Average Error: 58.6 → 0.7
Time: 4.8s
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 r101863 = 1.0;
        double r101864 = eps;
        double r101865 = r101863 - r101864;
        double r101866 = r101863 + r101864;
        double r101867 = r101865 / r101866;
        double r101868 = log(r101867);
        return r101868;
}
↓
double f(double eps) {
        double r101869 = 2.0;
        double r101870 = eps;
        double r101871 = 2.0;
        double r101872 = pow(r101870, r101871);
        double r101873 = 1.0;
        double r101874 = r101870 / r101873;
        double r101875 = fma(r101874, r101874, r101870);
        double r101876 = r101872 - r101875;
        double r101877 = r101869 * r101876;
        double r101878 = log(r101873);
        double r101879 = r101877 + r101878;
        return r101879;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020039 +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))))