Average Error: 29.0 → 0.1
Time: 4.3s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -10559.50745449947 \lor \neg \left(x \leq 11243.776937313149\right):\\ \;\;\;\;\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array} \]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -10559.50745449947 \lor \neg \left(x \leq 11243.776937313149\right):\\
\;\;\;\;\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\


\end{array}

(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -10559.50745449947) (not (<= x 11243.776937313149)))
   (- (/ -3.0 x) (+ (/ 1.0 (* x x)) (/ 3.0 (pow x 3.0))))
   (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0)))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -10559.50745449947) || !(x <= 11243.776937313149)) {
		tmp = (-3.0 / x) - ((1.0 / (x * x)) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
	}
	return tmp;
}

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 < -10559.5074544994695 or 11243.776937313149 < x

    1. Initial program 59.1

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

    2. Taylor expanded in x around inf 0.3

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

    3. Simplified0.0

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

    if -10559.5074544994695 < x < 11243.776937313149

    1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10559.50745449947 \lor \neg \left(x \leq 11243.776937313149\right):\\ \;\;\;\;\frac{-3}{x} - \left(\frac{1}{x \cdot x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{x - 1}\\ \end{array} \]

Reproduce

herbie shell --seed 2022003 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))