math.sqrt on complex, real part

Average Accuracy: 39.0% → 79.4%

Time: 12.0s

Precision: binary64

Cost: 13973

Math

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

↓

\[\begin{array}{l} t_0 := \frac{im}{\sqrt{-re}}\\ \mathbf{if}\;re \leq -1.45 \cdot 10^{+243}:\\ \;\;\;\;t_0 \cdot 0.5\\ \mathbf{elif}\;re \leq -1.6 \cdot 10^{+188}:\\ \;\;\;\;t_0 \cdot -0.5\\ \mathbf{elif}\;re \leq -9.8 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -0.25}\\ \mathbf{elif}\;re \leq -7.8 \cdot 10^{+51} \lor \neg \left(re \leq -4.5 \cdot 10^{-60}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \end{array} \]

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 (/ im (sqrt (- re)))))
   (if (<= re -1.45e+243)
     (* t_0 0.5)
     (if (<= re -1.6e+188)
       (* t_0 -0.5)
       (if (<= re -9.8e+154)
         (sqrt (* (* im (/ im re)) -0.25))
         (if (or (<= re -7.8e+51) (not (<= re -4.5e-60)))
           (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))
           (* 0.5 (sqrt (/ (- im) (/ re im))))))))))

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 = im / sqrt(-re);
	double tmp;
	if (re <= -1.45e+243) {
		tmp = t_0 * 0.5;
	} else if (re <= -1.6e+188) {
		tmp = t_0 * -0.5;
	} else if (re <= -9.8e+154) {
		tmp = sqrt(((im * (im / re)) * -0.25));
	} else if ((re <= -7.8e+51) || !(re <= -4.5e-60)) {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	} else {
		tmp = 0.5 * sqrt((-im / (re / im)));
	}
	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 = im / Math.sqrt(-re);
	double tmp;
	if (re <= -1.45e+243) {
		tmp = t_0 * 0.5;
	} else if (re <= -1.6e+188) {
		tmp = t_0 * -0.5;
	} else if (re <= -9.8e+154) {
		tmp = Math.sqrt(((im * (im / re)) * -0.25));
	} else if ((re <= -7.8e+51) || !(re <= -4.5e-60)) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	} else {
		tmp = 0.5 * Math.sqrt((-im / (re / im)));
	}
	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 = im / math.sqrt(-re)
	tmp = 0
	if re <= -1.45e+243:
		tmp = t_0 * 0.5
	elif re <= -1.6e+188:
		tmp = t_0 * -0.5
	elif re <= -9.8e+154:
		tmp = math.sqrt(((im * (im / re)) * -0.25))
	elif (re <= -7.8e+51) or not (re <= -4.5e-60):
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	else:
		tmp = 0.5 * math.sqrt((-im / (re / im)))
	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(im / sqrt(Float64(-re)))
	tmp = 0.0
	if (re <= -1.45e+243)
		tmp = Float64(t_0 * 0.5);
	elseif (re <= -1.6e+188)
		tmp = Float64(t_0 * -0.5);
	elseif (re <= -9.8e+154)
		tmp = sqrt(Float64(Float64(im * Float64(im / re)) * -0.25));
	elseif ((re <= -7.8e+51) || !(re <= -4.5e-60))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) / Float64(re / im))));
	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 = im / sqrt(-re);
	tmp = 0.0;
	if (re <= -1.45e+243)
		tmp = t_0 * 0.5;
	elseif (re <= -1.6e+188)
		tmp = t_0 * -0.5;
	elseif (re <= -9.8e+154)
		tmp = sqrt(((im * (im / re)) * -0.25));
	elseif ((re <= -7.8e+51) || ~((re <= -4.5e-60)))
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	else
		tmp = 0.5 * sqrt((-im / (re / im)));
	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[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[re, -1.45e+243], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[re, -1.6e+188], N[(t$95$0 * -0.5), $MachinePrecision], If[LessEqual[re, -9.8e+154], N[Sqrt[N[(N[(im * N[(im / re), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[re, -7.8e+51], N[Not[LessEqual[re, -4.5e-60]], $MachinePrecision]], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[((-im) / N[(re / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	39.0%
Target	47.0%
Herbie	79.4%

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

Derivation

1. Split input into 5 regimes

2. if re < -1.45000000000000003e243

   1. Initial program 0.0%

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

   2. Simplified 28.4%

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

      [Start] 0.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 0.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 28.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 48.0%

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

   4. Simplified 48.0%

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

      [Start] 48.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 48.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 48.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 47.8%

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

      [Start] 48.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \]
      add-sqr-sqrt [=>] 47.8	\[ \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}} \]
      pow2 [=>] 47.8	\[ \color{blue}{{\left(\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}\right)}^{2}} \]
      *-commutative [=>] 47.8	\[ {\left(\sqrt{\color{blue}{\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \cdot 0.5}}\right)}^{2} \]
      *-commutative [=>] 47.8	\[ {\left(\sqrt{\sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{im \cdot im}{re}\right)}} \cdot 0.5}\right)}^{2} \]
      associate-*r* [=>] 47.8	\[ {\left(\sqrt{\sqrt{\color{blue}{\left(2 \cdot -0.5\right) \cdot \frac{im \cdot im}{re}}} \cdot 0.5}\right)}^{2} \]
      associate-*r/ [=>] 47.8	\[ {\left(\sqrt{\sqrt{\color{blue}{\frac{\left(2 \cdot -0.5\right) \cdot \left(im \cdot im\right)}{re}}} \cdot 0.5}\right)}^{2} \]
      metadata-eval [=>] 47.8	\[ {\left(\sqrt{\sqrt{\frac{\color{blue}{-1} \cdot \left(im \cdot im\right)}{re}} \cdot 0.5}\right)}^{2} \]
      neg-mul-1 [<=] 47.8	\[ {\left(\sqrt{\sqrt{\frac{\color{blue}{-im \cdot im}}{re}} \cdot 0.5}\right)}^{2} \]
      distribute-rgt-neg-in [=>] 47.8	\[ {\left(\sqrt{\sqrt{\frac{\color{blue}{im \cdot \left(-im\right)}}{re}} \cdot 0.5}\right)}^{2} \]

   6. Applied egg-rr 63.2%

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

      [Start] 47.8	\[ {\left(\sqrt{\sqrt{\frac{im \cdot \left(-im\right)}{re}} \cdot 0.5}\right)}^{2} \]
      unpow2 [=>] 47.8	\[ \color{blue}{\sqrt{\sqrt{\frac{im \cdot \left(-im\right)}{re}} \cdot 0.5} \cdot \sqrt{\sqrt{\frac{im \cdot \left(-im\right)}{re}} \cdot 0.5}} \]
      add-sqr-sqrt [<=] 48.0	\[ \color{blue}{\sqrt{\frac{im \cdot \left(-im\right)}{re}} \cdot 0.5} \]
      frac-2neg [=>] 48.0	\[ \sqrt{\color{blue}{\frac{-im \cdot \left(-im\right)}{-re}}} \cdot 0.5 \]
      distribute-lft-neg-in [=>] 48.0	\[ \sqrt{\frac{\color{blue}{\left(-im\right) \cdot \left(-im\right)}}{-re}} \cdot 0.5 \]
      sqrt-div [=>] 71.7	\[ \color{blue}{\frac{\sqrt{\left(-im\right) \cdot \left(-im\right)}}{\sqrt{-re}}} \cdot 0.5 \]
      sqr-neg [=>] 71.7	\[ \frac{\sqrt{\color{blue}{im \cdot im}}}{\sqrt{-re}} \cdot 0.5 \]
      sqrt-unprod [<=] 52.2	\[ \frac{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}{\sqrt{-re}} \cdot 0.5 \]
      add-sqr-sqrt [<=] 63.2	\[ \frac{\color{blue}{im}}{\sqrt{-re}} \cdot 0.5 \]
      associate-*l/ [=>] 63.2	\[ \color{blue}{\frac{im \cdot 0.5}{\sqrt{-re}}} \]

   7. Simplified 63.2%

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

      [Start] 63.2	\[ \frac{im \cdot 0.5}{\sqrt{-re}} \]
      associate-/l* [=>] 63.2	\[ \color{blue}{\frac{im}{\frac{\sqrt{-re}}{0.5}}} \]
      associate-/r/ [=>] 63.2	\[ \color{blue}{\frac{im}{\sqrt{-re}} \cdot 0.5} \]

   if -1.45000000000000003e243 < re < -1.59999999999999985e188

   1. Initial program 0.0%

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

   2. Simplified 31.6%

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

      [Start] 0.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 0.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 31.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 49.6%

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

   4. Simplified 49.6%

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

      [Start] 49.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 49.6	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 49.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 47.5%

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

      [Start] 49.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \]
      pow1/2 [=>] 49.6	\[ 0.5 \cdot \color{blue}{{\left(2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)\right)}^{0.5}} \]
      pow-to-exp [=>] 47.5	\[ 0.5 \cdot \color{blue}{e^{\log \left(2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)\right) \cdot 0.5}} \]
      *-commutative [=>] 47.5	\[ 0.5 \cdot e^{\log \left(2 \cdot \color{blue}{\left(-0.5 \cdot \frac{im \cdot im}{re}\right)}\right) \cdot 0.5} \]
      associate-*r* [=>] 47.5	\[ 0.5 \cdot e^{\log \color{blue}{\left(\left(2 \cdot -0.5\right) \cdot \frac{im \cdot im}{re}\right)} \cdot 0.5} \]
      associate-*r/ [=>] 47.5	\[ 0.5 \cdot e^{\log \color{blue}{\left(\frac{\left(2 \cdot -0.5\right) \cdot \left(im \cdot im\right)}{re}\right)} \cdot 0.5} \]
      metadata-eval [=>] 47.5	\[ 0.5 \cdot e^{\log \left(\frac{\color{blue}{-1} \cdot \left(im \cdot im\right)}{re}\right) \cdot 0.5} \]
      neg-mul-1 [<=] 47.5	\[ 0.5 \cdot e^{\log \left(\frac{\color{blue}{-im \cdot im}}{re}\right) \cdot 0.5} \]
      distribute-rgt-neg-in [=>] 47.5	\[ 0.5 \cdot e^{\log \left(\frac{\color{blue}{im \cdot \left(-im\right)}}{re}\right) \cdot 0.5} \]

   6. Applied egg-rr 53.7%

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

      [Start] 47.5	\[ 0.5 \cdot e^{\log \left(\frac{im \cdot \left(-im\right)}{re}\right) \cdot 0.5} \]
      exp-to-pow [=>] 49.6	\[ 0.5 \cdot \color{blue}{{\left(\frac{im \cdot \left(-im\right)}{re}\right)}^{0.5}} \]
      unpow1/2 [=>] 49.6	\[ 0.5 \cdot \color{blue}{\sqrt{\frac{im \cdot \left(-im\right)}{re}}} \]
      frac-2neg [=>] 49.6	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-im \cdot \left(-im\right)}{-re}}} \]
      distribute-lft-neg-in [=>] 49.6	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{\left(-im\right) \cdot \left(-im\right)}}{-re}} \]
      sqrt-div [=>] 68.0	\[ 0.5 \cdot \color{blue}{\frac{\sqrt{\left(-im\right) \cdot \left(-im\right)}}{\sqrt{-re}}} \]
      sqrt-unprod [<=] 44.3	\[ 0.5 \cdot \frac{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}}{\sqrt{-re}} \]
      add-sqr-sqrt [<=] 53.7	\[ 0.5 \cdot \frac{\color{blue}{-im}}{\sqrt{-re}} \]
      neg-mul-1 [=>] 53.7	\[ 0.5 \cdot \frac{\color{blue}{-1 \cdot im}}{\sqrt{-re}} \]
      *-commutative [=>] 53.7	\[ 0.5 \cdot \frac{\color{blue}{im \cdot -1}}{\sqrt{-re}} \]
      associate-/l* [=>] 53.7	\[ 0.5 \cdot \color{blue}{\frac{im}{\frac{\sqrt{-re}}{-1}}} \]

   7. Simplified 53.7%

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

      [Start] 53.7	\[ 0.5 \cdot \frac{im}{\frac{\sqrt{-re}}{-1}} \]
      associate-/r/ [=>] 53.7	\[ 0.5 \cdot \color{blue}{\left(\frac{im}{\sqrt{-re}} \cdot -1\right)} \]
      rem-3cbrt-lft [<=] 53.0	\[ 0.5 \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right) \cdot \sqrt[3]{im}}}{\sqrt{-re}} \cdot -1\right) \]
      unpow2 [<=] 53.0	\[ 0.5 \cdot \left(\frac{\color{blue}{{\left(\sqrt[3]{im}\right)}^{2}} \cdot \sqrt[3]{im}}{\sqrt{-re}} \cdot -1\right) \]
      associate-*l/ [<=] 53.0	\[ 0.5 \cdot \left(\color{blue}{\left(\frac{{\left(\sqrt[3]{im}\right)}^{2}}{\sqrt{-re}} \cdot \sqrt[3]{im}\right)} \cdot -1\right) \]
      associate-/r/ [<=] 53.0	\[ 0.5 \cdot \left(\color{blue}{\frac{{\left(\sqrt[3]{im}\right)}^{2}}{\frac{\sqrt{-re}}{\sqrt[3]{im}}}} \cdot -1\right) \]
      associate-/r/ [<=] 53.0	\[ 0.5 \cdot \color{blue}{\frac{{\left(\sqrt[3]{im}\right)}^{2}}{\frac{\frac{\sqrt{-re}}{\sqrt[3]{im}}}{-1}}} \]
      associate-/l* [<=] 53.0	\[ 0.5 \cdot \color{blue}{\frac{{\left(\sqrt[3]{im}\right)}^{2} \cdot -1}{\frac{\sqrt{-re}}{\sqrt[3]{im}}}} \]
      *-commutative [=>] 53.0	\[ 0.5 \cdot \frac{\color{blue}{-1 \cdot {\left(\sqrt[3]{im}\right)}^{2}}}{\frac{\sqrt{-re}}{\sqrt[3]{im}}} \]
      associate-/l* [=>] 52.7	\[ 0.5 \cdot \color{blue}{\frac{-1}{\frac{\frac{\sqrt{-re}}{\sqrt[3]{im}}}{{\left(\sqrt[3]{im}\right)}^{2}}}} \]
      associate-/l/ [=>] 52.7	\[ 0.5 \cdot \frac{-1}{\color{blue}{\frac{\sqrt{-re}}{{\left(\sqrt[3]{im}\right)}^{2} \cdot \sqrt[3]{im}}}} \]
      unpow2 [=>] 52.7	\[ 0.5 \cdot \frac{-1}{\frac{\sqrt{-re}}{\color{blue}{\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right)} \cdot \sqrt[3]{im}}} \]
      rem-3cbrt-lft [=>] 53.3	\[ 0.5 \cdot \frac{-1}{\frac{\sqrt{-re}}{\color{blue}{im}}} \]
      associate-/r/ [=>] 53.6	\[ 0.5 \cdot \color{blue}{\left(\frac{-1}{\sqrt{-re}} \cdot im\right)} \]
      associate-*l/ [=>] 53.7	\[ 0.5 \cdot \color{blue}{\frac{-1 \cdot im}{\sqrt{-re}}} \]
      neg-mul-1 [<=] 53.7	\[ 0.5 \cdot \frac{\color{blue}{-im}}{\sqrt{-re}} \]

   if -1.59999999999999985e188 < re < -9.8000000000000003e154

   1. Initial program 0.0%

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

   2. Simplified 43.6%

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

      [Start] 0.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 0.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 43.6	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 48.1%

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

   4. Simplified 48.1%

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

      [Start] 48.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 48.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 48.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Applied egg-rr 48.0%

      \[\leadsto \color{blue}{\sqrt{\frac{im \cdot \left(-im\right)}{re} \cdot 0.25}} \]
      Proof

      [Start] 48.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \]
      add-sqr-sqrt [=>] 48.0	\[ \color{blue}{\sqrt{0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}} \cdot \sqrt{0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}}} \]
      sqrt-unprod [=>] 48.1	\[ \color{blue}{\sqrt{\left(0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right) \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)}} \]
      *-commutative [=>] 48.1	\[ \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right)} \]
      *-commutative [=>] 48.1	\[ \sqrt{\left(\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \cdot 0.5\right)}} \]
      swap-sqr [=>] 48.0	\[ \sqrt{\color{blue}{\left(\sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)} \cdot \sqrt{2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)}\right) \cdot \left(0.5 \cdot 0.5\right)}} \]
      add-sqr-sqrt [<=] 48.0	\[ \sqrt{\color{blue}{\left(2 \cdot \left(\frac{im \cdot im}{re} \cdot -0.5\right)\right)} \cdot \left(0.5 \cdot 0.5\right)} \]
      *-commutative [=>] 48.0	\[ \sqrt{\left(2 \cdot \color{blue}{\left(-0.5 \cdot \frac{im \cdot im}{re}\right)}\right) \cdot \left(0.5 \cdot 0.5\right)} \]
      associate-*r* [=>] 48.0	\[ \sqrt{\color{blue}{\left(\left(2 \cdot -0.5\right) \cdot \frac{im \cdot im}{re}\right)} \cdot \left(0.5 \cdot 0.5\right)} \]
      associate-*r/ [=>] 48.0	\[ \sqrt{\color{blue}{\frac{\left(2 \cdot -0.5\right) \cdot \left(im \cdot im\right)}{re}} \cdot \left(0.5 \cdot 0.5\right)} \]
      metadata-eval [=>] 48.0	\[ \sqrt{\frac{\color{blue}{-1} \cdot \left(im \cdot im\right)}{re} \cdot \left(0.5 \cdot 0.5\right)} \]
      neg-mul-1 [<=] 48.0	\[ \sqrt{\frac{\color{blue}{-im \cdot im}}{re} \cdot \left(0.5 \cdot 0.5\right)} \]
      distribute-rgt-neg-in [=>] 48.0	\[ \sqrt{\frac{\color{blue}{im \cdot \left(-im\right)}}{re} \cdot \left(0.5 \cdot 0.5\right)} \]
      metadata-eval [=>] 48.0	\[ \sqrt{\frac{im \cdot \left(-im\right)}{re} \cdot \color{blue}{0.25}} \]

   6. Simplified 49.7%

      \[\leadsto \color{blue}{\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -0.25}} \]
      Proof

      [Start] 48.0	\[ \sqrt{\frac{im \cdot \left(-im\right)}{re} \cdot 0.25} \]
      distribute-rgt-neg-out [=>] 48.0	\[ \sqrt{\frac{\color{blue}{-im \cdot im}}{re} \cdot 0.25} \]
      distribute-frac-neg [=>] 48.0	\[ \sqrt{\color{blue}{\left(-\frac{im \cdot im}{re}\right)} \cdot 0.25} \]
      associate-*r/ [<=] 49.7	\[ \sqrt{\left(-\color{blue}{im \cdot \frac{im}{re}}\right) \cdot 0.25} \]
      mul-1-neg [<=] 49.7	\[ \sqrt{\color{blue}{\left(-1 \cdot \left(im \cdot \frac{im}{re}\right)\right)} \cdot 0.25} \]
      *-commutative [=>] 49.7	\[ \sqrt{\color{blue}{\left(\left(im \cdot \frac{im}{re}\right) \cdot -1\right)} \cdot 0.25} \]
      associate-*l* [=>] 49.7	\[ \sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right) \cdot \left(-1 \cdot 0.25\right)}} \]
      metadata-eval [=>] 49.7	\[ \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-0.25}} \]

   if -9.8000000000000003e154 < re < -7.79999999999999968e51 or -4.50000000000000001e-60 < re

   1. Initial program 46.5%

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

   2. Simplified 88.7%

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

      [Start] 46.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 46.5	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 88.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   if -7.79999999999999968e51 < re < -4.50000000000000001e-60

   1. Initial program 26.1%

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

   2. Simplified 53.1%

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

      [Start] 26.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 26.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 53.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 27.5%

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

   4. Simplified 27.5%

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

      [Start] 27.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 27.5	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 27.5	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]

   5. Taylor expanded in im around 0 27.6%

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

   6. Simplified 28.8%

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

      [Start] 27.6	\[ 0.5 \cdot \sqrt{-1 \cdot \frac{{im}^{2}}{re}} \]
      mul-1-neg [=>] 27.6	\[ 0.5 \cdot \sqrt{\color{blue}{-\frac{{im}^{2}}{re}}} \]
      unpow2 [=>] 27.6	\[ 0.5 \cdot \sqrt{-\frac{\color{blue}{im \cdot im}}{re}} \]
      associate-/l* [=>] 28.8	\[ 0.5 \cdot \sqrt{-\color{blue}{\frac{im}{\frac{re}{im}}}} \]
      distribute-frac-neg [<=] 28.8	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-im}{\frac{re}{im}}}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{+243}:\\ \;\;\;\;\frac{im}{\sqrt{-re}} \cdot 0.5\\ \mathbf{elif}\;re \leq -1.6 \cdot 10^{+188}:\\ \;\;\;\;\frac{im}{\sqrt{-re}} \cdot -0.5\\ \mathbf{elif}\;re \leq -9.8 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -0.25}\\ \mathbf{elif}\;re \leq -7.8 \cdot 10^{+51} \lor \neg \left(re \leq -4.5 \cdot 10^{-60}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{\frac{re}{im}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	59.1%
Cost	8036

\[\begin{array}{l} t_0 := \frac{im}{\sqrt{-re}}\\ t_1 := t_0 \cdot -0.5\\ t_2 := t_0 \cdot 0.5\\ t_3 := 0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ t_4 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.08 \cdot 10^{-79}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq -2.5 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -3.6 \cdot 10^{-247}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;im \leq -3.4 \cdot 10^{-261}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-287}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;im \leq 5.2 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 3.7 \cdot 10^{-261}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;im \leq 6.5 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq 3.8 \cdot 10^{-168}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Accuracy	58.4%
Cost	7376

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -1.65 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 1.9 \cdot 10^{-225}:\\ \;\;\;\;\frac{im}{\sqrt{-re}} \cdot 0.5\\ \mathbf{elif}\;im \leq 1.35 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Accuracy	57.9%
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -2.8 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.7 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 3 \cdot 10^{-225}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 1.25 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 4
Accuracy	58.0%
Cost	7248

\[\begin{array}{l} t_0 := 0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{if}\;im \leq -3 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.8 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 4.2 \cdot 10^{-227}:\\ \;\;\;\;\frac{im}{\sqrt{-re}} \cdot 0.5\\ \mathbf{elif}\;im \leq 1.1 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5
Accuracy	58.5%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -3.6 \cdot 10^{-247}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 3.5 \cdot 10^{-163}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6
Accuracy	58.6%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -6.2 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 1.02 \cdot 10^{-162}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7
Accuracy	42.3%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;re \leq 3.15 \cdot 10^{-193}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 8
Accuracy	25.8%
Cost	6720

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

Reproduce

herbie shell --seed 2023133 
(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)))))