Average Error: 14.7 → 0.0
Time: 7.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6092285729639.72168 \lor \neg \left(x \le 1964.70844757560167\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6092285729639.72168 \lor \neg \left(x \le 1964.70844757560167\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 r97813 = x;
        double r97814 = r97813 * r97813;
        double r97815 = 1.0;
        double r97816 = r97814 + r97815;
        double r97817 = r97813 / r97816;
        return r97817;
}
↓
double f(double x) {
        double r97818 = x;
        double r97819 = -6092285729639.722;
        bool r97820 = r97818 <= r97819;
        double r97821 = 1964.7084475756017;
        bool r97822 = r97818 <= r97821;
        double r97823 = !r97822;
        bool r97824 = r97820 || r97823;
        double r97825 = 1.0;
        double r97826 = r97825 / r97818;
        double r97827 = 1.0;
        double r97828 = 5.0;
        double r97829 = pow(r97818, r97828);
        double r97830 = r97827 / r97829;
        double r97831 = 3.0;
        double r97832 = pow(r97818, r97831);
        double r97833 = r97827 / r97832;
        double r97834 = r97830 - r97833;
        double r97835 = r97826 + r97834;
        double r97836 = r97818 * r97818;
        double r97837 = r97836 + r97827;
        double r97838 = r97825 / r97837;
        double r97839 = r97818 * r97838;
        double r97840 = r97824 ? r97835 : r97839;
        return r97840;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6092285729639.722 or 1964.7084475756017 < x

    1. Initial program 30.2

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

    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 -6092285729639.722 < x < 1964.7084475756017

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6092285729639.72168 \lor \neg \left(x \le 1964.70844757560167\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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