Average Error: 29.6 → 0.1
Time: 3.8s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -9097.76265741964 \lor \neg \left(x \leq 13483.542986370416\right):\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{{x}^{3} - {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -9097.76265741964 \lor \neg \left(x \leq 13483.542986370416\right):\\
\;\;\;\;\left(\frac{\frac{-1}{x}}{x} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{{x}^{3} - {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\\

\end{array}
double code(double x) {
	return ((double) ((x / ((double) (x + 1.0))) - (((double) (x + 1.0)) / ((double) (x - 1.0)))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -9097.76265741964) || !(x <= 13483.542986370416))) {
		VAR = ((double) (((double) (((((double) -(1.0)) / x) / x) - (3.0 / x))) - (3.0 / ((double) pow(x, 3.0)))));
	} else {
		VAR = ((double) ((x / ((double) (x + 1.0))) - ((double) ((((double) (x + 1.0)) / ((double) (((double) pow(x, 3.0)) - ((double) pow(1.0, 3.0))))) * ((double) (((double) (x * x)) + ((double) (((double) (1.0 * 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 < -9097.7626574196402 or 13483.542986370416 < x

    1. Initial program 59.4

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

    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. Simplified0.3

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

    5. Applied distribute-lft-in0.3

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

    6. Simplified0.3

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

    7. Simplified0.0

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

    if -9097.7626574196402 < x < 13483.542986370416

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Applied associate-/r/0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9097.76265741964 \lor \neg \left(x \leq 13483.542986370416\right):\\ \;\;\;\;\left(\frac{\frac{-1}{x}}{x} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{x + 1}{{x}^{3} - {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))