Average Error: 29.4 → 0.1
Time: 14.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15268.755255591204:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 10987.67453528168:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -15268.755255591204:\\
\;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\

\mathbf{elif}\;x \le 10987.67453528168:\\
\;\;\;\;\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)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\

\end{array}
double f(double x) {
        double r4174978 = x;
        double r4174979 = 1.0;
        double r4174980 = r4174978 + r4174979;
        double r4174981 = r4174978 / r4174980;
        double r4174982 = r4174978 - r4174979;
        double r4174983 = r4174980 / r4174982;
        double r4174984 = r4174981 - r4174983;
        return r4174984;
}
↓
double f(double x) {
        double r4174985 = x;
        double r4174986 = -15268.755255591204;
        bool r4174987 = r4174985 <= r4174986;
        double r4174988 = -3.0;
        double r4174989 = r4174988 / r4174985;
        double r4174990 = -1.0;
        double r4174991 = r4174985 * r4174985;
        double r4174992 = r4174990 / r4174991;
        double r4174993 = r4174989 + r4174992;
        double r4174994 = r4174989 / r4174991;
        double r4174995 = r4174993 + r4174994;
        double r4174996 = 10987.67453528168;
        bool r4174997 = r4174985 <= r4174996;
        double r4174998 = 1.0;
        double r4174999 = r4174985 - r4174998;
        double r4175000 = r4174985 * r4174999;
        double r4175001 = r4174985 + r4174998;
        double r4175002 = r4175001 * r4175001;
        double r4175003 = r4175000 - r4175002;
        double r4175004 = r4174999 * r4175001;
        double r4175005 = r4175003 / r4175004;
        double r4175006 = r4174997 ? r4175005 : r4174995;
        double r4175007 = r4174987 ? r4174995 : r4175006;
        return r4175007;
}

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 < -15268.755255591204 or 10987.67453528168 < x

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -15268.755255591204 < x < 10987.67453528168

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15268.755255591204:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \mathbf{elif}\;x \le 10987.67453528168:\\ \;\;\;\;\frac{x \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-3}{x} + \frac{-1}{x \cdot x}\right) + \frac{\frac{-3}{x}}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))