Average Error: 29.2 → 0.1
Time: 4.8s
Precision: binary64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -13373.10811093312:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{elif}\;x \leq 10387.839376272022:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x + -1}\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x + -1}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x + -1}\right)}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -13373.10811093312:\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

\mathbf{elif}\;x \leq 10387.839376272022:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x + -1}\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x + -1}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x + -1}\right)}^{3}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\

\end{array}
(FPCore (x) :precision binary64 (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -13373.10811093312)
   (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow x 3.0))))
   (if (<= x 10387.839376272022)
     (*
      (cbrt (cbrt (pow (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 3.0)))
      (*
       (cbrt (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))))
       (cbrt (cbrt (pow (- (/ x (+ x 1.0)) (/ (+ x 1.0) (+ x -1.0))) 3.0)))))
     (- (/ -1.0 (* x x)) (+ (/ 3.0 x) (/ 3.0 (pow 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 <= -13373.10811093312) {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(x, 3.0)));
	} else if (x <= 10387.839376272022) {
		tmp = cbrt(cbrt(pow(((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))), 3.0))) * (cbrt((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))) * cbrt(cbrt(pow(((x / (x + 1.0)) - ((x + 1.0) / (x + -1.0))), 3.0))));
	} else {
		tmp = (-1.0 / (x * x)) - ((3.0 / x) + (3.0 / pow(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 < -13373.108110933121 or 10387.839376272022 < 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 -13373.108110933121 < x < 10387.839376272022

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt_binary640.1

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

    5. Applied add-cbrt-cube_binary640.1

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

    6. Simplified0.1

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

    8. Applied add-cbrt-cube_binary640.1

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

    9. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13373.10811093312:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{elif}\;x \leq 10387.839376272022:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x + -1}\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{x}{x + 1} - \frac{x + 1}{x + -1}} \cdot \sqrt[3]{\sqrt[3]{{\left(\frac{x}{x + 1} - \frac{x + 1}{x + -1}\right)}^{3}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))