Average Error: 38.5 → 22.6
Time: 4.3s
Precision: binary64
\[im \gt 0.0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8043609553613497340:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 5.88162362023062911 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 5.407517815502927 \cdot 10^{35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;re \le 5.74463349851708613 \cdot 10^{231}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot 0\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8043609553613497340:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le 5.88162362023062911 \cdot 10^{-84}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \le 5.407517815502927 \cdot 10^{35}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{elif}\;re \le 5.74463349851708613 \cdot 10^{231}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot 0\\

\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 <= -8.043609553613497e+18)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (-2.0 * re))))))));
	} else {
		double VAR_1;
		if ((re <= 5.881623620230629e-84)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
		} else {
			double VAR_2;
			if ((re <= 5.407517815502927e+35)) {
				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))))))))));
			} else {
				double VAR_3;
				if ((re <= 5.744633498517086e+231)) {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
				} else {
					VAR_3 = ((double) (0.5 * 0.0));
				}
				VAR_2 = VAR_3;
			}
			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 < -8043609553613497340

    1. Initial program 41.9

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

    2. Taylor expanded around -inf 13.8

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

    if -8043609553613497340 < re < 5.88162362023062911e-84 or 5.407517815502927e35 < re < 5.74463349851708613e231

    1. Initial program 33.7

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

    2. Taylor expanded around 0 22.1

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

    if 5.88162362023062911e-84 < re < 5.407517815502927e35

    1. Initial program 43.9

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

    3. Applied flip--43.9

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

    4. Simplified30.5

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

    if 5.74463349851708613e231 < re

    1. Initial program 64.0

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

    2. Taylor expanded around inf 47.4

      \[\leadsto 0.5 \cdot \color{blue}{0}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification22.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8043609553613497340:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le 5.88162362023062911 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 5.407517815502927 \cdot 10^{35}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;re \le 5.74463349851708613 \cdot 10^{231}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot 0\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(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)))))