Average Error: 38.8 → 6.0
Time: 3.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 9.9876162163965062 \cdot 10^{-278}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{elif}\;re \le 8.65125741163031176 \cdot 10^{177}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot 2}{\frac{re + \mathsf{hypot}\left(re, im\right)}{{im}^{\left(\frac{2}{2}\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 9.9876162163965062 \cdot 10^{-278}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\

\mathbf{elif}\;re \le 8.65125741163031176 \cdot 10^{177}:\\
\;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot 2}{\frac{re + \mathsf{hypot}\left(re, im\right)}{{im}^{\left(\frac{2}{2}\right)}}}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{re + \mathsf{hypot}\left(re, im\right)}}\\

\end{array}
double code(double re, double im) {
	return (0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))));
}
↓
double code(double re, double im) {
	double VAR;
	if ((re <= 9.987616216396506e-278)) {
		VAR = (0.5 * sqrt((2.0 * (hypot(im, re) - re))));
	} else {
		double VAR_1;
		if ((re <= 8.651257411630312e+177)) {
			VAR_1 = (0.5 * sqrt(((im * 2.0) / ((re + hypot(re, im)) / pow(im, (2.0 / 2.0))))));
		} else {
			VAR_1 = (0.5 * (sqrt((2.0 * pow(im, 2.0))) / sqrt((re + hypot(re, im)))));
		}
		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 3 regimes

  2. if re < 9.987616216396506e-278

    1. Initial program 31.1

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

    3. Applied +-commutative31.1

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

    4. Applied hypot-def0.1

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

    if 9.987616216396506e-278 < re < 8.651257411630312e+177

    1. Initial program 42.2

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

    3. Applied flip--42.0

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

    4. Simplified32.8

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

    5. Simplified31.7

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

    7. Applied sqr-pow31.7

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

    8. Applied associate-/l*10.5

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

    9. Applied associate-*r/10.5

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

    10. Simplified10.5

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

    if 8.651257411630312e+177 < re

    1. Initial program 64.0

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

    3. Applied flip--64.0

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

    4. Simplified49.4

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

    5. Simplified30.6

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

    7. Applied associate-*r/30.5

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

    8. Applied sqrt-div18.8

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 9.9876162163965062 \cdot 10^{-278}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(im, re\right) - re\right)}\\ \mathbf{elif}\;re \le 8.65125741163031176 \cdot 10^{177}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot 2}{\frac{re + \mathsf{hypot}\left(re, im\right)}{{im}^{\left(\frac{2}{2}\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot {im}^{2}}}{\sqrt{re + \mathsf{hypot}\left(re, im\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))