Average Error: 15.5 → 0.0
Time: 16.0s
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 r76685 = x;
        double r76686 = r76685 * r76685;
        double r76687 = 1.0;
        double r76688 = r76686 + r76687;
        double r76689 = r76685 / r76688;
        return r76689;
}
↓
double f(double x) {
        double r76690 = x;
        double r76691 = -3416824020916.5903;
        bool r76692 = r76690 <= r76691;
        double r76693 = 568.3518365384376;
        bool r76694 = r76690 <= r76693;
        double r76695 = !r76694;
        bool r76696 = r76692 || r76695;
        double r76697 = 1.0;
        double r76698 = r76697 / r76690;
        double r76699 = 1.0;
        double r76700 = 3.0;
        double r76701 = pow(r76690, r76700);
        double r76702 = r76699 / r76701;
        double r76703 = r76698 - r76702;
        double r76704 = 5.0;
        double r76705 = pow(r76690, r76704);
        double r76706 = r76699 / r76705;
        double r76707 = r76703 + r76706;
        double r76708 = fma(r76690, r76690, r76699);
        double r76709 = r76697 / r76708;
        double r76710 = r76690 * r76709;
        double r76711 = r76696 ? r76707 : r76710;
        return r76711;
}

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)))