Average Error: 28.6 → 0.1
Time: 2.8s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -11056.680698277803 \lor \neg \left(x \leq 14540.297240922848\right):\\ \;\;\;\;\frac{-1}{x} \cdot \frac{1}{x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -11056.680698277803 \lor \neg \left(x \leq 14540.297240922848\right):\\
\;\;\;\;\frac{-1}{x} \cdot \frac{1}{x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
double code(double x) {
	return ((double) ((x / ((double) (x + 1.0))) - (((double) (x + 1.0)) / ((double) (x - 1.0)))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -11056.680698277803) || !(x <= 14540.297240922848))) {
		VAR = ((double) (((double) ((-1.0 / x) * (1.0 / x))) - ((double) ((3.0 / x) + (3.0 / ((double) pow(x, 3.0)))))));
	} else {
		VAR = ((double) ((x / ((double) (x + 1.0))) + ((double) (((double) (x + 1.0)) * (-1.0 / ((double) (x - 1.0)))))));
	}
	return VAR;
}

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 < -11056.680698277803 or 14540.2972409228478 < x

    1. Initial program 59.3

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

    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. Simplified0.3

      \[\leadsto \color{blue}{\frac{-1}{x} \cdot \left(\frac{1}{x} + 3\right) - \frac{3}{{x}^{3}}}\]
    4. Using strategy rm

    5. Applied distribute-lft-in0.3

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

    6. Applied associate--l+0.3

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

    7. Simplified0.0

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

    if -11056.680698277803 < x < 14540.2972409228478

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11056.680698277803 \lor \neg \left(x \leq 14540.297240922848\right):\\ \;\;\;\;\frac{-1}{x} \cdot \frac{1}{x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \left(x + 1\right) \cdot \frac{-1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))