Average Error: 58.7 → 0.2
Time: 5.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\left(-\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}}\right) - \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\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)
double f(double eps) {
        double r63600 = 1.0;
        double r63601 = eps;
        double r63602 = r63600 - r63601;
        double r63603 = r63600 + r63601;
        double r63604 = r63602 / r63603;
        double r63605 = log(r63604);
        return r63605;
}
↓
double f(double eps) {
        double r63606 = 0.6666666666666666;
        double r63607 = eps;
        double r63608 = 3.0;
        double r63609 = pow(r63607, r63608);
        double r63610 = 1.0;
        double r63611 = pow(r63610, r63608);
        double r63612 = r63609 / r63611;
        double r63613 = r63606 * r63612;
        double r63614 = -r63613;
        double r63615 = 0.4;
        double r63616 = 5.0;
        double r63617 = pow(r63607, r63616);
        double r63618 = pow(r63610, r63616);
        double r63619 = r63617 / r63618;
        double r63620 = 2.0;
        double r63621 = r63620 * r63607;
        double r63622 = fma(r63615, r63619, r63621);
        double r63623 = r63614 - r63622;
        return r63623;
}

Error

Bits error versus eps

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 log-div58.7

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

  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

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020033 +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))))