Average Error: 58.5 → 0.2
Time: 28.3s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[(\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right))_*\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
(\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right))_*
double f(double eps) {
        double r6987692 = 1.0;
        double r6987693 = eps;
        double r6987694 = r6987692 - r6987693;
        double r6987695 = r6987692 + r6987693;
        double r6987696 = r6987694 / r6987695;
        double r6987697 = log(r6987696);
        return r6987697;
}
↓
double f(double eps) {
        double r6987698 = eps;
        double r6987699 = 5.0;
        double r6987700 = pow(r6987698, r6987699);
        double r6987701 = -0.4;
        double r6987702 = -2.0;
        double r6987703 = r6987698 * r6987702;
        double r6987704 = -0.6666666666666666;
        double r6987705 = r6987704 * r6987698;
        double r6987706 = r6987705 * r6987698;
        double r6987707 = r6987698 * r6987706;
        double r6987708 = r6987703 + r6987707;
        double r6987709 = fma(r6987700, r6987701, r6987708);
        return r6987709;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.5

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

  2. Taylor expanded around 0 0.2

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

  3. Simplified0.3

    \[\leadsto \color{blue}{(\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \frac{-2}{3}\right) \cdot \varepsilon - 2\right)\right))_*}\]
  4. Using strategy rm

  5. Applied sub-neg0.3

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

  6. Applied distribute-rgt-in0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

    \[\leadsto (\left({\varepsilon}^{5}\right) \cdot \frac{-2}{5} + \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right))_*\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))