Average Error: 58.7 → 0.2
Time: 15.0s
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 r46974 = 1.0;
        double r46975 = eps;
        double r46976 = r46974 - r46975;
        double r46977 = r46974 + r46975;
        double r46978 = r46976 / r46977;
        double r46979 = log(r46978);
        return r46979;
}
↓
double f(double eps) {
        double r46980 = eps;
        double r46981 = 3.0;
        double r46982 = pow(r46980, r46981);
        double r46983 = 1.0;
        double r46984 = pow(r46983, r46981);
        double r46985 = r46982 / r46984;
        double r46986 = -0.6666666666666666;
        double r46987 = r46985 * r46986;
        double r46988 = 0.4;
        double r46989 = 5.0;
        double r46990 = pow(r46980, r46989);
        double r46991 = pow(r46983, r46989);
        double r46992 = r46990 / r46991;
        double r46993 = r46988 * r46992;
        double r46994 = 2.0;
        double r46995 = r46994 * r46980;
        double r46996 = r46993 + r46995;
        double r46997 = r46987 - r46996;
        return r46997;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.2
Herbie0.2
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied div-inv58.7

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

  4. Applied log-prod58.7

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

  5. Simplified58.7

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

  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 2019208 
(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))))