Average Error: 29.1 → 0.1
Time: 10.5s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -0.9915391036973327 \lor \neg \left(x \leq 9825.804894356363\right):\\ \;\;\;\;\frac{-3}{x} - \left({x}^{-2} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\sqrt{x + 1}}{\frac{x - 1}{\sqrt{x + 1}}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -0.9915391036973327 \lor \neg \left(x \leq 9825.804894356363\right):\\
\;\;\;\;\frac{-3}{x} - \left({x}^{-2} + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x + 1} - \frac{\sqrt{x + 1}}{\frac{x - 1}{\sqrt{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 -0.9915391036973327) (not (<= x 9825.804894356363)))
   (- (/ -3.0 x) (+ (pow x -2.0) (/ 3.0 (pow x 3.0))))
   (- (/ x (+ x 1.0)) (/ (sqrt (+ x 1.0)) (/ (- x 1.0) (sqrt (+ x 1.0)))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -0.9915391036973327) || !(x <= 9825.804894356363)) {
		tmp = (-3.0 / x) - (pow(x, -2.0) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = (x / (x + 1.0)) - (sqrt(x + 1.0) / ((x - 1.0) / sqrt(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 < -0.99153910369733267 or 9825.8048943563626 < x

    1. Initial program 59.0

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

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.2

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

    5. Applied associate--l-_binary64_51310.2

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

    7. Applied pow2_binary64_52740.2

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

    8. Applied pow-flip_binary64_52670.2

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

    if -0.99153910369733267 < x < 9825.8048943563626

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary64_52150.1

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

    4. Applied associate-/l*_binary64_51380.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9915391036973327 \lor \neg \left(x \leq 9825.804894356363\right):\\ \;\;\;\;\frac{-3}{x} - \left({x}^{-2} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} - \frac{\sqrt{x + 1}}{\frac{x - 1}{\sqrt{x + 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))