Average Error: 58.6 → 0.2
Time: 31.7s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\varepsilon, -2, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\varepsilon, -2, {\varepsilon}^{5} \cdot \frac{-2}{5}\right)\right)
double f(double eps) {
        double r3412822 = 1.0;
        double r3412823 = eps;
        double r3412824 = r3412822 - r3412823;
        double r3412825 = r3412822 + r3412823;
        double r3412826 = r3412824 / r3412825;
        double r3412827 = log(r3412826);
        return r3412827;
}
↓
double f(double eps) {
        double r3412828 = eps;
        double r3412829 = r3412828 * r3412828;
        double r3412830 = r3412828 * r3412829;
        double r3412831 = -0.6666666666666666;
        double r3412832 = -2.0;
        double r3412833 = 5.0;
        double r3412834 = pow(r3412828, r3412833);
        double r3412835 = -0.4;
        double r3412836 = r3412834 * r3412835;
        double r3412837 = fma(r3412828, r3412832, r3412836);
        double r3412838 = fma(r3412830, r3412831, r3412837);
        return r3412838;
}

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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019138 +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))))