math.sqrt on complex, real part

Percentage Accurate: 40.6% → 80.8%

Time: 9.0s

Precision: binary64

Cost: 13709

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+224}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq -4.4 \cdot 10^{+72} \lor \neg \left(re \leq -1.06 \cdot 10^{-10}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \end{array} \]

FPCore

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

↓

(FPCore (re im)
 :precision binary64
 (if (<= re -1.05e+224)
   (* 0.5 (sqrt (* (- im) (/ im re))))
   (if (or (<= re -4.4e+72) (not (<= re -1.06e-10)))
     (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))
     (* 0.5 (sqrt (/ (* im im) (- re)))))))

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 tmp;
	if (re <= -1.05e+224) {
		tmp = 0.5 * sqrt((-im * (im / re)));
	} else if ((re <= -4.4e+72) || !(re <= -1.06e-10)) {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	} else {
		tmp = 0.5 * sqrt(((im * im) / -re));
	}
	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 tmp;
	if (re <= -1.05e+224) {
		tmp = 0.5 * Math.sqrt((-im * (im / re)));
	} else if ((re <= -4.4e+72) || !(re <= -1.06e-10)) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	} else {
		tmp = 0.5 * Math.sqrt(((im * im) / -re));
	}
	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):
	tmp = 0
	if re <= -1.05e+224:
		tmp = 0.5 * math.sqrt((-im * (im / re)))
	elif (re <= -4.4e+72) or not (re <= -1.06e-10):
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	else:
		tmp = 0.5 * math.sqrt(((im * im) / -re))
	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)
	tmp = 0.0
	if (re <= -1.05e+224)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(im / re))));
	elseif ((re <= -4.4e+72) || !(re <= -1.06e-10))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * im) / Float64(-re))));
	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)
	tmp = 0.0;
	if (re <= -1.05e+224)
		tmp = 0.5 * sqrt((-im * (im / re)));
	elseif ((re <= -4.4e+72) || ~((re <= -1.06e-10)))
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	else
		tmp = 0.5 * sqrt(((im * im) / -re));
	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_] := If[LessEqual[re, -1.05e+224], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, -4.4e+72], N[Not[LessEqual[re, -1.06e-10]], $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[(N[(im * im), $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	40.6%
Target	48.0%
Herbie	80.8%

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

Derivation

1. Split input into 3 regimes

2. if re < -1.0500000000000001e224

   1. Initial program 2.4%

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

   2. Simplified 45.8%

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

      [Start] 2.4	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 2.4	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 45.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 64.0%

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

   4. Simplified 88.0%

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

      [Start] 64.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 64.0	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 64.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]
      associate-/l* [=>] 88.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{im}{\frac{re}{im}}} \cdot -0.5\right)} \]

   5. Applied egg-rr 88.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} \]
      Step-by-step derivation

      [Start] 88.0	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)} \]
      add-log-exp [=>] 46.1	\[ 0.5 \cdot \color{blue}{\log \left(e^{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right)} \]
      *-un-lft-identity [=>] 46.1	\[ 0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right)} \]
      log-prod [=>] 46.1	\[ 0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right)\right)} \]
      metadata-eval [=>] 46.1	\[ 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right)\right) \]
      add-log-exp [<=] 88.0	\[ 0.5 \cdot \left(0 + \color{blue}{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right) \]
      *-commutative [=>] 88.0	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right) \]
      associate-*l* [=>] 88.0	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right) \]
      associate-/r/ [=>] 88.0	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right) \]
      *-commutative [<=] 88.0	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right) \]
      metadata-eval [=>] 88.0	\[ 0.5 \cdot \left(0 + \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right) \]

   6. Simplified 88.0%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{im \cdot \frac{-im}{re}}} \]
      Step-by-step derivation

      [Start] 88.0	\[ 0.5 \cdot \left(0 + \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right) \]
      +-lft-identity [=>] 88.0	\[ 0.5 \cdot \color{blue}{\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}} \]
      associate-*l* [=>] 88.0	\[ 0.5 \cdot \sqrt{\color{blue}{im \cdot \left(\frac{im}{re} \cdot -1\right)}} \]
      associate-*l/ [=>] 88.0	\[ 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{im \cdot -1}{re}}} \]
      *-commutative [<=] 88.0	\[ 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-1 \cdot im}}{re}} \]
      neg-mul-1 [<=] 88.0	\[ 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-im}}{re}} \]

   if -1.0500000000000001e224 < re < -4.4e72 or -1.06e-10 < re

   1. Initial program 46.1%

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

   2. Simplified 90.3%

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

      [Start] 46.1	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 46.1	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 90.3	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   if -4.4e72 < re < -1.06e-10

   1. Initial program 14.8%

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

   2. Simplified 23.9%

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

      [Start] 14.8	\[ 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
      +-commutative [=>] 14.8	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + \sqrt{re \cdot re + im \cdot im}\right)}} \]
      hypot-def [=>] 23.9	\[ 0.5 \cdot \sqrt{2 \cdot \left(re + \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   3. Taylor expanded in re around -inf 59.7%

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

   4. Simplified 59.7%

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

      [Start] 59.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(-0.5 \cdot \frac{{im}^{2}}{re}\right)} \]
      *-commutative [=>] 59.7	\[ 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\frac{{im}^{2}}{re} \cdot -0.5\right)}} \]
      unpow2 [=>] 59.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{\color{blue}{im \cdot im}}{re} \cdot -0.5\right)} \]
      associate-/l* [=>] 59.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\frac{im}{\frac{re}{im}}} \cdot -0.5\right)} \]

   5. Applied egg-rr 59.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(0 + \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} \]
      Step-by-step derivation

      [Start] 59.7	\[ 0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)} \]
      add-log-exp [=>] 16.9	\[ 0.5 \cdot \color{blue}{\log \left(e^{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right)} \]
      *-un-lft-identity [=>] 16.9	\[ 0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right)} \]
      log-prod [=>] 16.9	\[ 0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right)\right)} \]
      metadata-eval [=>] 16.9	\[ 0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right)\right) \]
      add-log-exp [<=] 59.7	\[ 0.5 \cdot \left(0 + \color{blue}{\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}}\right) \]
      *-commutative [=>] 59.7	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right) \]
      associate-*l* [=>] 59.7	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right) \]
      associate-/r/ [=>] 59.7	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right) \]
      *-commutative [<=] 59.7	\[ 0.5 \cdot \left(0 + \sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right) \]
      metadata-eval [=>] 59.7	\[ 0.5 \cdot \left(0 + \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right) \]

   6. Simplified 59.7%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{im \cdot \frac{-im}{re}}} \]
      Step-by-step derivation

      [Start] 59.7	\[ 0.5 \cdot \left(0 + \sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right) \]
      +-lft-identity [=>] 59.7	\[ 0.5 \cdot \color{blue}{\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}} \]
      associate-*l* [=>] 59.7	\[ 0.5 \cdot \sqrt{\color{blue}{im \cdot \left(\frac{im}{re} \cdot -1\right)}} \]
      associate-*l/ [=>] 59.7	\[ 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{im \cdot -1}{re}}} \]
      *-commutative [<=] 59.7	\[ 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-1 \cdot im}}{re}} \]
      neg-mul-1 [<=] 59.7	\[ 0.5 \cdot \sqrt{im \cdot \frac{\color{blue}{-im}}{re}} \]

   7. Applied egg-rr 59.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im}{-re}}} \]
      Step-by-step derivation

      [Start] 59.7	\[ 0.5 \cdot \sqrt{im \cdot \frac{-im}{re}} \]
      associate-*r/ [=>] 59.7	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot \left(-im\right)}{re}}} \]
      frac-2neg [=>] 59.7	\[ 0.5 \cdot \sqrt{\color{blue}{\frac{-im \cdot \left(-im\right)}{-re}}} \]
      distribute-lft-neg-out [<=] 59.7	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{\left(-im\right) \cdot \left(-im\right)}}{-re}} \]
      sqr-neg [=>] 59.7	\[ 0.5 \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{-re}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.05 \cdot 10^{+224}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{elif}\;re \leq -4.4 \cdot 10^{+72} \lor \neg \left(re \leq -1.06 \cdot 10^{-10}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	59.2%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -9 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(-2\right)}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 2
Accuracy	59.9%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;im \leq -4.3 \cdot 10^{-169}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3
Accuracy	30.7%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -1.5 \cdot 10^{-182}:\\ \;\;\;\;0.018518518518518517\\ \mathbf{elif}\;im \leq 10^{-187}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 4
Accuracy	43.7%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5
Accuracy	58.9%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;im \leq -8.6 \cdot 10^{-171}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(-2\right)}\\ \mathbf{elif}\;im \leq 2.9 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 6
Accuracy	8.8%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+94}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;4.885243819643157 \cdot 10^{-22}\\ \end{array} \]

Alternative 7
Accuracy	8.8%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{+95}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;4.923200210024256 \cdot 10^{-15}\\ \end{array} \]

Alternative 8
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -6 \cdot 10^{+98}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;4.96145150637606 \cdot 10^{-8}\\ \end{array} \]

Alternative 9
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -7 \cdot 10^{+94}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1.3395919067215363 \cdot 10^{-6}\\ \end{array} \]

Alternative 10
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;3.616898148148148 \cdot 10^{-5}\\ \end{array} \]

Alternative 11
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{+98}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.0001220703125\\ \end{array} \]

Alternative 12
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+95}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.0009765625\\ \end{array} \]

Alternative 13
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{+95}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.0023148148148148147\\ \end{array} \]

Alternative 14
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+94}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.0078125\\ \end{array} \]

Alternative 15
Accuracy	8.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{+94}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.018518518518518517\\ \end{array} \]

Alternative 16
Accuracy	6.2%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023159 
(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)))))