Average Error: 29.9 → 0.0
Time: 3.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3547768106768.249 \lor \neg \left(x \le 7926639408581.20996\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(3 \cdot x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3547768106768.249 \lor \neg \left(x \le 7926639408581.20996\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r131980 = x;
        double r131981 = 1.0;
        double r131982 = r131980 + r131981;
        double r131983 = r131980 / r131982;
        double r131984 = r131980 - r131981;
        double r131985 = r131982 / r131984;
        double r131986 = r131983 - r131985;
        return r131986;
}
↓
double f(double x) {
        double r131987 = x;
        double r131988 = -3547768106768.249;
        bool r131989 = r131987 <= r131988;
        double r131990 = 7926639408581.21;
        bool r131991 = r131987 <= r131990;
        double r131992 = !r131991;
        bool r131993 = r131989 || r131992;
        double r131994 = 1.0;
        double r131995 = -r131994;
        double r131996 = 2.0;
        double r131997 = pow(r131987, r131996);
        double r131998 = r131995 / r131997;
        double r131999 = 3.0;
        double r132000 = r131999 / r131987;
        double r132001 = r131998 - r132000;
        double r132002 = 3.0;
        double r132003 = pow(r131987, r132002);
        double r132004 = r131999 / r132003;
        double r132005 = r132001 - r132004;
        double r132006 = r131999 * r131987;
        double r132007 = r132006 + r131994;
        double r132008 = -r132007;
        double r132009 = r131987 * r131987;
        double r132010 = r131994 * r131994;
        double r132011 = r132009 - r132010;
        double r132012 = r132008 / r132011;
        double r132013 = r131993 ? r132005 : r132012;
        return r132013;
}

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 < -3547768106768.249 or 7926639408581.21 < x

    1. Initial program 60.3

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

    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 -3547768106768.249 < x < 7926639408581.21

    1. Initial program 0.8

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

    3. Applied frac-sub0.8

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

    4. Simplified0.8

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

    5. Taylor expanded around 0 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3547768106768.249 \lor \neg \left(x \le 7926639408581.20996\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\left(3 \cdot x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))