Asymptote C

Average Error: 28.6 → 0.0

Time: 2.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.473895295753532 \cdot 10^{+74} \lor \neg \left(x \le 130107.61295780109\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-3\right) - 1}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if (((x <= -2.473895295753532e+74) || !(x <= 130107.61295780109))) {
		VAR = ((double) (((double) (((double) -(1.0)) / ((double) (x * x)))) - ((double) (((double) (3.0 / x)) + ((double) (3.0 / ((double) pow(x, 3.0))))))));
	} else {
		VAR = ((double) (((double) (((double) (x * ((double) -(3.0)))) - 1.0)) / ((double) (((double) (x * x)) - ((double) (1.0 * 1.0))))));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -2.473895295753532e74 or 130107.61295780109 < x

   1. Initial program 59.8

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

   3. Applied add-cbrt-cube 61.2

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

   4. Applied add-cbrt-cube 62.9

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

   5. Applied cbrt-undiv 62.9

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

   6. Simplified 59.8

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

   7. Taylor expanded around inf 0.3

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

   8. Simplified 0.0

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

   if -2.473895295753532e74 < x < 130107.61295780109

   1. Initial program 5.5

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

   3. Applied frac-sub 5.5

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

   4. Simplified 5.5

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

   5. Taylor expanded around 0 0.0

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

   6. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.473895295753532 \cdot 10^{+74} \lor \neg \left(x \le 130107.61295780109\right):\\ \;\;\;\;\frac{-1}{x \cdot x} - \left(\frac{3}{x} + \frac{3}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-3\right) - 1}{x \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))