Average Error: 58.5 → 0.2
Time: 19.0s
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 r2954291 = 1.0;
        double r2954292 = eps;
        double r2954293 = r2954291 - r2954292;
        double r2954294 = r2954291 + r2954292;
        double r2954295 = r2954293 / r2954294;
        double r2954296 = log(r2954295);
        return r2954296;
}
↓
double f(double eps) {
        double r2954297 = eps;
        double r2954298 = r2954297 * r2954297;
        double r2954299 = r2954297 * r2954298;
        double r2954300 = -0.6666666666666666;
        double r2954301 = -2.0;
        double r2954302 = 5.0;
        double r2954303 = pow(r2954297, r2954302);
        double r2954304 = -0.4;
        double r2954305 = r2954303 * r2954304;
        double r2954306 = fma(r2954297, r2954301, r2954305);
        double r2954307 = fma(r2954299, r2954300, r2954306);
        return r2954307;
}

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 2019130 +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))))