Average Error: 39.4 → 20.8
Time: 5.1s
Precision: binary64
\[im \gt 0.0\]
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.5285776593939578 \cdot 10^{112}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -3.3615837238960323 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.14414430262349513 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{im - re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -4.5285776593939578 \cdot 10^{112}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\

\mathbf{elif}\;re \le -3.3615837238960323 \cdot 10^{-93}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\

\mathbf{elif}\;re \le 1.14414430262349513 \cdot 10^{-93}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{im - re}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {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 <= -4.528577659393958e+112)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (-2.0 * re))))))));
	} else {
		double VAR_1;
		if ((re <= -3.3615837238960323e-93)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (((double) fabs(((double) cbrt(((double) (((double) (re * re)) + ((double) (im * im)))))))) * ((double) sqrt(((double) cbrt(((double) (((double) (re * re)) + ((double) (im * im)))))))))) - re))))))));
		} else {
			double VAR_2;
			if ((re <= 1.1441443026234951e-93)) {
				VAR_2 = ((double) (0.5 * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) (((double) sqrt(((double) sqrt(2.0)))) * ((double) sqrt(((double) (im - re))))))))));
			} else {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) (0.0 + ((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 < -4.5285776593939578e112

    1. Initial program 55.2

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

    2. Taylor expanded around -inf 9.3

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

    if -4.5285776593939578e112 < re < -3.3615837238960323e-93

    1. Initial program 16.3

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

    3. Applied add-cube-cbrt16.5

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

    4. Applied sqrt-prod16.5

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

    5. Simplified16.5

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

    if -3.3615837238960323e-93 < re < 1.14414430262349513e-93

    1. Initial program 30.3

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

    2. Taylor expanded around 0 10.5

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

    4. Applied sqrt-prod10.8

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

    6. Applied add-sqr-sqrt10.8

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

    7. Applied sqrt-prod10.9

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

    8. Applied associate-*l*10.9

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

    if 1.14414430262349513e-93 < re

    1. Initial program 53.4

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

    3. Applied flip--53.4

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

    4. Simplified39.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification20.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.5285776593939578 \cdot 10^{112}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -3.3615837238960323 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left|\sqrt[3]{re \cdot re + im \cdot im}\right| \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 1.14414430262349513 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{im - re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{0 + {im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(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)))))