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

Average Error: 38.3 → 17.6

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -4.69108546213632265 \cdot 10^{113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \le -7.88936342124094937 \cdot 10^{-297}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 1.42254223392653965 \cdot 10^{-214}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 4.4095657239434897 \cdot 10^{116}:\\ \;\;\;\;0.5 \cdot \left(\frac{\left|im\right|}{\sqrt{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}} \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{re + re}}\right)\\ \end{array}\]

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 <= -4.6910854621363227e+113)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re * -2.0))))))));
	} else {
		double VAR_1;
		if ((re <= -7.88936342124095e-297)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))));
		} else {
			double VAR_2;
			if ((re <= 1.4225422339265396e-214)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
			} else {
				double VAR_3;
				if ((re <= 4.40956572394349e+116)) {
					VAR_3 = ((double) (0.5 * ((double) ((((double) fabs(im)) / ((double) sqrt(((double) sqrt(((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im))))))))))))) * (((double) sqrt(2.0)) / ((double) sqrt(((double) sqrt(((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))))))));
				} else {
					VAR_3 = ((double) (0.5 * ((double) (((double) fabs(im)) * ((double) sqrt((2.0 / ((double) (re + 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 5 regimes

2. if re < -4.69108546213632265e113

   1. Initial program 54.7

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

   2. Taylor expanded around -inf 10.2

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

   3. Simplified 10.2

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

   if -4.69108546213632265e113 < re < -7.88936342124094937e-297

   1. Initial program 19.9

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

   if -7.88936342124094937e-297 < re < 1.42254223392653965e-214

   1. Initial program 30.0

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

   2. Taylor expanded around 0 32.2

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

   if 1.42254223392653965e-214 < re < 4.4095657239434897e116

   1. Initial program 40.5

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

   3. Applied flip-- 40.4

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

   4. Applied associate-*r/ 40.4

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

   5. Applied sqrt-div 40.5

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

   6. Simplified 29.3

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

   7. Simplified 29.3

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

   9. Applied add-sqr-sqrt 29.3

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

   10. Applied sqrt-prod 29.4

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

   11. Applied sqrt-prod 29.4

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

   12. Applied times-frac 29.4

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

   13. Simplified 18.6

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

   14. Simplified 18.6

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

   if 4.4095657239434897e116 < re

   1. Initial program 62.1

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

   3. Applied flip-- 62.1

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

   4. Applied associate-*r/ 62.1

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

   5. Applied sqrt-div 62.1

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

   6. Simplified 46.5

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

   7. Simplified 46.5

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

   9. Applied *-un-lft-identity 46.5

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

   10. Applied sqrt-prod 46.5

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

   11. Applied sqrt-prod 46.5

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

   12. Applied times-frac 46.5

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

   13. Simplified 44.1

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

   14. Simplified 44.1

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

   16. Applied sqrt-undiv 44.1

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

   17. Taylor expanded around 0 10.1

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

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.69108546213632265 \cdot 10^{113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \le -7.88936342124094937 \cdot 10^{-297}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 1.42254223392653965 \cdot 10^{-214}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 4.4095657239434897 \cdot 10^{116}:\\ \;\;\;\;0.5 \cdot \left(\frac{\left|im\right|}{\sqrt{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}} \cdot \frac{\sqrt{2}}{\sqrt{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left|im\right| \cdot \sqrt{\frac{2}{re + re}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))))