Asymptote C

Average Error: 29.3 → 0.1

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -12471.6665092964158 \lor \neg \left(x \le 20005.7029934287893\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot x\right) \cdot x}{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]

C

double code(double x) {
	return ((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0)));
}

↓

double code(double x) {
	double temp;
	if (((x <= -12471.666509296416) || !(x <= 20005.70299342879))) {
		temp = (((-1.0 / pow(x, 2.0)) - (3.0 / x)) - (3.0 / pow(x, 3.0)));
	} else {
		temp = (((((x * x) * x) / (((x + 1.0) * (x + 1.0)) * (x + 1.0))) - pow(((x + 1.0) / (x - 1.0)), 3.0)) / ((((x + 1.0) / (x - 1.0)) * (((x + 1.0) / (x - 1.0)) + (x / (x + 1.0)))) + ((x / (x + 1.0)) * (x / (x + 1.0)))));
	}
	return temp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -12471.666509296416 or 20005.70299342879 < x

   1. Initial program 59.2

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

   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. Simplified 0.0

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

   if -12471.666509296416 < x < 20005.70299342879

   1. Initial program 0.1

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

   3. Applied flip3-- 0.1

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

   4. Simplified 0.1

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

   6. Applied add-cbrt-cube 0.1

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

   7. Applied add-cbrt-cube 0.1

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

   8. Applied cbrt-undiv 0.1

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

   9. Applied rem-cube-cbrt 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12471.6665092964158 \lor \neg \left(x \le 20005.7029934287893\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot x\right) \cdot x}{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right) \cdot \left(x + 1\right)} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\frac{x + 1}{x - 1} \cdot \left(\frac{x + 1}{x - 1} + \frac{x}{x + 1}\right) + \frac{x}{x + 1} \cdot \frac{x}{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))