Average Error: 58.4 → 0.2
Time: 58.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{-2}{3} \cdot \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) - 2 \cdot \varepsilon\right) - \frac{\frac{2}{5}}{{1}^{5}} \cdot {\varepsilon}^{5}\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\frac{-2}{3} \cdot \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) - 2 \cdot \varepsilon\right) - \frac{\frac{2}{5}}{{1}^{5}} \cdot {\varepsilon}^{5}
double f(double eps) {
        double r4491876 = 1.0;
        double r4491877 = eps;
        double r4491878 = r4491876 - r4491877;
        double r4491879 = r4491876 + r4491877;
        double r4491880 = r4491878 / r4491879;
        double r4491881 = log(r4491880);
        return r4491881;
}
↓
double f(double eps) {
        double r4491882 = -0.6666666666666666;
        double r4491883 = eps;
        double r4491884 = 1.0;
        double r4491885 = r4491883 / r4491884;
        double r4491886 = r4491885 * r4491885;
        double r4491887 = r4491885 * r4491886;
        double r4491888 = r4491882 * r4491887;
        double r4491889 = 2.0;
        double r4491890 = r4491889 * r4491883;
        double r4491891 = r4491888 - r4491890;
        double r4491892 = 0.4;
        double r4491893 = 5.0;
        double r4491894 = pow(r4491884, r4491893);
        double r4491895 = r4491892 / r4491894;
        double r4491896 = pow(r4491883, r4491893);
        double r4491897 = r4491895 * r4491896;
        double r4491898 = r4491891 - r4491897;
        return r4491898;
}

Error

Bits error versus eps

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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. Using strategy rm

  3. Applied log-div58.4

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

  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. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))