Average Error: 15.4 → 0.0
Time: 9.4s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2101569.4261024096:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x \cdot x}, \frac{1}{x}, \frac{1}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 471.29821069866506:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x \cdot x}, \frac{1}{x}, \frac{1}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2101569.4261024096:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{x \cdot x}, \frac{1}{x}, \frac{1}{x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 471.29821069866506:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{x \cdot x}, \frac{1}{x}, \frac{1}{x}\right) + \frac{1}{{x}^{5}}\\

\end{array}
double f(double x) {
        double r2271784 = x;
        double r2271785 = r2271784 * r2271784;
        double r2271786 = 1.0;
        double r2271787 = r2271785 + r2271786;
        double r2271788 = r2271784 / r2271787;
        return r2271788;
}
↓
double f(double x) {
        double r2271789 = x;
        double r2271790 = -2101569.4261024096;
        bool r2271791 = r2271789 <= r2271790;
        double r2271792 = -1.0;
        double r2271793 = r2271789 * r2271789;
        double r2271794 = r2271792 / r2271793;
        double r2271795 = 1.0;
        double r2271796 = r2271795 / r2271789;
        double r2271797 = fma(r2271794, r2271796, r2271796);
        double r2271798 = 5.0;
        double r2271799 = pow(r2271789, r2271798);
        double r2271800 = r2271795 / r2271799;
        double r2271801 = r2271797 + r2271800;
        double r2271802 = 471.29821069866506;
        bool r2271803 = r2271789 <= r2271802;
        double r2271804 = fma(r2271789, r2271789, r2271795);
        double r2271805 = r2271789 / r2271804;
        double r2271806 = r2271803 ? r2271805 : r2271801;
        double r2271807 = r2271791 ? r2271801 : r2271806;
        return r2271807;
}

Error

Bits error versus x

Target

Original15.4
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2101569.4261024096 or 471.29821069866506 < x

    1. Initial program 30.9

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

    2. Simplified30.9

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -2101569.4261024096 < x < 471.29821069866506

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2101569.4261024096:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x \cdot x}, \frac{1}{x}, \frac{1}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 471.29821069866506:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x \cdot x}, \frac{1}{x}, \frac{1}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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