Average Error: 58.7 → 0.2
Time: 21.7s
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 r97899 = 1.0;
        double r97900 = eps;
        double r97901 = r97899 - r97900;
        double r97902 = r97899 + r97900;
        double r97903 = r97901 / r97902;
        double r97904 = log(r97903);
        return r97904;
}
↓
double f(double eps) {
        double r97905 = 0.6666666666666666;
        double r97906 = eps;
        double r97907 = 3.0;
        double r97908 = pow(r97906, r97907);
        double r97909 = 1.0;
        double r97910 = pow(r97909, r97907);
        double r97911 = r97908 / r97910;
        double r97912 = 0.4;
        double r97913 = 5.0;
        double r97914 = pow(r97906, r97913);
        double r97915 = pow(r97909, r97913);
        double r97916 = r97914 / r97915;
        double r97917 = 2.0;
        double r97918 = r97917 * r97906;
        double r97919 = fma(r97912, r97916, r97918);
        double r97920 = fma(r97905, r97911, r97919);
        double r97921 = -r97920;
        return r97921;
}

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

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

  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 2019208 +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))))