Average Error: 58.4 → 0.2
Time: 22.8s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\frac{-2}{3} \cdot \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) - \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, \varepsilon \cdot 2\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\frac{-2}{3} \cdot \left(\frac{\varepsilon}{1} \cdot \left(\frac{\varepsilon}{1} \cdot \frac{\varepsilon}{1}\right)\right) - \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, \varepsilon \cdot 2\right)
double f(double eps) {
        double r3007718 = 1.0;
        double r3007719 = eps;
        double r3007720 = r3007718 - r3007719;
        double r3007721 = r3007718 + r3007719;
        double r3007722 = r3007720 / r3007721;
        double r3007723 = log(r3007722);
        return r3007723;
}
↓
double f(double eps) {
        double r3007724 = -0.6666666666666666;
        double r3007725 = eps;
        double r3007726 = 1.0;
        double r3007727 = r3007725 / r3007726;
        double r3007728 = r3007727 * r3007727;
        double r3007729 = r3007727 * r3007728;
        double r3007730 = r3007724 * r3007729;
        double r3007731 = 0.4;
        double r3007732 = 5.0;
        double r3007733 = pow(r3007725, r3007732);
        double r3007734 = pow(r3007726, r3007732);
        double r3007735 = r3007733 / r3007734;
        double r3007736 = 2.0;
        double r3007737 = r3007725 * r3007736;
        double r3007738 = fma(r3007731, r3007735, r3007737);
        double r3007739 = r3007730 - r3007738;
        return r3007739;
}

Error

Bits error versus eps

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

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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))))