Average Error: 28.5 → 0.1
Time: 6.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -13967.117865521159 \lor \neg \left(x \le 12664.160931741633\right):\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{x} - \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)\right) - 1}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 1}{x + 1} - \frac{x + 1}{x}}{\frac{x + 1}{x} \cdot \frac{x - 1}{x + 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -13967.117865521159 \lor \neg \left(x \le 12664.160931741633\right):\\
\;\;\;\;\frac{\left(2 \cdot \frac{1}{x} - \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)\right) - 1}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}}\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (x / ((double) (x + 1.0)))) - ((double) (((double) (x + 1.0)) / ((double) (x - 1.0))))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -13967.117865521159) || !(x <= 12664.160931741633))) {
		VAR = ((double) (((double) (((double) (((double) (2.0 * ((double) (1.0 / x)))) - ((double) (2.0 + ((double) (2.0 * ((double) (1.0 / ((double) pow(x, 2.0)))))))))) - 1.0)) / ((double) (((double) (x + 1.0)) * ((double) (((double) (x - 1.0)) / ((double) (x + 1.0))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (x - 1.0)) / ((double) (x + 1.0)))) - ((double) (((double) (x + 1.0)) / x)))) / ((double) (((double) (((double) (x + 1.0)) / x)) * ((double) (((double) (x - 1.0)) / ((double) (x + 1.0))))))));
	}
	return VAR;
}

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 < -13967.117865521159 or 12664.160931741633 < x

    1. Initial program 59.2

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

    3. Applied clear-num59.2

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

    4. Applied frac-sub58.7

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

    5. Simplified51.2

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

    6. Taylor expanded around inf 0.0

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

    if -13967.117865521159 < x < 12664.160931741633

    1. Initial program 0.1

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

    3. Applied clear-num0.1

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

    4. Applied clear-num0.1

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

    5. Applied frac-sub0.1

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

    6. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13967.117865521159 \lor \neg \left(x \le 12664.160931741633\right):\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{x} - \left(2 + 2 \cdot \frac{1}{{x}^{2}}\right)\right) - 1}{\left(x + 1\right) \cdot \frac{x - 1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - 1}{x + 1} - \frac{x + 1}{x}}{\frac{x + 1}{x} \cdot \frac{x - 1}{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))