Average Error: 28.3 → 0.1
Time: 2.1s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -13030.927938080447 \lor \neg \left(x \leq 14598.338637180132\right):\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \left(x + 1\right) \cdot \frac{1}{x + -1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -13030.927938080447 \lor \neg \left(x \leq 14598.338637180132\right):\\
\;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} - \left(x + 1\right) \cdot \frac{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 -13030.927938080447) (not (<= x 14598.338637180132)))
   (+ (- (/ -1.0 (* x x)) (/ 3.0 x)) (/ -3.0 (pow x 3.0)))
   (- (/ x (+ x 1.0)) (* (+ x 1.0) (/ 1.0 (+ x -1.0))))))
double code(double x) {
	return ((double) ((x / ((double) (x + 1.0))) - (((double) (x + 1.0)) / ((double) (x - 1.0)))));
}
↓
double code(double x) {
	double tmp;
	if (((x <= -13030.927938080447) || !(x <= 14598.338637180132))) {
		tmp = ((double) (((double) ((-1.0 / ((double) (x * x))) - (3.0 / x))) + (-3.0 / ((double) pow(x, 3.0)))));
	} else {
		tmp = ((double) ((x / ((double) (x + 1.0))) - ((double) (((double) (x + 1.0)) * (1.0 / ((double) (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 < -13030.927938080447 or 14598.338637180132 < 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 -13030.927938080447 < x < 14598.338637180132

    1. Initial program 0.1

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

    3. Applied div-inv_binary640.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13030.927938080447 \lor \neg \left(x \leq 14598.338637180132\right):\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \left(x + 1\right) \cdot \frac{1}{x + -1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))