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

Average Error: 38.4 → 23.6

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -426217874754138432:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -6.99753880270397086 \cdot 10^{-155}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\\ \mathbf{elif}\;im \le 5.7375817714329689 \cdot 10^{-160}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 3.6144863433458581 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;im \le 2.5127945524088855 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 1.7411669279487403 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{re + im}\right)}\\ \end{array}\]

C

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

↓

double code(double re, double im) {
	double VAR;
	if ((im <= -4.2621787475413843e+17)) {
		VAR = (0.5 * sqrt((2.0 * -(re + im))));
	} else {
		double VAR_1;
		if ((im <= -6.997538802703971e-155)) {
			VAR_1 = (0.5 * sqrt((2.0 * (fabs(im) * (fabs(im) / (sqrt(((re * re) + (im * im))) + re))))));
		} else {
			double VAR_2;
			if ((im <= 5.737581771432969e-160)) {
				VAR_2 = (0.5 * sqrt((2.0 * ((-1.0 * re) - re))));
			} else {
				double VAR_3;
				if ((im <= 3.614486343345858e-44)) {
					VAR_3 = (0.5 * (sqrt((2.0 * pow(im, 2.0))) / sqrt((sqrt(((re * re) + (im * im))) + re))));
				} else {
					double VAR_4;
					if ((im <= 2.5127945524088855e-16)) {
						VAR_4 = (0.5 * sqrt((2.0 * ((-1.0 * re) - re))));
					} else {
						double VAR_5;
						if ((im <= 1.7411669279487403e+143)) {
							VAR_5 = (0.5 * sqrt((2.0 * (fabs(im) * (fabs(im) / (sqrt(((re * re) + (im * im))) + re))))));
						} else {
							VAR_5 = (0.5 * sqrt((2.0 * (fabs(im) * (fabs(im) / (re + im))))));
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				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 5 regimes

2. if im < -4.2621787475413843e+17

   1. Initial program 42.4

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

   3. Applied flip-- 44.1

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

   4. Simplified 42.4

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

   5. Taylor expanded around -inf 14.1

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

   if -4.2621787475413843e+17 < im < -6.997538802703971e-155 or 2.5127945524088855e-16 < im < 1.7411669279487403e+143

   1. Initial program 23.9

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

   3. Applied flip-- 32.4

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

   4. Simplified 24.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 24.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}}\]

   7. Applied add-sqr-sqrt 24.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{\sqrt{{im}^{2}} \cdot \sqrt{{im}^{2}}}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\]

   8. Applied times-frac 24.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{\sqrt{{im}^{2}}}{1} \cdot \frac{\sqrt{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\]

   9. Simplified 24.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left|im\right|} \cdot \frac{\sqrt{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\]

   10. Simplified 24.6

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

   if -6.997538802703971e-155 < im < 5.737581771432969e-160 or 3.614486343345858e-44 < im < 2.5127945524088855e-16

   1. Initial program 41.3

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

   2. Taylor expanded around -inf 36.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} - re\right)}\]

   if 5.737581771432969e-160 < im < 3.614486343345858e-44

   1. Initial program 29.8

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

   3. Applied flip-- 41.3

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

   4. Simplified 30.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
   5. Using strategy rm

   6. Applied associate-*r/ 30.1

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

   7. Applied sqrt-div 27.8

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

   if 1.7411669279487403e+143 < im

   1. Initial program 61.3

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

   3. Applied flip-- 61.3

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

   4. Simplified 61.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 61.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\color{blue}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}}\]

   7. Applied add-sqr-sqrt 61.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{\sqrt{{im}^{2}} \cdot \sqrt{{im}^{2}}}}{1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}}\]

   8. Applied times-frac 61.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{\sqrt{{im}^{2}}}{1} \cdot \frac{\sqrt{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\]

   9. Simplified 61.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left|im\right|} \cdot \frac{\sqrt{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\]

   10. Simplified 59.9

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

   11. Taylor expanded around 0 9.0

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{\color{blue}{re + im}}\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 23.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -426217874754138432:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -6.99753880270397086 \cdot 10^{-155}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\\ \mathbf{elif}\;im \le 5.7375817714329689 \cdot 10^{-160}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 3.6144863433458581 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;im \le 2.5127945524088855 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 1.7411669279487403 \cdot 10^{143}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{\sqrt{re \cdot re + im \cdot im} + re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|im\right| \cdot \frac{\left|im\right|}{re + im}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))