math.sqrt on complex, real part

Percentage Accurate: 40.8% → 84.9%

Time: 9.8s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Initial Program: 40.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) + re)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Alternative 1: 84.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (* 0.5 (sqrt (/ (- (pow im 2.0)) re)))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * sqrt((-pow(im, 2.0) / re));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.sqrt((2.0 * (re + Math.sqrt(((re * re) + (im * im)))))) <= 0.0) {
		tmp = 0.5 * Math.sqrt((-Math.pow(im, 2.0) / re));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (re + math.sqrt(((re * re) + (im * im)))))) <= 0.0:
		tmp = 0.5 * math.sqrt((-math.pow(im, 2.0) / re))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (sqrt(Float64(2.0 * Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))))) <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-(im ^ 2.0)) / re)));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sqrt((2.0 * (re + sqrt(((re * re) + (im * im)))))) <= 0.0)
		tmp = 0.5 * sqrt((-(im ^ 2.0) / re));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[((-N[Power[im, 2.0], $MachinePrecision]) / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 10.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 10.7%

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

      2. hypot-def 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in re around -inf 61.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. mul-1-neg 61.6%

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

      2. distribute-neg-frac 61.6%

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

   6. Simplified 61.6%

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

   if 0.0 < (sqrt.f64 (*.f64 2 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 46.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 46.5%

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

      2. hypot-def 89.4%

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

   3. Simplified 89.4%

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

      1. sqrt-prod 89.8%

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

      2. +-commutative 89.8%

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

      3. hypot-def 46.3%

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

      4. sqrt-prod 46.5%

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

      5. add-sqr-sqrt 46.2%

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

      6. sqrt-unprod 46.5%

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

      7. *-commutative 46.5%

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

      8. *-commutative 46.5%

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

      9. swap-sqr 46.5%

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

   5. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot 0.25}} \]
   6. Step-by-step derivation

      1. *-commutative 89.4%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}} \]

      2. associate-*r* 90.3%

         \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

      3. metadata-eval 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 86.0%

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

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return sqrt((0.5 * (re + hypot(re, im))));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.2%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Step-by-step derivation

   1. +-commutative 41.2%

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

   2. hypot-def 77.7%

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

3. Simplified 77.7%

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

   1. sqrt-prod 78.1%

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

   2. +-commutative 78.1%

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

   3. hypot-def 41.0%

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

   4. sqrt-prod 41.2%

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

   5. add-sqr-sqrt 40.9%

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

   6. sqrt-unprod 41.2%

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

   7. *-commutative 41.2%

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

   8. *-commutative 41.2%

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

   9. swap-sqr 41.2%

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

5. Applied egg-rr 77.7%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right) \cdot 0.25}} \]
6. Step-by-step derivation

   1. *-commutative 77.7%

      \[\leadsto \sqrt{\color{blue}{0.25 \cdot \left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}} \]

   2. associate-*r* 78.5%

      \[\leadsto \sqrt{\color{blue}{\left(0.25 \cdot 2\right) \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]

   3. metadata-eval 78.5%

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

7. Simplified 78.5%

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

8. Final simplification 78.5%

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

Alternative 3: 41.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.36e+218)
   (* 0.5 (sqrt (* 2.0 (- re re))))
   (if (<= re 3.5e+39) (sqrt (* im 0.5)) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.36e+218) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else if (re <= 3.5e+39) {
		tmp = sqrt((im * 0.5));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.36d+218)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else if (re <= 3.5d+39) then
        tmp = sqrt((im * 0.5d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.36e+218) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else if (re <= 3.5e+39) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.36e+218:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	elif re <= 3.5e+39:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.36e+218)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	elseif (re <= 3.5e+39)
		tmp = sqrt(Float64(im * 0.5));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.36e+218)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	elseif (re <= 3.5e+39)
		tmp = sqrt((im * 0.5));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.36e+218], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.5e+39], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.36e218

   1. Initial program 2.4%

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

   2. Taylor expanded in re around -inf 37.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} + re\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 37.8%

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

   4. Simplified 37.8%

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

   if -1.36e218 < re < 3.5000000000000002e39

   1. Initial program 48.4%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 48.4%

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

      2. hypot-def 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in re around 0 28.3%

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

      1. associate-*r* 28.3%

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

   6. Simplified 28.3%

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

      1. add-sqr-sqrt 28.2%

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

      2. sqrt-unprod 28.3%

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

      3. associate-*l* 28.3%

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

      4. associate-*l* 28.3%

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

      5. swap-sqr 28.3%

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

      6. metadata-eval 28.3%

         \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot \left(\sqrt{im} \cdot \sqrt{2}\right)\right)} \]

      7. swap-sqr 28.2%

         \[\leadsto \sqrt{0.25 \cdot \color{blue}{\left(\left(\sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)}} \]

      8. add-sqr-sqrt 28.1%

         \[\leadsto \sqrt{0.25 \cdot \left(\color{blue}{im} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)} \]

      9. rem-square-sqrt 28.4%

         \[\leadsto \sqrt{0.25 \cdot \left(im \cdot \color{blue}{2}\right)} \]

   8. Applied egg-rr 28.4%

      \[\leadsto \color{blue}{\sqrt{0.25 \cdot \left(im \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 28.4%

         \[\leadsto \sqrt{\color{blue}{\left(im \cdot 2\right) \cdot 0.25}} \]

      2. associate-*l* 28.4%

         \[\leadsto \sqrt{\color{blue}{im \cdot \left(2 \cdot 0.25\right)}} \]

      3. metadata-eval 28.4%

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

   10. Simplified 28.4%

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

   if 3.5000000000000002e39 < re

   1. Initial program 34.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 34.5%

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

      2. hypot-def 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in im around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. unpow2 75.2%

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

      3. rem-square-sqrt 76.6%

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

      4. associate-*r* 76.6%

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

      5. metadata-eval 76.6%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

   6. Simplified 76.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{+218}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 4: 40.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.4e+41) (sqrt (* im 0.5)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4e+41) {
		tmp = sqrt((im * 0.5));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.4d+41) then
        tmp = sqrt((im * 0.5d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.4e+41) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.4e+41:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.4e+41)
		tmp = sqrt(Float64(im * 0.5));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.4e+41)
		tmp = sqrt((im * 0.5));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.4e+41], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.4e41

   1. Initial program 43.1%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 43.1%

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

      2. hypot-def 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in re around 0 25.3%

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

      1. associate-*r* 25.3%

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

   6. Simplified 25.3%

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

      1. add-sqr-sqrt 25.3%

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

      2. sqrt-unprod 25.3%

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

      3. associate-*l* 25.3%

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

      4. associate-*l* 25.3%

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

      5. swap-sqr 25.3%

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

      6. metadata-eval 25.3%

         \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot \left(\sqrt{im} \cdot \sqrt{2}\right)\right)} \]

      7. swap-sqr 25.3%

         \[\leadsto \sqrt{0.25 \cdot \color{blue}{\left(\left(\sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)}} \]

      8. add-sqr-sqrt 25.2%

         \[\leadsto \sqrt{0.25 \cdot \left(\color{blue}{im} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)} \]

      9. rem-square-sqrt 25.5%

         \[\leadsto \sqrt{0.25 \cdot \left(im \cdot \color{blue}{2}\right)} \]

   8. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{\sqrt{0.25 \cdot \left(im \cdot 2\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 25.5%

         \[\leadsto \sqrt{\color{blue}{\left(im \cdot 2\right) \cdot 0.25}} \]

      2. associate-*l* 25.5%

         \[\leadsto \sqrt{\color{blue}{im \cdot \left(2 \cdot 0.25\right)}} \]

      3. metadata-eval 25.5%

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

   10. Simplified 25.5%

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

   if 1.4e41 < re

   1. Initial program 34.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   2. Step-by-step derivation

      1. +-commutative 34.5%

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

      2. hypot-def 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in im around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. unpow2 75.2%

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

      3. rem-square-sqrt 76.6%

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

      4. associate-*r* 76.6%

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

      5. metadata-eval 76.6%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

   6. Simplified 76.6%

      \[\leadsto \color{blue}{1 \cdot \sqrt{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 5: 25.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{im \cdot 0.5} \end{array} \]

FPCore

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

C

double code(double re, double im) {
	return sqrt((im * 0.5));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt((im * 0.5d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
2. Step-by-step derivation

   1. +-commutative 41.2%

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

   2. hypot-def 77.7%

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

3. Simplified 77.7%

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

4. Taylor expanded in re around 0 22.1%

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

   1. associate-*r* 22.1%

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

6. Simplified 22.1%

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

   1. add-sqr-sqrt 22.1%

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

   2. sqrt-unprod 22.1%

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

   3. associate-*l* 22.1%

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

   4. associate-*l* 22.1%

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

   5. swap-sqr 22.1%

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

   6. metadata-eval 22.1%

      \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(\left(\sqrt{im} \cdot \sqrt{2}\right) \cdot \left(\sqrt{im} \cdot \sqrt{2}\right)\right)} \]

   7. swap-sqr 22.1%

      \[\leadsto \sqrt{0.25 \cdot \color{blue}{\left(\left(\sqrt{im} \cdot \sqrt{im}\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)}} \]

   8. add-sqr-sqrt 22.0%

      \[\leadsto \sqrt{0.25 \cdot \left(\color{blue}{im} \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)\right)} \]

   9. rem-square-sqrt 22.2%

      \[\leadsto \sqrt{0.25 \cdot \left(im \cdot \color{blue}{2}\right)} \]

8. Applied egg-rr 22.2%

   \[\leadsto \color{blue}{\sqrt{0.25 \cdot \left(im \cdot 2\right)}} \]
9. Step-by-step derivation

   1. *-commutative 22.2%

      \[\leadsto \sqrt{\color{blue}{\left(im \cdot 2\right) \cdot 0.25}} \]

   2. associate-*l* 22.2%

      \[\leadsto \sqrt{\color{blue}{im \cdot \left(2 \cdot 0.25\right)}} \]

   3. metadata-eval 22.2%

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

10. Simplified 22.2%

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

11. Final simplification 22.2%

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

Developer target: 47.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023305 
(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)))))