Average Error: 38.4 → 21.8
Time: 4.4min
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -9.2824486143826077 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -8.56597067240157711 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\ \mathbf{elif}\;re \le 4.2728634981077738 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\right)\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -9.2824486143826077 \cdot 10^{-135}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -8.56597067240157711 \cdot 10^{-251}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\

\mathbf{elif}\;re \le 4.2728634981077738 \cdot 10^{-190}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\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 ((re <= -9.282448614382608e-135)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (-2.0 * re))))))));
	} else {
		double VAR_1;
		if ((re <= -8.565970672401577e-251)) {
			VAR_1 = ((double) (0.5 * (((double) sqrt(2.0)) / (((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)))) / ((double) fabs(im))))));
		} else {
			double VAR_2;
			if ((re <= 4.272863498107774e-190)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im - re))))))));
			} else {
				double VAR_3;
				if ((re <= 3.466178613033722e+36)) {
					VAR_3 = ((double) (0.5 * (((double) sqrt(2.0)) / (((double) sqrt(((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) + re)))) / ((double) fabs(im))))));
				} else {
					VAR_3 = ((double) (0.5 * ((double) (((double) (((double) cbrt(((double) sqrt(2.0)))) * ((double) cbrt(((double) sqrt(2.0)))))) * ((double) (((double) cbrt(((double) sqrt(2.0)))) * (((double) fabs(im)) / ((double) sqrt(((double) (re + re)))))))))));
				}
				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 < -9.2824486143826077e-135

    1. Initial program 32.2

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

    2. Taylor expanded around -inf 21.8

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

    if -9.2824486143826077e-135 < re < -8.56597067240157711e-251 or 4.2728634981077738e-190 < re < 3.46617861303372194e36

    1. Initial program 35.0

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

    3. Applied flip--37.1

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

    4. Applied associate-*r/37.1

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

    5. Applied sqrt-div37.4

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

    6. Simplified32.1

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

    8. Applied sqrt-prod32.1

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

    9. Applied associate-/l*32.1

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

    10. Simplified22.6

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

    if -8.56597067240157711e-251 < re < 4.2728634981077738e-190

    1. Initial program 29.8

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

    2. Taylor expanded around 0 33.8

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

    if 3.46617861303372194e36 < re

    1. Initial program 58.8

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

    3. Applied flip--58.8

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

    4. Applied associate-*r/58.8

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

    5. Applied sqrt-div58.8

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

    6. Simplified40.4

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

    8. Applied *-un-lft-identity40.4

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

    9. Applied sqrt-prod40.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{im \cdot im}}}{1 \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    10. Applied times-frac40.4

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

    11. Simplified40.4

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

    12. Simplified36.0

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

    14. Applied add-cube-cbrt36.0

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

    15. Applied associate-*l*36.0

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

    16. Taylor expanded around inf 12.8

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

  4. Final simplification21.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -9.2824486143826077 \cdot 10^{-135}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -8.56597067240157711 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\ \mathbf{elif}\;re \le 4.2728634981077738 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 3.46617861303372194 \cdot 10^{36}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}{\left|im\right|}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\left|im\right|}{\sqrt{re + re}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))))