sqrt times

Average Error: 0.5 → 0.0

Time: 2.3s

Precision: 64

Math

\[\sqrt{x - 1} \cdot \sqrt{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 67840.3960161231953:\\ \;\;\;\;\sqrt{\left(x - 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\\ \end{array}\]

C

double code(double x) {
	return (sqrt((x - 1.0)) * sqrt(x));
}

↓

double code(double x) {
	double VAR;
	if ((x <= 67840.3960161232)) {
		VAR = sqrt(((x - 1.0) * x));
	} else {
		VAR = (x - (0.5 + (0.125 * (1.0 / x))));
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < 67840.3960161232

   1. Initial program 0.3

      \[\sqrt{x - 1} \cdot \sqrt{x}\]
   2. Using strategy rm

   3. Applied sqrt-unprod 0.1

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

   if 67840.3960161232 < x

   1. Initial program 0.5

      \[\sqrt{x - 1} \cdot \sqrt{x}\]

   2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 67840.3960161231953:\\ \;\;\;\;\sqrt{\left(x - 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1)) (sqrt x)))