Average Error: 58.5 → 0.6
Time: 8.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\log 1 + 2 \cdot \left(\varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right)
double f(double eps) {
        double r112844 = 1.0;
        double r112845 = eps;
        double r112846 = r112844 - r112845;
        double r112847 = r112844 + r112845;
        double r112848 = r112846 / r112847;
        double r112849 = log(r112848);
        return r112849;
}
↓
double f(double eps) {
        double r112850 = 1.0;
        double r112851 = log(r112850);
        double r112852 = 2.0;
        double r112853 = eps;
        double r112854 = r112853 * r112853;
        double r112855 = r112853 / r112850;
        double r112856 = fma(r112855, r112855, r112853);
        double r112857 = r112854 - r112856;
        double r112858 = r112852 * r112857;
        double r112859 = r112851 + r112858;
        return r112859;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 +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))))