Average Error: 58.7 → 0.6
Time: 7.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
2 \cdot \left({\varepsilon}^{2} - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right)\right) + \log 1
double f(double eps) {
        double r108886 = 1.0;
        double r108887 = eps;
        double r108888 = r108886 - r108887;
        double r108889 = r108886 + r108887;
        double r108890 = r108888 / r108889;
        double r108891 = log(r108890);
        return r108891;
}
↓
double f(double eps) {
        double r108892 = 2.0;
        double r108893 = eps;
        double r108894 = 2.0;
        double r108895 = pow(r108893, r108894);
        double r108896 = 1.0;
        double r108897 = r108893 / r108896;
        double r108898 = fma(r108897, r108897, r108893);
        double r108899 = r108895 - r108898;
        double r108900 = r108892 * r108899;
        double r108901 = log(r108896);
        double r108902 = r108900 + r108901;
        return r108902;
}

Error

Bits error versus eps

Target

Original58.7
Target0.2
Herbie0.6
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.7

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

  2. 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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020083 +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))))