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

Average Error: 38.7 → 21.4

Time: 3.7s

Precision: binary64

\[im \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1.21482900425984 \cdot 10^{120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \le -4.2023189404870015 \cdot 10^{67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le -1.62333323389319075 \cdot 10^{-113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re\right)}\\ \mathbf{elif}\;re \le 9.09680766621453191 \cdot 10^{85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \end{array}\]

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 <= -1.2148290042598397e+120)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re * -2.0))))))));
	} else {
		double VAR_1;
		if ((re <= -4.2023189404870015e+67)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
		} else {
			double VAR_2;
			if ((re <= -1.6233332338931907e-113)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) exp(((double) log(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) - re))))))));
			} else {
				double VAR_3;
				if ((re <= 9.096807666214532e+85)) {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * im))))));
				} else {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (im * im)) / ((double) (re + ((double) sqrt(((double) (((double) (re * re)) + ((double) (im * 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 < -1.21482900425984e120

   1. Initial program 56.6

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

   2. Taylor expanded around -inf 10.0

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

   3. Simplified 10.0

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

   if -1.21482900425984e120 < re < -4.2023189404870015e67

   1. Initial program 16.2

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

   2. Taylor expanded around 0 33.8

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

   if -4.2023189404870015e67 < re < -1.62333323389319075e-113

   1. Initial program 15.4

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

   3. Applied add-exp-log 17.6

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

   if -1.62333323389319075e-113 < re < 9.09680766621453191e85

   1. Initial program 34.3

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

   2. Taylor expanded around 0 16.9

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

   4. Applied add-exp-log 20.3

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

   6. Applied pow1/2 20.3

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

   7. Applied log-pow 20.3

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

   8. Applied exp-prod 20.6

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

   9. Taylor expanded around inf 20.0

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

   10. Simplified 16.5

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

   if 9.09680766621453191e85 < re

   1. Initial program 60.4

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

   3. Applied flip-- 60.4

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

   4. Simplified 44.5

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

   5. Simplified 44.5

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

4. Final simplification 21.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.21482900425984 \cdot 10^{120}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \le -4.2023189404870015 \cdot 10^{67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le -1.62333323389319075 \cdot 10^{-113}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} - re\right)}\\ \mathbf{elif}\;re \le 9.09680766621453191 \cdot 10^{85}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))