Asymptote C

Average Error: 29.1 → 0.3

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -22594.13715790783:\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{elif}\;x \le 49404672.017620057:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}, \frac{\sqrt[3]{x}}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{2}{x}}{x}\\ \end{array}\]

C

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

↓

double code(double x) {
	double VAR;
	if ((x <= -22594.137157907826)) {
		VAR = ((-1.0 / pow(x, 2.0)) - fma(3.0, (1.0 / x), (3.0 * (1.0 / pow(x, 3.0)))));
	} else {
		double VAR_1;
		if ((x <= 49404672.01762006)) {
			VAR_1 = fma(((cbrt(x) * cbrt(x)) / 1.0), (cbrt(x) / (x + 1.0)), -((x + 1.0) / (x - 1.0)));
		} else {
			VAR_1 = ((-3.0 / x) - ((2.0 / x) / x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -22594.137157907826

   1. Initial program 59.3

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

   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 0.3

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

   if -22594.137157907826 < x < 49404672.01762006

   1. Initial program 0.2

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

   3. Applied *-un-lft-identity 0.2

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

   4. Applied add-cube-cbrt 0.2

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

   5. Applied times-frac 0.2

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

   6. Applied fma-neg 0.2

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

   if 49404672.01762006 < x

   1. Initial program 59.8

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

   3. Applied *-un-lft-identity 59.8

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

   4. Applied add-cube-cbrt 60.9

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

   5. Applied times-frac 60.8

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

   6. Applied fma-neg 61.0

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

   7. Taylor expanded around inf 0.6

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

   8. Simplified 0.3

      \[\leadsto \color{blue}{\frac{-3}{x} - \frac{\frac{2}{x}}{x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -22594.13715790783:\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{elif}\;x \le 49404672.017620057:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}, \frac{\sqrt[3]{x}}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{2}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))