Average Error: 15.3 → 0.0
Time: 13.0s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -36392348527513.984375 \lor \neg \left(x \le 2834276.9391883411444723606109619140625\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -36392348527513.984375 \lor \neg \left(x \le 2834276.9391883411444723606109619140625\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 r97717 = x;
        double r97718 = r97717 * r97717;
        double r97719 = 1.0;
        double r97720 = r97718 + r97719;
        double r97721 = r97717 / r97720;
        return r97721;
}
↓
double f(double x) {
        double r97722 = x;
        double r97723 = -36392348527513.984;
        bool r97724 = r97722 <= r97723;
        double r97725 = 2834276.939188341;
        bool r97726 = r97722 <= r97725;
        double r97727 = !r97726;
        bool r97728 = r97724 || r97727;
        double r97729 = 1.0;
        double r97730 = r97729 / r97722;
        double r97731 = 1.0;
        double r97732 = 3.0;
        double r97733 = pow(r97722, r97732);
        double r97734 = r97731 / r97733;
        double r97735 = r97730 - r97734;
        double r97736 = 5.0;
        double r97737 = pow(r97722, r97736);
        double r97738 = r97731 / r97737;
        double r97739 = r97735 + r97738;
        double r97740 = fma(r97722, r97722, r97731);
        double r97741 = r97722 / r97740;
        double r97742 = r97728 ? r97739 : r97741;
        return r97742;
}

Error

Bits error versus x

Target

Original15.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -36392348527513.984 or 2834276.939188341 < x

    1. Initial program 31.7

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

    2. Simplified31.7

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

    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 -36392348527513.984 < x < 2834276.939188341

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -36392348527513.984375 \lor \neg \left(x \le 2834276.9391883411444723606109619140625\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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