Average Error: 29.0 → 0.1
Time: 4.7s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -12732.029291456316 \lor \neg \left(x \leq 10696.905064826944\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{x}{x + 1}\right)}^{6} - {\left(\frac{x + 1}{x + -1}\right)}^{6}}{{\left(\frac{x}{x + 1}\right)}^{4} + \left({\left(\frac{x + 1}{x + -1}\right)}^{4} + {\left(\frac{x}{x + 1}\right)}^{2} \cdot {\left(\frac{x + 1}{x + -1}\right)}^{2}\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x + -1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -12732.029291456316 \lor \neg \left(x \leq 10696.905064826944\right):\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{x}{x + 1}\right)}^{6} - {\left(\frac{x + 1}{x + -1}\right)}^{6}}{{\left(\frac{x}{x + 1}\right)}^{4} + \left({\left(\frac{x + 1}{x + -1}\right)}^{4} + {\left(\frac{x}{x + 1}\right)}^{2} \cdot {\left(\frac{x + 1}{x + -1}\right)}^{2}\right)}}{\frac{x}{x + 1} + \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 -12732.029291456316) (not (<= x 10696.905064826944)))
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (/
    (/
     (- (pow (/ x (+ x 1.0)) 6.0) (pow (/ (+ x 1.0) (+ x -1.0)) 6.0))
     (+
      (pow (/ x (+ x 1.0)) 4.0)
      (+
       (pow (/ (+ x 1.0) (+ x -1.0)) 4.0)
       (* (pow (/ x (+ x 1.0)) 2.0) (pow (/ (+ x 1.0) (+ x -1.0)) 2.0)))))
    (+ (/ x (+ x 1.0)) (/ (+ 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 <= -12732.029291456316) || !(x <= 10696.905064826944)) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else {
		tmp = ((pow((x / (x + 1.0)), 6.0) - pow(((x + 1.0) / (x + -1.0)), 6.0)) / (pow((x / (x + 1.0)), 4.0) + (pow(((x + 1.0) / (x + -1.0)), 4.0) + (pow((x / (x + 1.0)), 2.0) * pow(((x + 1.0) / (x + -1.0)), 2.0))))) / ((x / (x + 1.0)) + ((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 < -12732.0292914563161 or 10696.905064826944 < x

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.0

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

    if -12732.0292914563161 < x < 10696.905064826944

    1. Initial program 0.1

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

    3. Applied flip--_binary64_17580.1

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

    5. Applied flip3--_binary64_17870.1

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

    6. Simplified0.1

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

    7. Simplified0.1

      \[\leadsto \frac{\frac{{\left(\frac{x}{x + 1}\right)}^{6} - {\left(\frac{x + 1}{x - 1}\right)}^{6}}{\color{blue}{{\left(\frac{x}{x + 1}\right)}^{4} + \left({\left(\frac{x + 1}{x - 1}\right)}^{4} + {\left(\frac{x}{x + 1}\right)}^{2} \cdot {\left(\frac{x + 1}{x - 1}\right)}^{2}\right)}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12732.029291456316 \lor \neg \left(x \leq 10696.905064826944\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{x}{x + 1}\right)}^{6} - {\left(\frac{x + 1}{x + -1}\right)}^{6}}{{\left(\frac{x}{x + 1}\right)}^{4} + \left({\left(\frac{x + 1}{x + -1}\right)}^{4} + {\left(\frac{x}{x + 1}\right)}^{2} \cdot {\left(\frac{x + 1}{x + -1}\right)}^{2}\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x + -1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))