Average Error: 14.8 → 0.0
Time: 16.5s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12032692145665368408206631501824 \lor \neg \left(x \le 470.1828671092736726677685510367155075073\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12032692145665368408206631501824 \lor \neg \left(x \le 470.1828671092736726677685510367155075073\right):\\
\;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

\end{array}
double f(double x) {
        double r93550 = x;
        double r93551 = r93550 * r93550;
        double r93552 = 1.0;
        double r93553 = r93551 + r93552;
        double r93554 = r93550 / r93553;
        return r93554;
}
↓
double f(double x) {
        double r93555 = x;
        double r93556 = -1.2032692145665368e+31;
        bool r93557 = r93555 <= r93556;
        double r93558 = 470.1828671092737;
        bool r93559 = r93555 <= r93558;
        double r93560 = !r93559;
        bool r93561 = r93557 || r93560;
        double r93562 = 1.0;
        double r93563 = r93562 / r93555;
        double r93564 = 1.0;
        double r93565 = 3.0;
        double r93566 = pow(r93555, r93565);
        double r93567 = r93564 / r93566;
        double r93568 = r93563 - r93567;
        double r93569 = 5.0;
        double r93570 = pow(r93555, r93569);
        double r93571 = r93564 / r93570;
        double r93572 = r93568 + r93571;
        double r93573 = fma(r93555, r93555, r93564);
        double r93574 = r93555 / r93573;
        double r93575 = r93561 ? r93572 : r93574;
        return r93575;
}

Error

Bits error versus x

Target

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.2032692145665368e+31 or 470.1828671092737 < x

    1. Initial program 31.6

      \[\frac{x}{x \cdot x + 1}\]

    2. Simplified31.6

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]

    3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{{x}^{5}} + \frac{1}{x}\right) - 1 \cdot \frac{1}{{x}^{3}}}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}}\]

    if -1.2032692145665368e+31 < x < 470.1828671092737

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12032692145665368408206631501824 \lor \neg \left(x \le 470.1828671092736726677685510367155075073\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))