math.sqrt on complex, real part

Average Error: 39.1 → 27.3

Time: 3.9s

Precision: binary64

Math

\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} + re \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} + re \le 1.45354764161396553 \cdot 10^{152}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array}\]

C

double code(double re, double im) {
	return ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))));
}

↓

double code(double re, double im) {
	double VAR;
	if ((((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)) <= 0.0)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) pow(im, 2.0)) / ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)) <= 1.4535476416139655e+152)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))));
		} else {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im + re))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	39.1
Target	33.9
Herbie	27.3

\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ (sqrt (+ (* re re) (* im im))) re) < 0.0

   1. Initial program 58.8

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
   2. Using strategy rm

   3. Applied flip-+ 58.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

   4. Simplified 36.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

   if 0.0 < (+ (sqrt (+ (* re re) (* im im))) re) < 1.45354764161396553e152

   1. Initial program 4.3

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

   if 1.45354764161396553e152 < (+ (sqrt (+ (* re re) (* im im))) re)

   1. Initial program 63.4

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

   2. Taylor expanded around 0 44.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} + re\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} + re \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} + re \le 1.45354764161396553 \cdot 10^{152}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))