Average Error: 15.5 → 0.0
Time: 16.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 568.3518365384376238580443896353244781494\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 568.3518365384376238580443896353244781494\right):\\
\;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\

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

\end{array}
double f(double x) {
        double r67713 = x;
        double r67714 = r67713 * r67713;
        double r67715 = 1.0;
        double r67716 = r67714 + r67715;
        double r67717 = r67713 / r67716;
        return r67717;
}
↓
double f(double x) {
        double r67718 = x;
        double r67719 = -3416824020916.5903;
        bool r67720 = r67718 <= r67719;
        double r67721 = 568.3518365384376;
        bool r67722 = r67718 <= r67721;
        double r67723 = !r67722;
        bool r67724 = r67720 || r67723;
        double r67725 = 1.0;
        double r67726 = r67725 / r67718;
        double r67727 = 1.0;
        double r67728 = 3.0;
        double r67729 = pow(r67718, r67728);
        double r67730 = r67727 / r67729;
        double r67731 = r67726 - r67730;
        double r67732 = 5.0;
        double r67733 = pow(r67718, r67732);
        double r67734 = r67727 / r67733;
        double r67735 = r67731 + r67734;
        double r67736 = fma(r67718, r67718, r67727);
        double r67737 = r67725 / r67736;
        double r67738 = r67718 * r67737;
        double r67739 = r67724 ? r67735 : r67738;
        return r67739;
}

Error

Bits error versus x

Target

Original15.5
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3416824020916.5903 or 568.3518365384376 < x

    1. Initial program 31.4

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

    2. Simplified31.4

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

    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 -3416824020916.5903 < x < 568.3518365384376

    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. Using strategy rm

    4. Applied div-inv0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 568.3518365384376238580443896353244781494\right):\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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