_divideComplex, real part

Average Error: 25.5 → 24.2

Time: 3.4s

Precision: 64

Math

\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} = -\infty \lor \neg \left(\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.483779865357748 \cdot 10^{302}\right):\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double VAR;
	if ((((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= -inf.0) || !((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1.4837798653577477e+302))) {
		VAR = (x_46_im / sqrt(((y_46_re * y_46_re) + (y_46_im * y_46_im))));
	} else {
		VAR = ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / sqrt(((y_46_re * y_46_re) + (y_46_im * y_46_im)))) / sqrt(((y_46_re * y_46_re) + (y_46_im * y_46_im))));
	}
	return VAR;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 2 regimes

2. if (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))) < -inf.0 or 1.4837798653577477e+302 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

   1. Initial program 63.7

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 63.7

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied associate-/r* 63.7

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   5. Taylor expanded around 0 59.0

      \[\leadsto \frac{\color{blue}{x.im}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   if -inf.0 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))) < 1.4837798653577477e+302

   1. Initial program 11.1

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 11.1

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied associate-/r* 11.0

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} = -\infty \lor \neg \left(\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le 1.483779865357748 \cdot 10^{302}\right):\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))