Average Error: 58.6 → 0.2
Time: 33.0s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\left({\varepsilon}^{5}\right), \frac{-2}{5}, \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\left({\varepsilon}^{5}\right), \frac{-2}{5}, \left(\varepsilon \cdot -2 + \varepsilon \cdot \left(\left(\frac{-2}{3} \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\right)
double f(double eps) {
        double r5639731 = 1.0;
        double r5639732 = eps;
        double r5639733 = r5639731 - r5639732;
        double r5639734 = r5639731 + r5639732;
        double r5639735 = r5639733 / r5639734;
        double r5639736 = log(r5639735);
        return r5639736;
}
↓
double f(double eps) {
        double r5639737 = eps;
        double r5639738 = 5.0;
        double r5639739 = pow(r5639737, r5639738);
        double r5639740 = -0.4;
        double r5639741 = -2.0;
        double r5639742 = r5639737 * r5639741;
        double r5639743 = -0.6666666666666666;
        double r5639744 = r5639743 * r5639737;
        double r5639745 = r5639744 * r5639737;
        double r5639746 = r5639737 * r5639745;
        double r5639747 = r5639742 + r5639746;
        double r5639748 = fma(r5639739, r5639740, r5639747);
        return r5639748;
}

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. 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. Simplified0.2

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

  5. Applied sub-neg0.2

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

  6. Applied distribute-rgt-in0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019124 +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))))