Asymptote C

Average Error: 28.9 → 0.1

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8055.65433913428842 \lor \neg \left(x \le 6807.5677494866923\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{{x}^{3}}, \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}, \frac{\sqrt[3]{x}}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]

C

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

↓

double code(double x) {
	double temp;
	if (((x <= -8055.654339134288) || !(x <= 6807.567749486692))) {
		temp = ((-1.0 / pow(x, 2.0)) - fma(3.0, (1.0 / pow(x, 3.0)), (3.0 / x)));
	} else {
		temp = fma(((cbrt(x) * cbrt(x)) / 1.0), (cbrt(x) / (x + 1.0)), -((x + 1.0) / (x - 1.0)));
	}
	return temp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -8055.654339134288 or 6807.567749486692 < x

   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

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

   4. Taylor expanded around 0 0.3

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

   5. Simplified 0.0

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

   if -8055.654339134288 < x < 6807.567749486692

   1. Initial program 0.1

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

   3. Applied *-un-lft-identity 0.1

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

   4. Applied add-cube-cbrt 0.1

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

   5. Applied times-frac 0.1

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

   6. Applied fma-neg 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8055.65433913428842 \lor \neg \left(x \le 6807.5677494866923\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{{x}^{3}}, \frac{3}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}, \frac{\sqrt[3]{x}}{x + 1}, -\frac{x + 1}{x - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))