Average Error: 38.8 → 26.3
Time: 4.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 1.99373440133803904 \cdot 10^{-165}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 8.81639119557174445 \cdot 10^{152}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \le 0.0:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 1.99373440133803904 \cdot 10^{-165}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 8.81639119557174445 \cdot 10^{152}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\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 ((((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)) <= 0.0)) {
		VAR = ((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))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)) <= 1.993734401338039e-165)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (-2.0 * re))))))));
		} else {
			double VAR_2;
			if ((((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)) <= 8.816391195571744e+152)) {
				VAR_2 = ((double) (0.5 * ((double) (((double) sqrt(2.0)) * ((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re))))))));
			} else {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (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 (- (sqrt (+ (* re re) (* im im))) re) < 0.0

    1. Initial program 58.5

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

    3. Applied flip--58.0

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

    4. Simplified35.7

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

    if 0.0 < (- (sqrt (+ (* re re) (* im im))) re) < 1.993734401338039e-165

    1. Initial program 54.7

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

    2. Taylor expanded around -inf 31.0

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

    if 1.993734401338039e-165 < (- (sqrt (+ (* re re) (* im im))) re) < 8.816391195571744e+152

    1. Initial program 0.6

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

    3. Applied sqrt-prod1.1

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

    if 8.816391195571744e+152 < (- (sqrt (+ (* re re) (* im im))) re)

    1. Initial program 63.6

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

    2. Taylor expanded around 0 44.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{im} - re\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} - re \le 0.0:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 1.99373440133803904 \cdot 10^{-165}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} - re \le 8.81639119557174445 \cdot 10^{152}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020129 
(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)))))