Average Error: 58.6 → 0.2
Time: 4.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, -\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, -\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}}\right) - 2 \cdot \varepsilon
double f(double eps) {
        double r70773 = 1.0;
        double r70774 = eps;
        double r70775 = r70773 - r70774;
        double r70776 = r70773 + r70774;
        double r70777 = r70775 / r70776;
        double r70778 = log(r70777);
        return r70778;
}
↓
double f(double eps) {
        double r70779 = -0.6666666666666666;
        double r70780 = eps;
        double r70781 = 3.0;
        double r70782 = pow(r70780, r70781);
        double r70783 = 1.0;
        double r70784 = pow(r70783, r70781);
        double r70785 = r70782 / r70784;
        double r70786 = 0.4;
        double r70787 = 5.0;
        double r70788 = pow(r70780, r70787);
        double r70789 = pow(r70783, r70787);
        double r70790 = r70788 / r70789;
        double r70791 = r70786 * r70790;
        double r70792 = -r70791;
        double r70793 = fma(r70779, r70785, r70792);
        double r70794 = 2.0;
        double r70795 = r70794 * r70780;
        double r70796 = r70793 - r70795;
        return r70796;
}

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. Using strategy rm

  3. Applied log-div58.6

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

  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. Using strategy rm

  7. Applied fma-udef0.2

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

  8. Applied associate--r+0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020018 +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))))