Average Error: 58.6 → 0.2
Time: 12.7s
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 r2702792 = 1.0;
        double r2702793 = eps;
        double r2702794 = r2702792 - r2702793;
        double r2702795 = r2702792 + r2702793;
        double r2702796 = r2702794 / r2702795;
        double r2702797 = log(r2702796);
        return r2702797;
}
↓
double f(double eps) {
        double r2702798 = eps;
        double r2702799 = r2702798 * r2702798;
        double r2702800 = r2702798 * r2702799;
        double r2702801 = -0.6666666666666666;
        double r2702802 = -2.0;
        double r2702803 = 5.0;
        double r2702804 = pow(r2702798, r2702803);
        double r2702805 = -0.4;
        double r2702806 = r2702804 * r2702805;
        double r2702807 = fma(r2702798, r2702802, r2702806);
        double r2702808 = fma(r2702800, r2702801, r2702807);
        return r2702808;
}

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

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

  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 2019133 +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))))