Average Error: 28.8 → 0.0
Time: 3.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7336262872.6649323 \lor \neg \left(x \le 96405.2938384949375\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot x - 1 \cdot 1}{3 \cdot x + 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7336262872.6649323 \lor \neg \left(x \le 96405.2938384949375\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{\frac{x \cdot x - 1 \cdot 1}{3 \cdot x + 1}}\\

\end{array}
double code(double x) {
	return ((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0)));
}
↓
double code(double x) {
	double temp;
	if (((x <= -7336262872.664932) || !(x <= 96405.29383849494))) {
		temp = (((-1.0 / pow(x, 2.0)) - (3.0 / x)) - (3.0 / pow(x, 3.0)));
	} else {
		temp = (-1.0 / (((x * x) - (1.0 * 1.0)) / ((3.0 * x) + 1.0)));
	}
	return temp;
}

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 < -7336262872.664932 or 96405.29383849494 < x

    1. Initial program 59.8

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -7336262872.664932 < x < 96405.29383849494

    1. Initial program 0.3

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

    3. Applied frac-sub0.3

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

    4. Simplified0.3

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

    5. Taylor expanded around 0 0.0

      \[\leadsto \frac{\color{blue}{-\left(3 \cdot x + 1\right)}}{x \cdot x - 1 \cdot 1}\]
    6. Using strategy rm

    7. Applied neg-mul-10.0

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

    8. Applied associate-/l*0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7336262872.6649323 \lor \neg \left(x \le 96405.2938384949375\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot x - 1 \cdot 1}{3 \cdot x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))