Average Error: 38.7 → 28.1
Time: 3.6s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.4182869934259443 \cdot 10^{116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -2.7347411849077974 \cdot 10^{-191}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{3} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\ \mathbf{elif}\;re \le 4.3336418922841928 \cdot 10^{-154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.4182869934259443 \cdot 10^{116}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -2.7347411849077974 \cdot 10^{-191}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{3} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\

\mathbf{elif}\;re \le 4.3336418922841928 \cdot 10^{-154}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\end{array}
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 <= -1.4182869934259443e+116)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (-2.0 * re))))))));
	} else {
		double VAR_1;
		if ((re <= -2.7347411849077974e-191)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (((double) pow(((double) sqrt(((double) sqrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))), 3.0)) * ((double) sqrt(((double) sqrt(((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))))) - re))))))));
		} else {
			double VAR_2;
			if ((re <= 4.333641892284193e-154)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
			} else {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) pow(im, 2.0)) / ((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re))))))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if re < -1.4182869934259443e116

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 10.3

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

    if -1.4182869934259443e116 < re < -2.7347411849077974e-191

    1. Initial program 16.8

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

    3. Applied add-sqr-sqrt16.8

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

    4. Applied sqrt-prod16.9

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

    6. Applied add-sqr-sqrt16.9

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

    7. Applied sqrt-prod16.9

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

    8. Applied sqrt-prod16.9

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

    9. Applied associate-*r*16.9

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

    10. Simplified17.0

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

    if -2.7347411849077974e-191 < re < 4.3336418922841928e-154

    1. Initial program 31.9

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

    2. Taylor expanded around 0 35.6

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

    if 4.3336418922841928e-154 < re

    1. Initial program 51.2

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

    3. Applied flip--51.2

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

    4. Simplified38.1

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{{im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification28.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.4182869934259443 \cdot 10^{116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -2.7347411849077974 \cdot 10^{-191}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left({\left(\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}}\right)}^{3} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im}}} - re\right)}\\ \mathbf{elif}\;re \le 4.3336418922841928 \cdot 10^{-154}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(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)))))