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

Average Error: 38.7 → 24.1

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;im \le -1.2671946284215746 \cdot 10^{83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -1.651091264846107 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\\ \mathbf{elif}\;im \le -2.0514133168585066 \cdot 10^{-182}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le -1.84433611703650748 \cdot 10^{-231}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le 1.45766144586139333 \cdot 10^{-304}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{2 \cdot re}\right)}\\ \mathbf{elif}\;im \le 7.28289994053803203 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 1.22568432920739315 \cdot 10^{73}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im - re}\right)\\ \end{array}\]

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 temp;
	if ((im <= -1.2671946284215746e+83)) {
		temp = (0.5 * sqrt((2.0 * -(re + im))));
	} else {
		double temp_1;
		if ((im <= -1.651091264846107e-119)) {
			temp_1 = (0.5 * (sqrt(2.0) * sqrt((sqrt(((re * re) + (im * im))) - re))));
		} else {
			double temp_2;
			if ((im <= -2.0514133168585066e-182)) {
				temp_2 = (0.5 * sqrt((2.0 * ((-1.0 * re) - re))));
			} else {
				double temp_3;
				if ((im <= -1.8443361170365075e-231)) {
					temp_3 = (0.5 * sqrt((2.0 * -(re + im))));
				} else {
					double temp_4;
					if ((im <= 1.4576614458613933e-304)) {
						temp_4 = (0.5 * sqrt((2.0 * (im * (im / (2.0 * re))))));
					} else {
						double temp_5;
						if ((im <= 7.282899940538032e-158)) {
							temp_5 = (0.5 * sqrt((2.0 * ((-1.0 * re) - re))));
						} else {
							double temp_6;
							if ((im <= 1.2256843292073932e+73)) {
								temp_6 = (0.5 * (sqrt(2.0) * sqrt((sqrt(((re * re) + (im * im))) - re))));
							} else {
								temp_6 = (0.5 * (sqrt(2.0) * sqrt((im - re))));
							}
							temp_5 = temp_6;
						}
						temp_4 = temp_5;
					}
					temp_3 = temp_4;
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if im < -1.2671946284215746e+83 or -2.0514133168585066e-182 < im < -1.8443361170365075e-231

   1. Initial program 49.3

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

   3. Applied flip-- 53.0

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

   4. Simplified 51.1

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

   5. Taylor expanded around -inf 18.1

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

   if -1.2671946284215746e+83 < im < -1.651091264846107e-119 or 7.282899940538032e-158 < im < 1.2256843292073932e+73

   1. Initial program 24.1

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

   3. Applied sqrt-prod 24.4

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

   if -1.651091264846107e-119 < im < -2.0514133168585066e-182 or 1.4576614458613933e-304 < im < 7.282899940538032e-158

   1. Initial program 42.3

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

   2. Taylor expanded around -inf 37.5

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

   if -1.8443361170365075e-231 < im < 1.4576614458613933e-304

   1. Initial program 42.0

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

   3. Applied flip-- 59.0

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

   4. Simplified 49.8

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

   6. Applied *-un-lft-identity 49.8

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

   7. Applied add-sqr-sqrt 63.1

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

   8. Applied unpow-prod-down 63.1

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

   9. Applied times-frac 63.1

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

   10. Simplified 63.1

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

   11. Simplified 49.2

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

   12. Taylor expanded around inf 46.6

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

   if 1.2256843292073932e+73 < im

   1. Initial program 48.3

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

   3. Applied sqrt-prod 48.4

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

   4. Taylor expanded around 0 10.5

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

4. Final simplification 24.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.2671946284215746 \cdot 10^{83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le -1.651091264846107 \cdot 10^{-119}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\\ \mathbf{elif}\;im \le -2.0514133168585066 \cdot 10^{-182}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le -1.84433611703650748 \cdot 10^{-231}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;im \le 1.45766144586139333 \cdot 10^{-304}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \frac{im}{2 \cdot re}\right)}\\ \mathbf{elif}\;im \le 7.28289994053803203 \cdot 10^{-158}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;im \le 1.22568432920739315 \cdot 10^{73}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im - re}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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)))))