Average Error: 38.1 → 25.5
Time: 4.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -8.4005577757922215 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{-1}{2} \cdot \frac{im}{re} + -2 \cdot \frac{re}{im}}}\\ \mathbf{elif}\;re \le 1.7249368426738814 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -8.4005577757922215 \cdot 10^{86}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{-1}{2} \cdot \frac{im}{re} + -2 \cdot \frac{re}{im}}}\\

\mathbf{elif}\;re \le 1.7249368426738814 \cdot 10^{-51}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{4 \cdot 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 <= -8.400557775792221e+86)) {
		VAR = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im / ((double) (((double) (-0.5 * ((double) (im / re)))) + ((double) (-2.0 * ((double) (re / im))))))))))))));
	} else {
		double VAR_1;
		if ((re <= 1.7249368426738814e-51)) {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (2.0 * ((double) (im / ((double) (((double) (((double) sqrt(((double) (((double) (re * re)) + ((double) (im * im)))))) - re)) / im))))))))));
		} else {
			VAR_1 = ((double) (0.5 * ((double) sqrt(((double) (4.0 * re))))));
		}
		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

Original38.1
Target33.1
Herbie25.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -8.400557775792221e+86

    1. Initial program 60.2

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

    3. Applied flip-+60.2

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

    4. Simplified45.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied associate-/l*44.6

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

    7. Taylor expanded around -inf 25.3

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

    8. Simplified25.3

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

    if -8.400557775792221e+86 < re < 1.7249368426738814e-51

    1. Initial program 32.0

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

    3. Applied flip-+35.7

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

    4. Simplified31.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]
    5. Using strategy rm

    6. Applied associate-/l*29.7

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

    if 1.7249368426738814e-51 < re

    1. Initial program 35.9

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

    3. Applied flip-+57.2

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

    4. Simplified56.8

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

    5. Taylor expanded around 0 17.6

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

  4. Final simplification25.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -8.4005577757922215 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{-1}{2} \cdot \frac{im}{re} + -2 \cdot \frac{re}{im}}}\\ \mathbf{elif}\;re \le 1.7249368426738814 \cdot 10^{-51}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} - re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{4 \cdot re}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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)))))