Average Error: 58.6 → 0.3
Time: 17.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\frac{-2}{5}, {\varepsilon}^{5}, -2 \cdot \varepsilon\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \frac{-2}{3}, \mathsf{fma}\left(\frac{-2}{5}, {\varepsilon}^{5}, -2 \cdot \varepsilon\right)\right)
double f(double eps) {
        double r2099634 = 1.0;
        double r2099635 = eps;
        double r2099636 = r2099634 - r2099635;
        double r2099637 = r2099634 + r2099635;
        double r2099638 = r2099636 / r2099637;
        double r2099639 = log(r2099638);
        return r2099639;
}
↓
double f(double eps) {
        double r2099640 = eps;
        double r2099641 = r2099640 * r2099640;
        double r2099642 = r2099640 * r2099641;
        double r2099643 = -0.6666666666666666;
        double r2099644 = -0.4;
        double r2099645 = 5.0;
        double r2099646 = pow(r2099640, r2099645);
        double r2099647 = -2.0;
        double r2099648 = r2099647 * r2099640;
        double r2099649 = fma(r2099644, r2099646, r2099648);
        double r2099650 = fma(r2099642, r2099643, r2099649);
        return r2099650;
}

Error

Bits error versus eps

Target

Original58.6
Target0.3
Herbie0.3
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 +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))))