Average Error: 28.6 → 0.0
Time: 13.2s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2954.202156251063 \lor \neg \left(x \leq 1889.84365421045\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} + \frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2954.202156251063 \lor \neg \left(x \leq 1889.84365421045\right):\\
\;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} + \frac{x + 1}{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 -2954.202156251063) (not (<= x 1889.84365421045)))
   (-
    (- (/ -3.0 x) (/ 1.0 (* x x)))
    (+ (/ 3.0 (pow x 3.0)) (/ 1.0 (pow x 4.0))))
   (/
    (-
     (* (cbrt (pow (/ x (+ x 1.0)) 3.0)) (cbrt (pow (/ x (+ x 1.0)) 3.0)))
     (* (/ (+ x 1.0) (- x 1.0)) (/ (+ x 1.0) (- x 1.0))))
    (+ (cbrt (pow (/ x (+ x 1.0)) 3.0)) (/ (+ x 1.0) (- 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 <= -2954.202156251063) || !(x <= 1889.84365421045)) {
		tmp = ((-3.0 / x) - (1.0 / (x * x))) - ((3.0 / pow(x, 3.0)) + (1.0 / pow(x, 4.0)));
	} else {
		tmp = ((cbrt(pow((x / (x + 1.0)), 3.0)) * cbrt(pow((x / (x + 1.0)), 3.0))) - (((x + 1.0) / (x - 1.0)) * ((x + 1.0) / (x - 1.0)))) / (cbrt(pow((x / (x + 1.0)), 3.0)) + ((x + 1.0) / (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 < -2954.20215625106312 or 1889.84365421045 < x

    1. Initial program 59.1

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{-\left(\frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + \left(\frac{1}{{x}^{4}} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}\]

    3. Simplified0.0

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

    if -2954.20215625106312 < x < 1889.84365421045

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube_binary64_18190.1

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

    4. Simplified0.1

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

    6. Applied flip--_binary64_17580.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2954.202156251063 \lor \neg \left(x \leq 1889.84365421045\right):\\ \;\;\;\;\left(\frac{-3}{x} - \frac{1}{x \cdot x}\right) - \left(\frac{3}{{x}^{3}} + \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\sqrt[3]{{\left(\frac{x}{x + 1}\right)}^{3}} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))