Average Error: 29.0 → 0.2
Time: 9.9s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -550341021.66475821:\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{elif}\;x \le 86764.4481716747541:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\frac{{\left({\left(x \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right) - \left(x + 1\right) \cdot 1\right)}^{3}\right)}^{3}}{{\left({\left(\left(x + 1\right) \cdot \left(x - 1\right)\right)}^{3}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -550341021.66475821:\\
\;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\

\mathbf{elif}\;x \le 86764.4481716747541:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{\frac{{\left({\left(x \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right) - \left(x + 1\right) \cdot 1\right)}^{3}\right)}^{3}}{{\left({\left(\left(x + 1\right) \cdot \left(x - 1\right)\right)}^{3}\right)}^{3}}}}\\

\mathbf{else}:\\
\;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\

\end{array}
double code(double x) {
	return ((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0)));
}
↓
double code(double x) {
	double temp;
	if ((x <= -550341021.6647582)) {
		temp = ((-1.0 / pow(x, 2.0)) - fma(3.0, (1.0 / x), (3.0 * (1.0 / pow(x, 3.0)))));
	} else {
		double temp_1;
		if ((x <= 86764.44817167475)) {
			temp_1 = cbrt(cbrt((pow(pow(((x * ((x - 1.0) - (x + 1.0))) - ((x + 1.0) * 1.0)), 3.0), 3.0) / pow(pow(((x + 1.0) * (x - 1.0)), 3.0), 3.0))));
		} else {
			temp_1 = -((1.0 * (1.0 / pow(x, 2.0))) + ((3.0 * (1.0 / x)) + (3.0 * (1.0 / pow(x, 3.0)))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -550341021.6647582

    1. Initial program 60.0

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

    if -550341021.6647582 < x < 86764.44817167475

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube0.2

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

    4. Simplified0.2

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

    6. Applied add-cbrt-cube0.2

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

    7. Simplified0.2

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

    9. Applied frac-sub0.2

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

    10. Applied cube-div0.2

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

    11. Applied cube-div0.2

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

    13. Applied distribute-lft-in0.2

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

    14. Applied associate--r+0.2

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

    15. Simplified0.1

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

    if 86764.44817167475 < x

    1. Initial program 59.4

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

    3. Applied add-cbrt-cube59.4

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

    4. Simplified59.4

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

    6. Applied add-cbrt-cube59.4

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

    7. Simplified59.4

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

    8. Taylor expanded around inf 0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -550341021.66475821:\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{elif}\;x \le 86764.4481716747541:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\frac{{\left({\left(x \cdot \left(\left(x - 1\right) - \left(x + 1\right)\right) - \left(x + 1\right) \cdot 1\right)}^{3}\right)}^{3}}{{\left({\left(\left(x + 1\right) \cdot \left(x - 1\right)\right)}^{3}\right)}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;-\left(1 \cdot \frac{1}{{x}^{2}} + \left(3 \cdot \frac{1}{x} + 3 \cdot \frac{1}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))