69854;
        double r69856 = -r69855;
        return r69856;
}

Error

Bits error versus eps

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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. Using strategy rm

  3. Applied log-div58.6

    \[\leadsto \color{blue}{\log \left(1 - \varepsilon\right) - \log \left(1 + \varepsilon\right)}\]

  4. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]
  5. Using strategy rm

  6. Applied associate-+r+0.2

    \[\leadsto -\color{blue}{\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)}\]

  7. Final simplification0.2

    \[\leadsto -\left(\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) + 2 \cdot \varepsilon\right)\]

Reproduce

herbie shell --seed 2020018 
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))