Average Error: 29.3 → 0.1
Time: 3.5s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -14488.508192621899 \lor \neg \left(x \leq 13924.25804052536\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x + -1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{-1 + x \cdot x}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -14488.508192621899 \lor \neg \left(x \leq 13924.25804052536\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

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

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -14488.508192621899) (not (<= x 13924.25804052536)))
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (/ (- (* x (+ x -1.0)) (* (+ x 1.0) (+ x 1.0))) (+ -1.0 (* x x)))))
double code(double x) {
	return (x / (x + 1.0)) - ((x + 1.0) / (x - 1.0));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -14488.508192621899) || !(x <= 13924.25804052536)) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = ((x * (x + -1.0)) - ((x + 1.0) * (x + 1.0))) / (-1.0 + (x * x));
	}
	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 < -14488.508192621899 or 13924.2580405253593 < x

    1. Initial program 59.3

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

    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

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

    if -14488.508192621899 < x < 13924.2580405253593

    1. Initial program 0.1

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

    3. Applied frac-sub_binary64_52020.1

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

    4. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -14488.508192621899 \lor \neg \left(x \leq 13924.25804052536\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x + -1\right) - \left(x + 1\right) \cdot \left(x + 1\right)}{-1 + x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))