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

Average Error: 38.6 → 23.1

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -8.95791604585242049 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -8.12032175799812403 \cdot 10^{-235}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}{im}}\right)}\\ \mathbf{elif}\;re \le 7.222181096583517 \cdot 10^{-225}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{re + im}{im}}\right)}\\ \mathbf{elif}\;re \le 1.33865600224557405 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2 \cdot \left(1 \cdot \frac{1}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)} \cdot \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{re + re}{im}}\right)}\\ \end{array}\]

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 ((re <= -8.95791604585242e-50)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (-1.0 * re)) - re))))))));
	} else {
		double VAR_1;
		if ((re <= -8.120321757998124e-235)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (1.0 * ((double) (im / ((double) (((double) (((double) pow(((double) (((double) (re * re)) + ((double) (im * im)))), 0.5)) + re)) / im))))))))))));
		} else {
			double VAR_2;
			if ((re <= 7.222181096583517e-225)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (1.0 * ((double) (im / ((double) (((double) (re + im)) / im))))))))))));
			} else {
				double VAR_3;
				if ((re <= 1.338656002245574e+154)) {
					VAR_3 = ((double) (0.5 * ((double) (((double) sqrt(((double) (2.0 * ((double) (1.0 * ((double) (1.0 / ((double) (((double) pow(((double) (((double) (re * re)) + ((double) (im * im)))), 0.5)) + re)))))))))) * ((double) fabs(im))))));
				} else {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (1.0 * ((double) (im / ((double) (((double) (re + re)) / im))))))))))));
				}
				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 re < -8.95791604585242e-50

   1. Initial program 35.6

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

   3. Applied pow1 35.6

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

   4. Taylor expanded around -inf 17.8

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

   if -8.95791604585242e-50 < re < -8.120321757998124e-235

   1. Initial program 22.9

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

   3. Applied pow1 22.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{{\left(re \cdot re + im \cdot im\right)}^{1}}} - re\right)}\]
   4. Using strategy rm

   5. Applied flip-- 36.0

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

   6. Simplified 36.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{{\left(re \cdot re + im \cdot im\right)}^{1}} + re}}\]

   7. Simplified 36.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}}}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 36.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{1 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}}}\]

   10. Applied *-un-lft-identity 36.0

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{1 \cdot \left({im}^{2} + 0\right)}}{1 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}}\]

   11. Applied times-frac 36.0

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{{im}^{2} + 0}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)}}\]

   12. Simplified 36.0

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \frac{{im}^{2} + 0}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)}\]

   13. Simplified 35.8

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

   if -8.120321757998124e-235 < re < 7.222181096583517e-225

   1. Initial program 31.2

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

   3. Applied pow1 31.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{{\left(re \cdot re + im \cdot im\right)}^{1}}} - re\right)}\]
   4. Using strategy rm

   5. Applied flip-- 31.5

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

   6. Simplified 31.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{{\left(re \cdot re + im \cdot im\right)}^{1}} + re}}\]

   7. Simplified 31.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}}}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 31.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{1 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}}}\]

   10. Applied *-un-lft-identity 31.5

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{1 \cdot \left({im}^{2} + 0\right)}}{1 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}}\]

   11. Applied times-frac 31.5

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{{im}^{2} + 0}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)}}\]

   12. Simplified 31.5

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \frac{{im}^{2} + 0}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)}\]

   13. Simplified 30.8

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

   14. Taylor expanded around 0 33.9

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

   if 7.222181096583517e-225 < re < 1.338656002245574e+154

   1. Initial program 41.4

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

   3. Applied pow1 41.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{{\left(re \cdot re + im \cdot im\right)}^{1}}} - re\right)}\]
   4. Using strategy rm

   5. Applied flip-- 41.3

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

   6. Simplified 30.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{{\left(re \cdot re + im \cdot im\right)}^{1}} + re}}\]

   7. Simplified 30.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}}}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 30.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{1 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}}}\]

   10. Applied *-un-lft-identity 30.1

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{1 \cdot \left({im}^{2} + 0\right)}}{1 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}}\]

   11. Applied times-frac 30.1

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{{im}^{2} + 0}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)}}\]

   12. Simplified 30.1

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \frac{{im}^{2} + 0}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)}\]

   13. Simplified 27.9

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

   15. Applied div-inv 27.9

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

   16. Applied *-un-lft-identity 27.9

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

   17. Applied times-frac 30.1

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

   18. Applied associate-*r* 30.1

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

   19. Applied associate-*r* 30.1

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

   20. Applied sqrt-prod 28.9

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

   21. Simplified 17.6

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

   if 1.338656002245574e+154 < re

   1. Initial program 64.0

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

   3. Applied pow1 64.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{{\left(re \cdot re + im \cdot im\right)}^{1}}} - re\right)}\]
   4. Using strategy rm

   5. Applied flip-- 64.0

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

   6. Simplified 52.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2} + 0}}{\sqrt{{\left(re \cdot re + im \cdot im\right)}^{1}} + re}}\]

   7. Simplified 52.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}}}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 52.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} + 0}{\color{blue}{1 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}}}\]

   10. Applied *-un-lft-identity 52.0

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{1 \cdot \left({im}^{2} + 0\right)}}{1 \cdot \left({\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re\right)}}\]

   11. Applied times-frac 52.0

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{{im}^{2} + 0}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)}}\]

   12. Simplified 52.0

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \frac{{im}^{2} + 0}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)}\]

   13. Simplified 51.6

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

   14. Taylor expanded around inf 23.7

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{\color{blue}{re} + re}{im}}\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.95791604585242049 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -8.12032175799812403 \cdot 10^{-235}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}{im}}\right)}\\ \mathbf{elif}\;re \le 7.222181096583517 \cdot 10^{-225}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{re + im}{im}}\right)}\\ \mathbf{elif}\;re \le 1.33865600224557405 \cdot 10^{154}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2 \cdot \left(1 \cdot \frac{1}{{\left(re \cdot re + im \cdot im\right)}^{\frac{1}{2}} + re}\right)} \cdot \left|im\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \frac{im}{\frac{re + re}{im}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(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)))))