Average Error: 28.7 → 0.0
Time: 2.5s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -10771490727067154 \lor \neg \left(x \leq 268163.1256190682\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - \left(1 + \left(x + 1\right)\right)\right) - 1 \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -10771490727067154 \lor \neg \left(x \leq 268163.1256190682\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -10771490727067154.0) (not (<= x 268163.1256190682)))
   (- (/ (- 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)))
    (- (* x x) (* 1.0 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 <= -10771490727067154.0) || !(x <= 268163.1256190682))) {
		tmp = ((double) ((((double) -(1.0)) / ((double) (x * x))) - ((double) ((3.0 / x) + (3.0 / ((double) pow(x, 3.0)))))));
	} else {
		tmp = (((double) (((double) (x * ((double) (x - ((double) (1.0 + ((double) (x + 1.0)))))))) - ((double) (1.0 * ((double) (x + 1.0)))))) / ((double) (((double) (x * x)) - ((double) (1.0 * 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 < -10771490727067154 or 268163.125619068218 < x

    1. Initial program Error: 59.9 bits

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

    3. Applied frac-subError: 62.2 bits

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

    4. SimplifiedError: 62.2 bits

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

    5. Taylor expanded around inf Error: 0.3 bits

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

    6. SimplifiedError: 0.0 bits

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

    if -10771490727067154 < x < 268163.125619068218

    1. Initial program Error: 0.6 bits

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

    3. Applied frac-subError: 0.6 bits

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

    4. SimplifiedError: 0.6 bits

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

    6. Applied distribute-lft-inError: 0.6 bits

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

    7. Applied associate--r+Error: 0.6 bits

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

    8. SimplifiedError: 0.0 bits

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

  4. Final simplificationError: 0.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10771490727067154 \lor \neg \left(x \leq 268163.1256190682\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x - \left(1 + \left(x + 1\right)\right)\right) - 1 \cdot \left(x + 1\right)}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))