Average Error: 15.5 → 0.0
Time: 9.8s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 780.2110112890140953822992742061614990234\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 780.2110112890140953822992742061614990234\right):\\
\;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\

\end{array}
double f(double x) {
        double r47814 = x;
        double r47815 = r47814 * r47814;
        double r47816 = 1.0;
        double r47817 = r47815 + r47816;
        double r47818 = r47814 / r47817;
        return r47818;
}
↓
double f(double x) {
        double r47819 = x;
        double r47820 = -3416824020916.5903;
        bool r47821 = r47819 <= r47820;
        double r47822 = 780.2110112890141;
        bool r47823 = r47819 <= r47822;
        double r47824 = !r47823;
        bool r47825 = r47821 || r47824;
        double r47826 = 1.0;
        double r47827 = r47826 / r47819;
        double r47828 = 1.0;
        double r47829 = 5.0;
        double r47830 = pow(r47819, r47829);
        double r47831 = r47828 / r47830;
        double r47832 = 3.0;
        double r47833 = pow(r47819, r47832);
        double r47834 = r47828 / r47833;
        double r47835 = r47831 - r47834;
        double r47836 = r47827 + r47835;
        double r47837 = r47819 * r47819;
        double r47838 = r47837 + r47828;
        double r47839 = r47826 / r47838;
        double r47840 = r47819 * r47839;
        double r47841 = r47825 ? r47836 : r47840;
        return r47841;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 780.2110112890141 < x

    1. Initial program 31.5

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -3416824020916.5903 < x < 780.2110112890141

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied div-inv0.0

      \[\leadsto \color{blue}{x \cdot \frac{1}{x \cdot x + 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 780.2110112890140953822992742061614990234\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]

Reproduce

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

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

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