Average Error: 58.6 → 0.6
Time: 16.4s
Precision: 64
\[\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)\]
↓
\[\mathsf{fma}\left(2, \varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \log 1\right)\]
\log \left(\frac{1 - \varepsilon}{1 + \varepsilon}\right)
↓
\mathsf{fma}\left(2, \varepsilon \cdot \varepsilon - \mathsf{fma}\left(\frac{\varepsilon}{1}, \frac{\varepsilon}{1}, \varepsilon\right), \log 1\right)
double f(double eps) {
        double r3264333 = 1.0;
        double r3264334 = eps;
        double r3264335 = r3264333 - r3264334;
        double r3264336 = r3264333 + r3264334;
        double r3264337 = r3264335 / r3264336;
        double r3264338 = log(r3264337);
        return r3264338;
}
↓
double f(double eps) {
        double r3264339 = 2.0;
        double r3264340 = eps;
        double r3264341 = r3264340 * r3264340;
        double r3264342 = 1.0;
        double r3264343 = r3264340 / r3264342;
        double r3264344 = fma(r3264343, r3264343, r3264340);
        double r3264345 = r3264341 - r3264344;
        double r3264346 = log(r3264342);
        double r3264347 = fma(r3264339, r3264345, r3264346);
        return r3264347;
}

Error

Bits error versus eps

Target

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

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (eps)
  :name "logq (problem 3.4.3)"

  :herbie-target
  (* -2.0 (+ (+ eps (/ (pow eps 3.0) 3.0)) (/ (pow eps 5.0) 5.0)))

  (log (/ (- 1.0 eps) (+ 1.0 eps))))