Average Error: 30.0 → 0.0
Time: 2.4s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -13944.835699645984 \lor \neg \left(x \leq 13458.95778492134\right):\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x + 1} - \frac{x + 1}{x + -1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -13944.835699645984 \lor \neg \left(x \leq 13458.95778492134\right):\\
\;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{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 -13944.835699645984) (not (<= x 13458.95778492134)))
   (+ (- (/ -1.0 (* x x)) (/ 3.0 x)) (/ -3.0 (pow x 3.0)))
   (- (* x (/ 1.0 (+ 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 <= -13944.835699645984) || !(x <= 13458.95778492134)) {
		tmp = ((-1.0 / (x * x)) - (3.0 / x)) + (-3.0 / pow(x, 3.0));
	} else {
		tmp = (x * (1.0 / (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 < -13944.835699645984 or 13458.9577849213401 < x

    1. Initial program 59.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -13944.835699645984 < x < 13458.9577849213401

    1. Initial program 0.1

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

    3. Applied div-inv_binary640.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13944.835699645984 \lor \neg \left(x \leq 13458.95778492134\right):\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x + 1} - \frac{x + 1}{x + -1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))