math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.4 → 25.8

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 1.9085851189822194 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 3.89707012593855238 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 1.61667196582586895 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 5.5638916685512744 \cdot 10^{75}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

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) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))) <= 1.9085851189822194e-149)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (0.0 + ((double) pow(im, 2.0)))) / ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))))));
	} else {
		double VAR_1;
		if ((((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))) <= 3.8970701259385524e-120)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
		} else {
			double VAR_2;
			if ((((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))) <= 1.616671965825869e-76)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (-2.0 * re))))))));
			} else {
				double VAR_3;
				if ((((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)))))) <= 5.563891668551274e+75)) {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))));
				} else {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 1.9085851189822194e-149

   1. Initial program 57.1

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

   3. Applied flip-- 57.1

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

   4. Simplified 28.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

   if 1.9085851189822194e-149 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 3.89707012593855238e-120 or 5.5638916685512744e75 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))

   1. Initial program 63.0

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

   2. Taylor expanded around 0 45.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)}\]

   if 3.89707012593855238e-120 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 1.61667196582586895e-76

   1. Initial program 43.9

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

   2. Taylor expanded around -inf 21.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]

   if 1.61667196582586895e-76 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 5.5638916685512744e75

   1. Initial program 0.2

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 25.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 1.9085851189822194 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 3.89707012593855238 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 1.61667196582586895 \cdot 10^{-76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 5.5638916685512744 \cdot 10^{75}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))