Average Error: 58.6 → 0.2
Time: 14.5s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\frac{{\varepsilon}^{3}}{{1}^{3}} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\frac{{\varepsilon}^{3}}{{1}^{3}} \cdot \frac{-2}{3} - \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)
double f(double eps) {
        double r46886 = 1.0;
        double r46887 = eps;
        double r46888 = r46886 - r46887;
        double r46889 = r46886 + r46887;
        double r46890 = r46888 / r46889;
        double r46891 = log(r46890);
        return r46891;
}
↓
double f(double eps) {
        double r46892 = eps;
        double r46893 = 3.0;
        double r46894 = pow(r46892, r46893);
        double r46895 = 1.0;
        double r46896 = pow(r46895, r46893);
        double r46897 = r46894 / r46896;
        double r46898 = -0.6666666666666666;
        double r46899 = r46897 * r46898;
        double r46900 = 0.4;
        double r46901 = 5.0;
        double r46902 = pow(r46892, r46901);
        double r46903 = pow(r46895, r46901);
        double r46904 = r46902 / r46903;
        double r46905 = r46900 * r46904;
        double r46906 = 2.0;
        double r46907 = r46906 * r46892;
        double r46908 = r46905 + r46907;
        double r46909 = r46899 - r46908;
        return r46909;
}

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. Using strategy rm

  3. Applied div-inv58.6

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

  4. Applied log-prod58.6

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

  5. Simplified58.6

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

  6. Taylor expanded around 0 0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019198 
(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))))