Average Error: 58.6 → 0.6
Time: 10.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left(\varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left(\varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1
double f(double eps) {
        double r102918 = 1.0;
        double r102919 = eps;
        double r102920 = r102918 - r102919;
        double r102921 = r102918 + r102919;
        double r102922 = r102920 / r102921;
        double r102923 = log(r102922);
        return r102923;
}
↓
double f(double eps) {
        double r102924 = 2.0;
        double r102925 = eps;
        double r102926 = r102925 * r102925;
        double r102927 = 1.0;
        double r102928 = r102925 / r102927;
        double r102929 = fma(r102928, r102928, r102925);
        double r102930 = r102926 - r102929;
        double r102931 = r102924 * r102930;
        double r102932 = log(r102927);
        double r102933 = r102931 + r102932;
        return r102933;
}

Error

Bits error versus eps

Target

Original58.6
Target0.2
Herbie0.6
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

    \[\leadsto 2 \cdot \left(\varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]

Reproduce

herbie shell --seed 2020042 +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))))