Average Error: 29.4 → 0.1
Time: 4.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -12879.754469549705 \lor \neg \left(x \le 12427.129501950301\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -12879.754469549705 \lor \neg \left(x \le 12427.129501950301\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\

\end{array}
double f(double x) {
        double r134834 = x;
        double r134835 = 1.0;
        double r134836 = r134834 + r134835;
        double r134837 = r134834 / r134836;
        double r134838 = r134834 - r134835;
        double r134839 = r134836 / r134838;
        double r134840 = r134837 - r134839;
        return r134840;
}
↓
double f(double x) {
        double r134841 = x;
        double r134842 = -12879.754469549705;
        bool r134843 = r134841 <= r134842;
        double r134844 = 12427.129501950301;
        bool r134845 = r134841 <= r134844;
        double r134846 = !r134845;
        bool r134847 = r134843 || r134846;
        double r134848 = 1.0;
        double r134849 = -r134848;
        double r134850 = 2.0;
        double r134851 = pow(r134841, r134850);
        double r134852 = r134849 / r134851;
        double r134853 = 3.0;
        double r134854 = r134853 / r134841;
        double r134855 = r134852 - r134854;
        double r134856 = 3.0;
        double r134857 = pow(r134841, r134856);
        double r134858 = r134853 / r134857;
        double r134859 = r134855 - r134858;
        double r134860 = r134841 - r134848;
        double r134861 = r134841 * r134860;
        double r134862 = r134841 + r134848;
        double r134863 = r134862 * r134862;
        double r134864 = r134861 - r134863;
        double r134865 = r134841 * r134841;
        double r134866 = r134848 * r134848;
        double r134867 = r134865 - r134866;
        double r134868 = r134864 / r134867;
        double r134869 = r134847 ? r134859 : r134868;
        return r134869;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -12879.754469549705 or 12427.129501950301 < x

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}}\]

    if -12879.754469549705 < x < 12427.129501950301

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12879.754469549705 \lor \neg \left(x \le 12427.129501950301\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))