Average Error: 29.6 → 0.0
Time: 2.1s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.7941879771824913 \cdot 10^{+55} \lor \neg \left(x \leq 179310016481811.53\right):\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1 + x \cdot x}{-1 - x \cdot 3}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.7941879771824913 \cdot 10^{+55} \lor \neg \left(x \leq 179310016481811.53\right):\\
\;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -1.7941879771824913e+55) (not (<= x 179310016481811.53)))
   (+ (- (/ -1.0 (* x x)) (/ 3.0 x)) (/ -3.0 (pow x 3.0)))
   (/ 1.0 (/ (+ -1.0 (* x x)) (- -1.0 (* x 3.0))))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -1.7941879771824913e+55) || !(x <= 179310016481811.53)) {
		tmp = ((-1.0 / (x * x)) - (3.0 / x)) + (-3.0 / pow(x, 3.0));
	} else {
		tmp = 1.0 / ((-1.0 + (x * x)) / (-1.0 - (x * 3.0)));
	}
	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 < -1.7941879771824913e55 or 179310016481811.531 < x

    1. Initial program 60.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -1.7941879771824913e55 < x < 179310016481811.531

    1. Initial program 4.5

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

    3. Applied frac-sub_binary644.4

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

    4. Simplified4.4

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

    5. Taylor expanded around 0 0.0

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

    6. Simplified0.0

      \[\leadsto \frac{\color{blue}{-1 - x \cdot 3}}{x \cdot x + -1}\]
    7. Using strategy rm

    8. Applied clear-num_binary640.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + -1}{-1 - x \cdot 3}}}\]

    9. Simplified0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7941879771824913 \cdot 10^{+55} \lor \neg \left(x \leq 179310016481811.53\right):\\ \;\;\;\;\left(\frac{-1}{x \cdot x} - \frac{3}{x}\right) + \frac{-3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1 + x \cdot x}{-1 - x \cdot 3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))