Average Error: 58.6 → 0.6
Time: 20.8s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(-2, \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \mathsf{fma}\left(2, {\varepsilon}^{2}, \log 1\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(-2, \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \mathsf{fma}\left(2, {\varepsilon}^{2}, \log 1\right)\right)
double f(double eps) {
        double r77629 = 1.0;
        double r77630 = eps;
        double r77631 = r77629 - r77630;
        double r77632 = r77629 + r77630;
        double r77633 = r77631 / r77632;
        double r77634 = log(r77633);
        return r77634;
}
↓
double f(double eps) {
        double r77635 = 2.0;
        double r77636 = -r77635;
        double r77637 = eps;
        double r77638 = 1.0;
        double r77639 = r77637 / r77638;
        double r77640 = fma(r77639, r77639, r77637);
        double r77641 = 2.0;
        double r77642 = pow(r77637, r77641);
        double r77643 = log(r77638);
        double r77644 = fma(r77635, r77642, r77643);
        double r77645 = fma(r77636, r77640, r77644);
        return r77645;
}

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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019212 +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))))