Average Error: 58.5 → 0.2
Time: 12.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(\frac{-2}{3} \cdot {\left(\frac{\varepsilon}{1}\right)}^{3} - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(\frac{-2}{3} \cdot {\left(\frac{\varepsilon}{1}\right)}^{3} - \frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon
double f(double eps) {
        double r83964 = 1.0;
        double r83965 = eps;
        double r83966 = r83964 - r83965;
        double r83967 = r83964 + r83965;
        double r83968 = r83966 / r83967;
        double r83969 = log(r83968);
        return r83969;
}
↓
double f(double eps) {
        double r83970 = -0.6666666666666666;
        double r83971 = eps;
        double r83972 = 1.0;
        double r83973 = r83971 / r83972;
        double r83974 = 3.0;
        double r83975 = pow(r83973, r83974);
        double r83976 = r83970 * r83975;
        double r83977 = 0.4;
        double r83978 = 5.0;
        double r83979 = pow(r83971, r83978);
        double r83980 = pow(r83972, r83978);
        double r83981 = r83979 / r83980;
        double r83982 = r83977 * r83981;
        double r83983 = r83976 - r83982;
        double r83984 = 2.0;
        double r83985 = r83984 * r83971;
        double r83986 = r83983 - r83985;
        return r83986;
}

Error

Bits error versus eps

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

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

  3. Applied add-exp-log58.5

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

  4. Applied add-exp-log58.5

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

  5. Applied div-exp58.5

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

  6. Applied rem-log-exp58.5

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

  7. Taylor expanded around 0 0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 
(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))))