Average Error: 58.5 → 0.2
Time: 15.9s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \frac{-2}{3}, \left(\mathsf{fma}\left(\varepsilon, -2, \left({\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \frac{-2}{3}, \left(\mathsf{fma}\left(\varepsilon, -2, \left({\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)\right)\right)
double f(double eps) {
        double r1584500 = 1.0;
        double r1584501 = eps;
        double r1584502 = r1584500 - r1584501;
        double r1584503 = r1584500 + r1584501;
        double r1584504 = r1584502 / r1584503;
        double r1584505 = log(r1584504);
        return r1584505;
}
↓
double f(double eps) {
        double r1584506 = eps;
        double r1584507 = r1584506 * r1584506;
        double r1584508 = r1584506 * r1584507;
        double r1584509 = -0.6666666666666666;
        double r1584510 = -2.0;
        double r1584511 = 5.0;
        double r1584512 = pow(r1584506, r1584511);
        double r1584513 = -0.4;
        double r1584514 = r1584512 * r1584513;
        double r1584515 = fma(r1584506, r1584510, r1584514);
        double r1584516 = fma(r1584508, r1584509, r1584515);
        return r1584516;
}

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. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019132 +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))))