math.sqrt on complex, real part

Average Error: 38.8 → 26.8

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 6.200882643143995 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

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) sqrt(((double) (2.0 * ((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))) <= 0.0)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * (((double) (im * im)) / ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))))));
	} else {
		double VAR_1;
		if ((((double) sqrt(((double) (2.0 * ((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))) <= 6.200882643143995e+75)) {
			VAR_1 = ((double) (((double) sqrt(((double) (2.0 * ((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))) * 0.5));
		} else {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + im))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Target

Original	38.8
Target	33.9
Herbie	26.8

\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 0.0

   1. Initial program 57.7

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

   3. Applied flip-+ 57.7

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

   4. Simplified 31.5

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

   if 0.0 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 6.2008826431439952e75

   1. Initial program 4.7

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

   if 6.2008826431439952e75 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))

   1. Initial program 63.3

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

   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 26.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 6.200882643143995 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))))