Average Error: 38.2 → 27.1
Time: 4.6s
Precision: binary64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.5723644386022114 \cdot 10^{-302}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le 2.73482328897160755 \cdot 10^{-169}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 5.11600267670621171 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re + \sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{re + \sqrt{im \cdot im + re \cdot re}}\right)}\\ \mathbf{elif}\;re \le 4.73579724108509554 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 9.20722563241957548 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.5723644386022114 \cdot 10^{-302}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\

\mathbf{elif}\;re \le 2.73482328897160755 \cdot 10^{-169}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 5.11600267670621171 \cdot 10^{-144}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re + \sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{re + \sqrt{im \cdot im + re \cdot re}}\right)}\\

\mathbf{elif}\;re \le 4.73579724108509554 \cdot 10^{-105}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 9.20722563241957548 \cdot 10^{119}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + 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 ((re <= -2.5723644386022114e-302)) {
		VAR = ((double) (0.5 * ((double) (((double) sqrt(((double) (((double) (im * im)) * 2.0)))) / ((double) sqrt(((double) (((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re)))))) - re))))))));
	} else {
		double VAR_1;
		if ((re <= 2.7348232889716076e-169)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + im))))))));
		} else {
			double VAR_2;
			if ((re <= 5.1160026767062117e-144)) {
				VAR_2 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (((double) sqrt(((double) (re + ((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re)))))))))) * ((double) sqrt(((double) (re + ((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re))))))))))))))))));
			} else {
				double VAR_3;
				if ((re <= 4.7357972410850955e-105)) {
					VAR_3 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + im))))))));
				} else {
					double VAR_4;
					if ((re <= 9.207225632419575e+119)) {
						VAR_4 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + ((double) sqrt(((double) (((double) (im * im)) + ((double) (re * re))))))))))))));
					} else {
						VAR_4 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (re + re))))))));
					}
					VAR_3 = VAR_4;
				}
				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

Target

Original38.2
Target33.3
Herbie27.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -2.5723644386022114e-302

    1. Initial program 45.3

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

    3. Applied flip-+45.2

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

    4. Applied associate-*r/45.2

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

    5. Applied sqrt-div45.3

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

    6. Simplified34.5

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

    if -2.5723644386022114e-302 < re < 2.73482328897160755e-169 or 5.11600267670621171e-144 < re < 4.73579724108509554e-105

    1. Initial program 26.1

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

    2. Taylor expanded around 0 35.2

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

    if 2.73482328897160755e-169 < re < 5.11600267670621171e-144

    1. Initial program 20.3

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

    3. Applied add-sqr-sqrt20.5

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

    4. Simplified20.5

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

    5. Simplified20.5

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

    if 4.73579724108509554e-105 < re < 9.20722563241957548e119

    1. Initial program 15.6

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

    if 9.20722563241957548e119 < re

    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 10.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.5723644386022114 \cdot 10^{-302}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2}}{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}\\ \mathbf{elif}\;re \le 2.73482328897160755 \cdot 10^{-169}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 5.11600267670621171 \cdot 10^{-144}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re + \sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{re + \sqrt{im \cdot im + re \cdot re}}\right)}\\ \mathbf{elif}\;re \le 4.73579724108509554 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 9.20722563241957548 \cdot 10^{119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{im \cdot im + re \cdot re}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))