Average Error: 58.4 → 0.7
Time: 21.2s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(-2, \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \mathsf{fma}\left(2, {\varepsilon}^{2}, \log 1\right)\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(-2, \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \mathsf{fma}\left(2, {\varepsilon}^{2}, \log 1\right)\right)
double f(double eps) {
        double r96293 = 1.0;
        double r96294 = eps;
        double r96295 = r96293 - r96294;
        double r96296 = r96293 + r96294;
        double r96297 = r96295 / r96296;
        double r96298 = log(r96297);
        return r96298;
}
↓
double f(double eps) {
        double r96299 = 2.0;
        double r96300 = -r96299;
        double r96301 = eps;
        double r96302 = 1.0;
        double r96303 = r96301 / r96302;
        double r96304 = fma(r96303, r96303, r96301);
        double r96305 = 2.0;
        double r96306 = pow(r96301, r96305);
        double r96307 = log(r96302);
        double r96308 = fma(r96299, r96306, r96307);
        double r96309 = fma(r96300, r96304, r96308);
        return r96309;
}

Error

Bits error versus eps

Target

Original58.4
Target0.2
Herbie0.7
\[-2 \cdot \left(\left(\varepsilon + \frac{{\varepsilon}^{3}}{3}\right) + \frac{{\varepsilon}^{5}}{5}\right)\]

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.7

    \[\leadsto \color{blue}{\left(2 \cdot {\varepsilon}^{2} + \log 1\right) - \left(2 \cdot \frac{{\varepsilon}^{2}}{{1}^{2}} + 2 \cdot \varepsilon\right)}\]

  3. Simplified0.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \mathsf{fma}\left(2, {\varepsilon}^{2}, \log 1\right)\right)}\]

  4. Final simplification0.7

    \[\leadsto \mathsf{fma}\left(-2, \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \mathsf{fma}\left(2, {\varepsilon}^{2}, \log 1\right)\right)\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"
  :precision binary64

  :herbie-target
  (* -2 (+ (+ eps (/ (pow eps 3) 3)) (/ (pow eps 5) 5)))

  (log (/ (- 1 eps) (+ 1 eps))))