Average Error: 58.6 → 0.2
Time: 43.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[{\varepsilon}^{5} \cdot \frac{-2}{5} - \left(2 + \log \left(e^{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon}\right)\right) \cdot \varepsilon\]
double f(double eps) {
        double r6045954 = 1.0;
        double r6045955 = eps;
        double r6045956 = r6045954 - r6045955;
        double r6045957 = r6045954 + r6045955;
        double r6045958 = r6045956 / r6045957;
        double r6045959 = log(r6045958);
        return r6045959;
}
↓
double f(double eps) {
        double r6045960 = eps;
        double r6045961 = 5.0;
        double r6045962 = pow(r6045960, r6045961);
        double r6045963 = -0.4;
        double r6045964 = r6045962 * r6045963;
        double r6045965 = 2.0;
        double r6045966 = 0.6666666666666666;
        double r6045967 = r6045966 * r6045960;
        double r6045968 = r6045967 * r6045960;
        double r6045969 = exp(r6045968);
        double r6045970 = log(r6045969);
        double r6045971 = r6045965 + r6045970;
        double r6045972 = r6045971 * r6045960;
        double r6045973 = r6045964 - r6045972;
        return r6045973;
}


\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
{\varepsilon}^{5} \cdot \frac{-2}{5} - \left(2 + \log \left(e^{\left(\frac{2}{3} \cdot \varepsilon\right) \cdot \varepsilon}\right)\right) \cdot \varepsilon

Error

Bits error versus eps

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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  5. Applied add-log-exp0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019101 
(FPCore (eps)
  :name "logq (problem 3.4.3)"

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

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