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

Average Error: 38.5 → 20.3

Time: 3.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.5272377132889611 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \le 2.61800145549090708 \cdot 10^{93}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|im\right| \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\frac{\sqrt{2}}{re + \sqrt{re \cdot re + im \cdot im}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{re + re}}\right)\\ \end{array}\]

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 ((re <= -1.527237713288961e-52)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re * -2.0))))))));
	} else {
		double VAR_1;
		if ((re <= 2.618001455490907e+93)) {
			VAR_1 = ((double) (0.5 * ((double) (((double) (((double) fabs(im)) * ((double) sqrt(((double) sqrt(2.0)))))) * ((double) sqrt(((double) (((double) sqrt(2.0)) / ((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))))))))))));
		} else {
			VAR_1 = ((double) (0.5 * ((double) (((double) fabs(im)) * ((double) sqrt(((double) (2.0 / ((double) (re + re))))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 3 regimes

2. if re < -1.5272377132889611e-52

   1. Initial program 36.5

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

   2. Taylor expanded around -inf 17.0

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

   3. Simplified 17.0

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

   if -1.5272377132889611e-52 < re < 2.61800145549090708e93

   1. Initial program 32.2

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

   3. Applied flip-- 35.6

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

   4. Applied associate-*r/ 35.6

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

   5. Applied sqrt-div 35.9

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

   6. Simplified 31.3

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

   7. Simplified 31.3

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

   9. Applied *-un-lft-identity 31.3

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

   10. Applied sqrt-prod 31.3

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

   11. Applied sqrt-prod 31.3

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{im \cdot im} \cdot \sqrt{2}}}{\sqrt{1} \cdot \sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\]

   12. Applied times-frac 31.3

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

   13. Simplified 25.7

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

   15. Applied sqrt-undiv 25.6

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

   17. Applied *-un-lft-identity 25.6

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

   18. Applied add-sqr-sqrt 25.8

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

   19. Applied times-frac 25.7

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

   20. Applied sqrt-prod 25.6

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

   21. Applied associate-*r* 25.6

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

   22. Simplified 25.6

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

   if 2.61800145549090708e93 < re

   1. Initial program 60.3

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

   3. Applied flip-- 60.3

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

   4. Applied associate-*r/ 60.3

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

   5. Applied sqrt-div 60.3

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

   6. Simplified 43.2

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

   7. Simplified 43.2

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

   9. Applied *-un-lft-identity 43.2

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

   10. Applied sqrt-prod 43.2

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

   11. Applied sqrt-prod 43.2

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{im \cdot im} \cdot \sqrt{2}}}{\sqrt{1} \cdot \sqrt{re + \sqrt{re \cdot re + im \cdot im}}}\]

   12. Applied times-frac 43.2

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

   13. Simplified 40.1

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

   15. Applied sqrt-undiv 40.0

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

   16. Taylor expanded around inf 10.4

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

4. Final simplification 20.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.5272377132889611 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \le 2.61800145549090708 \cdot 10^{93}:\\ \;\;\;\;0.5 \cdot \left(\left(\left|im\right| \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\frac{\sqrt{2}}{re + \sqrt{re \cdot re + im \cdot im}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{re + re}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))))