Average Error: 28.3 → 0.1
Time: 2.5s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -12666.995240131017 \lor \neg \left(x \leq 14038.924336122142\right):\\ \;\;\;\;-\left(\frac{1}{x \cdot x} + \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{x}{\left(x + 1\right) \cdot \left(x - 1\right)} - \frac{x + 1}{x - 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -12666.995240131017 \lor \neg \left(x \leq 14038.924336122142\right):\\
\;\;\;\;-\left(\frac{1}{x \cdot x} + \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x - 1\right) \cdot \frac{x}{\left(x + 1\right) \cdot \left(x - 1\right)} - \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 -12666.995240131017) (not (<= x 14038.924336122142)))
   (- (+ (/ 1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0)))))
   (- (* (- x 1.0) (/ x (* (+ x 1.0) (- x 1.0)))) (/ (+ x 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 <= -12666.995240131017) || !(x <= 14038.924336122142))) {
		tmp = ((double) -(((double) ((1.0 / ((double) (x * x))) + ((double) ((3.0 / x) + (3.0 / ((double) pow(x, 3.0)))))))));
	} else {
		tmp = ((double) (((double) (((double) (x - 1.0)) * (x / ((double) (((double) (x + 1.0)) * ((double) (x - 1.0))))))) - (((double) (x + 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 < -12666.9952401310165 or 14038.924336122142 < x

    1. Initial program 59.2

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

    3. Applied div-inv_binary6459.4

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

    4. Taylor expanded around inf 0.3

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

    5. Simplified0.0

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

    if -12666.9952401310165 < x < 14038.924336122142

    1. Initial program 0.1

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

    3. Applied flip-+_binary640.1

      \[\leadsto \frac{x}{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} - \frac{x + 1}{x - 1}\]

    4. Applied associate-/r/_binary640.1

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

    5. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12666.995240131017 \lor \neg \left(x \leq 14038.924336122142\right):\\ \;\;\;\;-\left(\frac{1}{x \cdot x} + \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \frac{x}{\left(x + 1\right) \cdot \left(x - 1\right)} - \frac{x + 1}{x - 1}\\ \end{array}\]

Reproduce

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