Average Error: 58.6 → 0.2
Time: 6.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r84610 = 1.0;
        double r84611 = eps;
        double r84612 = r84610 - r84611;
        double r84613 = r84610 + r84611;
        double r84614 = r84612 / r84613;
        double r84615 = log(r84614);
        return r84615;
}
↓
double f(double eps) {
        double r84616 = eps;
        double r84617 = 1.0;
        double r84618 = r84616 / r84617;
        double r84619 = 3.0;
        double r84620 = pow(r84618, r84619);
        double r84621 = -0.6666666666666666;
        double r84622 = 0.4;
        double r84623 = 5.0;
        double r84624 = pow(r84616, r84623);
        double r84625 = pow(r84617, r84623);
        double r84626 = r84624 / r84625;
        double r84627 = 2.0;
        double r84628 = r84627 * r84616;
        double r84629 = fma(r84622, r84626, r84628);
        double r84630 = -r84629;
        double r84631 = fma(r84620, r84621, r84630);
        return r84631;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.6

    \[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
  2. Using strategy rm

  3. Applied div-inv58.6

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

  4. Applied log-prod58.6

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

  5. Simplified58.6

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

  6. Taylor expanded around 0 0.2

    \[\leadsto \color{blue}{-\left(\frac{2}{3} \cdot \frac{{\varepsilon}^{3}}{{1}^{3}} + \left(\frac{2}{5} \cdot \frac{{\varepsilon}^{5}}{{1}^{5}} + 2 \cdot \varepsilon\right)\right)}\]

  7. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)}\]

  8. Final simplification0.2

    \[\leadsto \mathsf{fma}\left({\left(\frac{\varepsilon}{1}\right)}^{3}, \frac{-2}{3}, -\mathsf{fma}\left(\frac{2}{5}, \frac{{\varepsilon}^{5}}{{1}^{5}}, 2 \cdot \varepsilon\right)\right)\]

Reproduce

herbie shell --seed 2020045 +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))))