Average Error: 58.6 → 0.2
Time: 19.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[-\mathsf{fma}\left(\frac{2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
-\mathsf{fma}\left(\frac{2}{3}, \frac{{\varepsilon}^{3}}{{1}^{3}}, \mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r51865 = 1.0;
        double r51866 = eps;
        double r51867 = r51865 - r51866;
        double r51868 = r51865 + r51866;
        double r51869 = r51867 / r51868;
        double r51870 = log(r51869);
        return r51870;
}
↓
double f(double eps) {
        double r51871 = 0.6666666666666666;
        double r51872 = eps;
        double r51873 = 3.0;
        double r51874 = pow(r51872, r51873);
        double r51875 = 1.0;
        double r51876 = pow(r51875, r51873);
        double r51877 = r51874 / r51876;
        double r51878 = 0.4;
        double r51879 = 5.0;
        double r51880 = pow(r51872, r51879);
        double r51881 = pow(r51875, r51879);
        double r51882 = r51880 / r51881;
        double r51883 = 2.0;
        double r51884 = r51883 * r51872;
        double r51885 = fma(r51878, r51882, r51884);
        double r51886 = fma(r51871, r51877, r51885);
        double r51887 = -r51886;
        return r51887;
}

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

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019198 +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))))