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

Average Error: 38.8 → 21.7

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.43155255567854579 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -3.73226044257471103 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \left(\left|\left|im\right| \cdot \sqrt{\frac{1}{re + \sqrt{{re}^{2} + {im}^{2}}}}\right| \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \le -6.6319531765329248 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 7.1737243468109888 \cdot 10^{-240}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 6.10690435627845302 \cdot 10^{-174}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 5.0041734007880612 \cdot 10^{108}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{\left|im\right|}{\sqrt{re + re}}\right|\right)\\ \end{array}\]

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 ((re <= -1.4315525556785458e-07)) {
		temp = (0.5 * sqrt((2.0 * (-2.0 * re))));
	} else {
		double temp_1;
		if ((re <= -3.732260442574711e-107)) {
			temp_1 = (0.5 * (fabs((fabs(im) * sqrt((1.0 / (re + sqrt((pow(re, 2.0) + pow(im, 2.0)))))))) * sqrt(2.0)));
		} else {
			double temp_2;
			if ((re <= -6.631953176532925e-267)) {
				temp_2 = (0.5 * sqrt((2.0 * -(re + im))));
			} else {
				double temp_3;
				if ((re <= 7.173724346810989e-240)) {
					temp_3 = (0.5 * sqrt((2.0 * (im - re))));
				} else {
					double temp_4;
					if ((re <= 6.106904356278453e-174)) {
						temp_4 = (0.5 * sqrt((2.0 * -(re + im))));
					} else {
						double temp_5;
						if ((re <= 5.004173400788061e+108)) {
							temp_5 = (0.5 * (sqrt(sqrt(2.0)) * (sqrt(sqrt(2.0)) * fabs((fabs(im) / sqrt((sqrt(((re * re) + (im * im))) + re)))))));
						} else {
							temp_5 = (0.5 * (sqrt(2.0) * fabs((fabs(im) / sqrt((re + re))))));
						}
						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 6 regimes

2. if re < -1.4315525556785458e-07

   1. Initial program 40.5

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

   2. Taylor expanded around -inf 14.7

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

   if -1.4315525556785458e-07 < re < -3.732260442574711e-107

   1. Initial program 14.9

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

   3. Applied flip-- 39.0

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

   4. Simplified 39.0

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

   6. Applied add-sqr-sqrt 39.1

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

   7. Applied add-sqr-sqrt 39.1

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

   8. Applied times-frac 39.1

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

   9. Simplified 39.1

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

   10. Simplified 38.7

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

   12. Applied sqrt-prod 38.8

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

   13. Simplified 38.8

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

   14. Taylor expanded around 0 38.8

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

   if -3.732260442574711e-107 < re < -6.631953176532925e-267 or 7.173724346810989e-240 < re < 6.106904356278453e-174

   1. Initial program 25.9

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

   3. Applied flip-- 31.1

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

   4. Simplified 31.1

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

   5. Taylor expanded around -inf 35.8

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

   if -6.631953176532925e-267 < re < 7.173724346810989e-240

   1. Initial program 30.9

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

   2. Taylor expanded around 0 32.0

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

   if 6.106904356278453e-174 < re < 5.004173400788061e+108

   1. Initial program 42.4

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

   3. Applied flip-- 42.4

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

   4. Simplified 31.0

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

   6. Applied add-sqr-sqrt 31.1

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

   7. Applied add-sqr-sqrt 31.1

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

   8. Applied times-frac 31.1

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

   9. Simplified 31.1

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

   10. Simplified 28.1

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

   12. Applied sqrt-prod 28.2

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

   13. Simplified 16.5

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

   15. Applied add-sqr-sqrt 16.5

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

   16. Applied sqrt-prod 16.5

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

   17. Applied associate-*l* 16.5

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

   if 5.004173400788061e+108 < re

   1. Initial program 61.4

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

   3. Applied flip-- 61.4

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

   4. Simplified 46.2

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

   6. Applied add-sqr-sqrt 46.3

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

   7. Applied add-sqr-sqrt 46.3

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

   8. Applied times-frac 46.3

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

   9. Simplified 46.3

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

   10. Simplified 45.8

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

   12. Applied sqrt-prod 45.9

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

   13. Simplified 42.3

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

   14. Taylor expanded around inf 9.9

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

4. Final simplification 21.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.43155255567854579 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -3.73226044257471103 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \left(\left|\left|im\right| \cdot \sqrt{\frac{1}{re + \sqrt{{re}^{2} + {im}^{2}}}}\right| \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \le -6.6319531765329248 \cdot 10^{-267}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 7.1737243468109888 \cdot 10^{-240}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 6.10690435627845302 \cdot 10^{-174}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 5.0041734007880612 \cdot 10^{108}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \left|\frac{\left|im\right|}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \left|\frac{\left|im\right|}{\sqrt{re + re}}\right|\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(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)))))