Average Error: 58.4 → 0.3
Time: 5.8s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)
double f(double eps) {
        double r79679 = 1.0;
        double r79680 = eps;
        double r79681 = r79679 - r79680;
        double r79682 = r79679 + r79680;
        double r79683 = r79681 / r79682;
        double r79684 = log(r79683);
        return r79684;
}
↓
double f(double eps) {
        double r79685 = 2.0;
        double r79686 = eps;
        double r79687 = r79685 * r79686;
        double r79688 = -r79687;
        double r79689 = 0.6666666666666666;
        double r79690 = 3.0;
        double r79691 = pow(r79686, r79690);
        double r79692 = 0.4;
        double r79693 = 5.0;
        double r79694 = pow(r79686, r79693);
        double r79695 = r79692 * r79694;
        double r79696 = fma(r79689, r79691, r79695);
        double r79697 = r79688 - r79696;
        return r79697;
}

Error

Bits error versus eps

Target

Original58.4
Target0.3
Herbie0.3
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

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 0.3

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

  5. Simplified0.3

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

  6. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{-\left(2 \cdot \varepsilon + \left(0.66666666666666663 \cdot {\varepsilon}^{3} + 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\right)}\]

  7. Simplified0.3

    \[\leadsto \color{blue}{\left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)}\]

  8. Final simplification0.3

    \[\leadsto \left(-2 \cdot \varepsilon\right) - \mathsf{fma}\left(0.66666666666666663, {\varepsilon}^{3}, 0.40000000000000002 \cdot {\varepsilon}^{5}\right)\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(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))))