math.sqrt on complex, real part

Average Accuracy: 39.7% → 82.2%

Time: 11.4s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{if}\;re \leq -9.2 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq -1.4 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;re \leq -1.05 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -6.3 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot e^{0.5 \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right) + \log \left(\frac{-1}{re}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

↓

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))
   (if (<= re -9.2e+150)
     (* 0.5 (sqrt (* im (/ im (- re)))))
     (if (<= re -1.4e+84)
       t_0
       (if (<= re -1.05e+45)
         (* 0.5 (/ im (sqrt (- re))))
         (if (<= re -6.3e+16)
           (*
            0.5
            (exp (* 0.5 (+ (* -2.0 (log (/ -1.0 im))) (log (/ -1.0 re))))))
           t_0))))))

C

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 t_0 = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	double tmp;
	if (re <= -9.2e+150) {
		tmp = 0.5 * sqrt((im * (im / -re)));
	} else if (re <= -1.4e+84) {
		tmp = t_0;
	} else if (re <= -1.05e+45) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (re <= -6.3e+16) {
		tmp = 0.5 * exp((0.5 * ((-2.0 * log((-1.0 / im))) + log((-1.0 / re)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) + re)));
}

↓

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	double tmp;
	if (re <= -9.2e+150) {
		tmp = 0.5 * Math.sqrt((im * (im / -re)));
	} else if (re <= -1.4e+84) {
		tmp = t_0;
	} else if (re <= -1.05e+45) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (re <= -6.3e+16) {
		tmp = 0.5 * Math.exp((0.5 * ((-2.0 * Math.log((-1.0 / im))) + Math.log((-1.0 / re)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	return 0.5 * math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) + re)))

↓

def code(re, im):
	t_0 = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	tmp = 0
	if re <= -9.2e+150:
		tmp = 0.5 * math.sqrt((im * (im / -re)))
	elif re <= -1.4e+84:
		tmp = t_0
	elif re <= -1.05e+45:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif re <= -6.3e+16:
		tmp = 0.5 * math.exp((0.5 * ((-2.0 * math.log((-1.0 / im))) + math.log((-1.0 / re)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re))))
end

↓

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))))
	tmp = 0.0
	if (re <= -9.2e+150)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / Float64(-re)))));
	elseif (re <= -1.4e+84)
		tmp = t_0;
	elseif (re <= -1.05e+45)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (re <= -6.3e+16)
		tmp = Float64(0.5 * exp(Float64(0.5 * Float64(Float64(-2.0 * log(Float64(-1.0 / im))) + log(Float64(-1.0 / re))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((2.0 * (sqrt(((re * re) + (im * im))) + re)));
end

↓

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	tmp = 0.0;
	if (re <= -9.2e+150)
		tmp = 0.5 * sqrt((im * (im / -re)));
	elseif (re <= -1.4e+84)
		tmp = t_0;
	elseif (re <= -1.05e+45)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (re <= -6.3e+16)
		tmp = 0.5 * exp((0.5 * ((-2.0 * log((-1.0 / im))) + log((-1.0 / re)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -9.2e+150], N[(0.5 * N[Sqrt[N[(im * N[(im / (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.4e+84], t$95$0, If[LessEqual[re, -1.05e+45], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -6.3e+16], N[(0.5 * N[Exp[N[(0.5 * N[(N[(-2.0 * N[Log[N[(-1.0 / im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Target

Original	39.7%
Target	47.7%
Herbie	82.2%

\[\begin{array}{l} \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if re < -9.20000000000000004e150

   1. Initial program 0.1%

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

   2. Simplified 34.2%

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

      [Start] 0.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 0.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 34.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 54.8%

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

   4. Simplified 54.8%

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

      [Start] 54.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 54.8	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 54.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Taylor expanded in im around 0 54.9%

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

   6. Simplified 65.3%

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

      [Start] 54.9	\[ 0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re}} \]
      unpow2 [=>] 54.9	\[ 0.5 \cdot \sqrt{-1 \cdot \frac{\color{blue}{im \cdot im}}{re}} \]
      associate-*r/ [=>] 54.9	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-1 \cdot \left(im \cdot im\right)}{re}}} \]
      associate-*l/ [<=] 54.9	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-1}{re} \cdot \left(im \cdot im\right)}} \]
      /-rgt-identity [<=] 54.9	\[ 0.5 \cdot \sqrt{\frac{-1}{re} \cdot \color{blue}{\frac{im \cdot im}{1}}} \]
      associate-*r/ [=>] 54.9	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{\frac{-1}{re} \cdot \left(im \cdot im\right)}{1}}} \]
      associate-/r/ [<=] 53.8	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{\frac{-1}{\frac{re}{im \cdot im}}}}{1}} \]
      associate-/l/ [=>] 53.8	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-1}{1 \cdot \frac{re}{im \cdot im}}}} \]
      metadata-eval [<=] 53.8	\[ 0.5 \cdot \sqrt{\frac{-1}{\color{blue}{\frac{-1}{-1}} \cdot \frac{re}{im \cdot im}}} \]
      associate-/r* [=>] 64.3	\[ 0.5 \cdot \sqrt{\frac{-1}{\frac{-1}{-1} \cdot \color{blue}{\frac{\frac{re}{im}}{im}}}} \]
      times-frac [<=] 64.3	\[ 0.5 \cdot \sqrt{\frac{-1}{\color{blue}{\frac{-1 \cdot \frac{re}{im}}{-1 \cdot im}}}} \]
      neg-mul-1 [<=] 64.3	\[ 0.5 \cdot \sqrt{\frac{-1}{\frac{\color{blue}{-\frac{re}{im}}}{-1 \cdot im}}} \]
      mul-1-neg [=>] 64.3	\[ 0.5 \cdot \sqrt{\frac{-1}{\frac{-\frac{re}{im}}{\color{blue}{-im}}}} \]
      associate-/l* [<=] 65.3	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-1 \cdot \left(-im\right)}{-\frac{re}{im}}}} \]
      mul-1-neg [=>] 65.3	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{-\left(-im\right)}}{-\frac{re}{im}}} \]
      remove-double-neg [=>] 65.3	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{im}}{-\frac{re}{im}}} \]
      distribute-neg-frac [=>] 65.3	\[ 0.5 \cdot \sqrt{\frac{im}{\color{blue}{\frac{-re}{im}}}} \]
      associate-/r/ [=>] 65.3	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{im}{-re} \cdot im}} \]
      *-commutative [=>] 65.3	\[ 0.5 \cdot \sqrt{\color{blue}{im \cdot \frac{im}{-re}}} \]

   if -9.20000000000000004e150 < re < -1.39999999999999991e84 or -6.3e16 < re

   1. Initial program 46.8%

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

   2. Simplified 88.5%

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

      [Start] 46.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 46.8	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 88.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   if -1.39999999999999991e84 < re < -1.04999999999999997e45

   1. Initial program 23.9%

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

   2. Simplified 53.2%

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

      [Start] 23.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 23.9	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 53.2	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 33.0%

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

   4. Simplified 33.0%

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

      [Start] 33.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 33.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 33.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 31.3%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(\frac{-im \cdot im}{re}\right) \cdot 0.5}} \]

   6. Applied egg-rr 33.1%

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

   if -1.04999999999999997e45 < re < -6.3e16

   1. Initial program 22.6%

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

   2. Simplified 49.9%

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

      [Start] 22.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 22.6	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 49.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 36.9%

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

   4. Simplified 36.9%

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

      [Start] 36.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 36.9	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 36.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 34.8%

      \[\leadsto 0.5 \cdot \color{blue}{e^{\log \left(\frac{-im \cdot im}{re}\right) \cdot 0.5}} \]

   6. Taylor expanded in im around -inf 23.6%

      \[\leadsto 0.5 \cdot e^{\color{blue}{\left(-2 \cdot \log \left(\frac{-1}{im}\right) + \log \left(\frac{-1}{re}\right)\right)} \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq -1.4 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{elif}\;re \leq -1.05 \cdot 10^{+45}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq -6.3 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot e^{0.5 \cdot \left(-2 \cdot \log \left(\frac{-1}{im}\right) + \log \left(\frac{-1}{re}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.9%
Cost	27401

\[\begin{array}{l} t_0 := re + \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-287} \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \end{array} \]

Alternative 2
Accuracy	58.5%
Cost	8036

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ t_2 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -6.8 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.0052:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;im \leq -9 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-243}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 3.6 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Accuracy	58.0%
Cost	8036

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{if}\;im \leq -4.4 \cdot 10^{+46}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -0.005:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;im \leq -1.32 \cdot 10^{-110}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(\frac{re}{\frac{im}{re}} \cdot -0.5 - im\right)\right)}\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{-243}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 1.72 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3.2 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;im \leq 1.56 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4
Accuracy	59.3%
Cost	7772

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ t_1 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.3 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.4 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 3.1 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-108}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;im \leq 2 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5
Accuracy	59.8%
Cost	7772

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{if}\;im \leq -3 \cdot 10^{-114}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 9 \cdot 10^{-243}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.4 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-107}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6
Accuracy	59.1%
Cost	7644

\[\begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot 2}\\ t_1 := 0.5 \cdot \frac{im}{\sqrt{-re}}\\ t_2 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -7 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-241}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 1.85 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 7.5 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 2.4 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.8 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	58.8%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -1.95 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 8
Accuracy	41.6%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;im \leq 1.05 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 9
Accuracy	26.3%
Cost	6720

\[0.5 \cdot \sqrt{im \cdot 2} \]

Reproduce

herbie shell --seed 2023129 
(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)))))