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

Average Error: 38.4 → 22.8

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.79730493784702906 \cdot 10^{146}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 7.96106282919079009 \cdot 10^{-250}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 2.2522826674058156 \cdot 10^{-185}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 2.1169694866595254 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right)\\ \mathbf{elif}\;re \le 2.4265458997480397 \cdot 10^{117}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{re + re}\right)}\\ \end{array}\]

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 <= -1.797304937847029e+146)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (-2.0 * re))))))));
	} else {
		double VAR_1;
		if ((re <= 7.96106282919079e-250)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (((double) sqrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) * ((double) sqrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) - re))))))));
		} else {
			double VAR_2;
			if ((re <= 2.2522826674058156e-185)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) -(((double) (re + im))))))))));
			} else {
				double VAR_3;
				if ((re <= 2.1169694866595254e-22)) {
					VAR_3 = ((double) (0.5 * ((double) (((double) (((double) cbrt(((double) sqrt(((double) (2.0 * ((double) (im * ((double) (im / ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)))))))))))) * ((double) cbrt(((double) sqrt(((double) (2.0 * ((double) (im * ((double) (im / ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)))))))))))))) * ((double) cbrt(((double) sqrt(((double) (2.0 * ((double) (im * ((double) (im / ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))))))))))));
				} else {
					double VAR_4;
					if ((re <= 2.4265458997480397e+117)) {
						VAR_4 = ((double) (0.5 * ((double) (((double) sqrt(((double) (2.0 * ((double) pow(im, 2.0)))))) / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))));
					} else {
						VAR_4 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im * ((double) (im / ((double) (re + re))))))))))));
					}
					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 6 regimes

2. if re < -1.797304937847029e+146

   1. Initial program 61.7

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

   2. Taylor expanded around -inf 9.2

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

   if -1.797304937847029e+146 < re < 7.96106282919079e-250

   1. Initial program 21.8

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

   3. Applied add-sqr-sqrt 21.8

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

   4. Applied sqrt-prod 21.9

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

   if 7.96106282919079e-250 < re < 2.2522826674058156e-185

   1. Initial program 31.0

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

   3. Applied flip-- 30.3

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

   4. Simplified 30.3

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

   5. Taylor expanded around -inf 39.0

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

   if 2.2522826674058156e-185 < re < 2.1169694866595254e-22

   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 flip-- 35.5

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

   4. Simplified 29.9

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

   6. Applied *-un-lft-identity 29.9

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

   7. Applied add-sqr-sqrt 46.9

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

   8. Applied unpow-prod-down 46.9

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

   9. Applied times-frac 45.6

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

   10. Simplified 45.6

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

   11. Simplified 26.6

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

   13. Applied add-cube-cbrt 27.4

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right)}\]

   if 2.1169694866595254e-22 < re < 2.4265458997480397e+117

   1. Initial program 47.3

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

   3. Applied flip-- 47.3

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

   4. Simplified 30.9

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

   6. Applied associate-*r/ 30.9

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

   7. Applied sqrt-div 28.7

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

   if 2.4265458997480397e+117 < re

   1. Initial program 62.4

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

   3. Applied flip-- 62.4

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

   4. Simplified 47.4

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

   6. Applied *-un-lft-identity 47.4

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

   7. Applied add-sqr-sqrt 55.5

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

   8. Applied unpow-prod-down 55.5

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

   9. Applied times-frac 55.3

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

   10. Simplified 55.3

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

   11. Simplified 47.0

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

   12. Taylor expanded around inf 23.6

       \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{\color{blue}{re} + re}\right)}\]
3. Recombined 6 regimes into one program.

4. Final simplification 22.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.79730493784702906 \cdot 10^{146}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 7.96106282919079009 \cdot 10^{-250}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 2.2522826674058156 \cdot 10^{-185}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 2.1169694866595254 \cdot 10^{-22}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right) \cdot \sqrt[3]{\sqrt{2 \cdot \left(im \cdot \frac{im}{\sqrt{re \cdot re + im \cdot im} + re}\right)}}\right)\\ \mathbf{elif}\;re \le 2.4265458997480397 \cdot 10^{117}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{re + re}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(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)))))