Average Error: 58.6 → 0.2
Time: 12.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, -2, \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\frac{-2}{3}, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon, \mathsf{fma}\left(\varepsilon, -2, \frac{-2}{5} \cdot {\varepsilon}^{5}\right)\right)
double f(double eps) {
        double r1314899 = 1.0;
        double r1314900 = eps;
        double r1314901 = r1314899 - r1314900;
        double r1314902 = r1314899 + r1314900;
        double r1314903 = r1314901 / r1314902;
        double r1314904 = log(r1314903);
        return r1314904;
}
↓
double f(double eps) {
        double r1314905 = -0.6666666666666666;
        double r1314906 = eps;
        double r1314907 = r1314906 * r1314906;
        double r1314908 = r1314907 * r1314906;
        double r1314909 = -2.0;
        double r1314910 = -0.4;
        double r1314911 = 5.0;
        double r1314912 = pow(r1314906, r1314911);
        double r1314913 = r1314910 * r1314912;
        double r1314914 = fma(r1314906, r1314909, r1314913);
        double r1314915 = fma(r1314905, r1314908, r1314914);
        return r1314915;
}

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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))