math.sqrt on complex, real part

Percentage Accurate: 42.0% → 83.2%

Time: 5.6s

Alternatives: 6

Speedup: 1.8×

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: 42.0% 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: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{+64}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.2e+64)
   (* 0.5 (sqrt (* im (/ (- im) re))))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.2e+64) {
		tmp = 0.5 * sqrt((im * (-im / re)));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.2e+64) {
		tmp = 0.5 * Math.sqrt((im * (-im / re)));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.2e+64:
		tmp = 0.5 * math.sqrt((im * (-im / re)))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.2e+64)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(-im) / 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 (re <= -8.2e+64)
		tmp = 0.5 * sqrt((im * (-im / re)));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.2e+64], N[(0.5 * N[Sqrt[N[(im * N[((-im) / re), $MachinePrecision]), $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 re < -8.19999999999999956e64

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-def 26.6%

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

   3. Simplified 26.6%

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

   4. Taylor expanded in re around -inf 54.4%

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

      1. *-commutative 54.4%

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

      2. unpow2 54.4%

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

      3. associate-/l* 59.3%

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

   6. Simplified 59.3%

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

      1. expm1-log1p-u 58.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 30.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 30.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 58.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 59.5%

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

      3. *-commutative 59.5%

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

      4. neg-mul-1 59.5%

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

      5. distribute-rgt-neg-in 59.5%

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

      6. distribute-frac-neg 59.5%

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

   10. Simplified 59.5%

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

   if -8.19999999999999956e64 < re

   1. Initial program 48.1%

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

      1. +-commutative 48.1%

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

      2. hypot-def 91.7%

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

   3. Simplified 91.7%

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

      1. add-sqr-sqrt 90.9%

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

      2. sqrt-unprod 91.7%

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

      3. *-commutative 91.7%

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

      4. *-commutative 91.7%

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

      5. swap-sqr 91.7%

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

      6. add-sqr-sqrt 91.7%

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

      7. *-commutative 91.7%

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

      8. metadata-eval 91.7%

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

   5. Applied egg-rr 91.7%

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

      1. associate-*l* 91.7%

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

      2. metadata-eval 91.7%

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

   7. Simplified 91.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{+64}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 41.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{+205}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.8e+205)
   (* 0.5 (sqrt (/ im (/ re im))))
   (if (<= re 3.9e-101)
     (* 0.5 (sqrt (* im 2.0)))
     (if (<= re 3e-76)
       (sqrt re)
       (if (<= re 2.05e+101) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.8e+205) {
		tmp = 0.5 * sqrt((im / (re / im)));
	} else if (re <= 3.9e-101) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else if (re <= 3e-76) {
		tmp = sqrt(re);
	} else if (re <= 2.05e+101) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} 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 <= (-7.8d+205)) then
        tmp = 0.5d0 * sqrt((im / (re / im)))
    else if (re <= 3.9d-101) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else if (re <= 3d-76) then
        tmp = sqrt(re)
    else if (re <= 2.05d+101) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.8e+205) {
		tmp = 0.5 * Math.sqrt((im / (re / im)));
	} else if (re <= 3.9e-101) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else if (re <= 3e-76) {
		tmp = Math.sqrt(re);
	} else if (re <= 2.05e+101) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.8e+205:
		tmp = 0.5 * math.sqrt((im / (re / im)))
	elif re <= 3.9e-101:
		tmp = 0.5 * math.sqrt((im * 2.0))
	elif re <= 3e-76:
		tmp = math.sqrt(re)
	elif re <= 2.05e+101:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.8e+205)
		tmp = Float64(0.5 * sqrt(Float64(im / Float64(re / im))));
	elseif (re <= 3.9e-101)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	elseif (re <= 3e-76)
		tmp = sqrt(re);
	elseif (re <= 2.05e+101)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.8e+205)
		tmp = 0.5 * sqrt((im / (re / im)));
	elseif (re <= 3.9e-101)
		tmp = 0.5 * sqrt((im * 2.0));
	elseif (re <= 3e-76)
		tmp = sqrt(re);
	elseif (re <= 2.05e+101)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.8e+205], N[(0.5 * N[Sqrt[N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.9e-101], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e-76], N[Sqrt[re], $MachinePrecision], If[LessEqual[re, 2.05e+101], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -7.7999999999999997e205

   1. Initial program 2.3%

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

      1. +-commutative 2.3%

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

      2. hypot-def 23.5%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in re around -inf 61.9%

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

      1. *-commutative 61.9%

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

      2. unpow2 61.9%

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

      3. associate-/l* 73.7%

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

   6. Simplified 73.7%

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

      1. expm1-log1p-u 72.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 38.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 38.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 38.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 38.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 38.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 38.0%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 38.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 72.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 73.5%

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

      3. rem-square-sqrt 73.5%

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

      4. fabs-sqr 73.5%

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

      5. rem-square-sqrt 73.5%

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

      6. fabs-mul 73.5%

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

      7. rem-square-sqrt 21.7%

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

      8. fabs-sqr 21.7%

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

      9. metadata-eval 21.7%

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

      10. rem-square-sqrt 21.7%

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

      11. *-rgt-identity 21.7%

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

      12. *-commutative 21.7%

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

      13. associate-/r/ 21.7%

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

   10. Simplified 21.7%

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

   if -7.7999999999999997e205 < re < 3.90000000000000015e-101

   1. Initial program 41.1%

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

      1. +-commutative 41.1%

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

      2. hypot-def 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in re around 0 33.5%

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

      1. *-commutative 33.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 33.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 3.90000000000000015e-101 < re < 3.00000000000000024e-76 or 2.05e101 < re

   1. Initial program 35.0%

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

      1. +-commutative 35.0%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.3%

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

      1. associate-*r* 91.3%

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

      2. unpow2 91.3%

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

      3. rem-square-sqrt 93.1%

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

      4. metadata-eval 93.1%

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

      5. *-lft-identity 93.1%

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

   6. Simplified 93.1%

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

   if 3.00000000000000024e-76 < re < 2.05e101

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 27.7%

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

      1. distribute-lft-out 27.7%

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

      2. +-commutative 27.7%

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

      3. *-commutative 27.7%

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

      4. +-commutative 27.7%

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

   6. Simplified 27.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.8 \cdot 10^{+205}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 3: 49.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 10^{-75}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 1.36 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.3e+61)
   (* 0.5 (sqrt (* im (/ (- im) re))))
   (if (<= re 3.9e-101)
     (* 0.5 (sqrt (* im 2.0)))
     (if (<= re 1e-75)
       (sqrt re)
       (if (<= re 1.36e+101) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.3e+61) {
		tmp = 0.5 * sqrt((im * (-im / re)));
	} else if (re <= 3.9e-101) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else if (re <= 1e-75) {
		tmp = sqrt(re);
	} else if (re <= 1.36e+101) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} 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 <= (-2.3d+61)) then
        tmp = 0.5d0 * sqrt((im * (-im / re)))
    else if (re <= 3.9d-101) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else if (re <= 1d-75) then
        tmp = sqrt(re)
    else if (re <= 1.36d+101) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.3e+61) {
		tmp = 0.5 * Math.sqrt((im * (-im / re)));
	} else if (re <= 3.9e-101) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else if (re <= 1e-75) {
		tmp = Math.sqrt(re);
	} else if (re <= 1.36e+101) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.3e+61:
		tmp = 0.5 * math.sqrt((im * (-im / re)))
	elif re <= 3.9e-101:
		tmp = 0.5 * math.sqrt((im * 2.0))
	elif re <= 1e-75:
		tmp = math.sqrt(re)
	elif re <= 1.36e+101:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.3e+61)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(-im) / re))));
	elseif (re <= 3.9e-101)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	elseif (re <= 1e-75)
		tmp = sqrt(re);
	elseif (re <= 1.36e+101)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.3e+61)
		tmp = 0.5 * sqrt((im * (-im / re)));
	elseif (re <= 3.9e-101)
		tmp = 0.5 * sqrt((im * 2.0));
	elseif (re <= 1e-75)
		tmp = sqrt(re);
	elseif (re <= 1.36e+101)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.3e+61], N[(0.5 * N[Sqrt[N[(im * N[((-im) / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.9e-101], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-75], N[Sqrt[re], $MachinePrecision], If[LessEqual[re, 1.36e+101], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if re < -2.3e61

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-def 26.6%

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

   3. Simplified 26.6%

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

   4. Taylor expanded in re around -inf 54.4%

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

      1. *-commutative 54.4%

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

      2. unpow2 54.4%

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

      3. associate-/l* 59.3%

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

   6. Simplified 59.3%

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

      1. expm1-log1p-u 58.7%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)\right)} \]

      2. expm1-udef 30.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\right)} - 1\right)} \]

      3. *-commutative 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{\frac{re}{im}} \cdot -0.5\right) \cdot 2}}\right)} - 1\right) \]

      4. associate-*l* 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{im}{\frac{re}{im}} \cdot \left(-0.5 \cdot 2\right)}}\right)} - 1\right) \]

      5. associate-/r/ 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\frac{im}{re} \cdot im\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      6. *-commutative 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(im \cdot \frac{im}{re}\right)} \cdot \left(-0.5 \cdot 2\right)}\right)} - 1\right) \]

      7. metadata-eval 30.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot \color{blue}{-1}}\right)} - 1\right) \]

   8. Applied egg-rr 30.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-def 58.9%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(im \cdot \frac{im}{re}\right) \cdot -1}\right)\right)} \]

      2. expm1-log1p 59.5%

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

      3. *-commutative 59.5%

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

      4. neg-mul-1 59.5%

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

      5. distribute-rgt-neg-in 59.5%

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

      6. distribute-frac-neg 59.5%

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

   10. Simplified 59.5%

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

   if -2.3e61 < re < 3.90000000000000015e-101

   1. Initial program 51.2%

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

      1. +-commutative 51.2%

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

      2. hypot-def 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in re around 0 39.8%

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

      1. *-commutative 39.8%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 39.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 3.90000000000000015e-101 < re < 9.9999999999999996e-76 or 1.35999999999999998e101 < re

   1. Initial program 35.0%

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

      1. +-commutative 35.0%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.3%

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

      1. associate-*r* 91.3%

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

      2. unpow2 91.3%

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

      3. rem-square-sqrt 93.1%

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

      4. metadata-eval 93.1%

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

      5. *-lft-identity 93.1%

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

   6. Simplified 93.1%

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

   if 9.9999999999999996e-76 < re < 1.35999999999999998e101

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 27.7%

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

      1. distribute-lft-out 27.7%

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

      2. +-commutative 27.7%

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

      3. *-commutative 27.7%

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

      4. +-commutative 27.7%

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

   6. Simplified 27.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{-im}{re}}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 10^{-75}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 1.36 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 4: 39.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 3.2e-101)
   (* 0.5 (sqrt (* im 2.0)))
   (if (<= re 2.8e-76)
     (sqrt re)
     (if (<= re 1.25e+101) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 3.2e-101) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else if (re <= 2.8e-76) {
		tmp = sqrt(re);
	} else if (re <= 1.25e+101) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} 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 <= 3.2d-101) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else if (re <= 2.8d-76) then
        tmp = sqrt(re)
    else if (re <= 1.25d+101) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 3.2e-101) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else if (re <= 2.8e-76) {
		tmp = Math.sqrt(re);
	} else if (re <= 1.25e+101) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 3.2e-101:
		tmp = 0.5 * math.sqrt((im * 2.0))
	elif re <= 2.8e-76:
		tmp = math.sqrt(re)
	elif re <= 1.25e+101:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 3.2e-101)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	elseif (re <= 2.8e-76)
		tmp = sqrt(re);
	elseif (re <= 1.25e+101)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 3.2e-101)
		tmp = 0.5 * sqrt((im * 2.0));
	elseif (re <= 2.8e-76)
		tmp = sqrt(re);
	elseif (re <= 1.25e+101)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 3.2e-101], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e-76], N[Sqrt[re], $MachinePrecision], If[LessEqual[re, 1.25e+101], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < 3.19999999999999978e-101

   1. Initial program 35.6%

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

      1. +-commutative 35.6%

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

      2. hypot-def 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in re around 0 29.1%

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

      1. *-commutative 29.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 29.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 3.19999999999999978e-101 < re < 2.8000000000000001e-76 or 1.24999999999999997e101 < re

   1. Initial program 35.0%

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

      1. +-commutative 35.0%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.3%

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

      1. associate-*r* 91.3%

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

      2. unpow2 91.3%

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

      3. rem-square-sqrt 93.1%

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

      4. metadata-eval 93.1%

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

      5. *-lft-identity 93.1%

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

   6. Simplified 93.1%

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

   if 2.8000000000000001e-76 < re < 1.24999999999999997e101

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in re around 0 27.7%

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

      1. distribute-lft-out 27.7%

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

      2. +-commutative 27.7%

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

      3. *-commutative 27.7%

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

      4. +-commutative 27.7%

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

   6. Simplified 27.7%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.2 \cdot 10^{-101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;re \leq 1.25 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 5: 38.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{-101} \lor \neg \left(re \leq 3.9 \cdot 10^{-76}\right) \land re \leq 1.25 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re 3.9e-101) (and (not (<= re 3.9e-76)) (<= re 1.25e+101)))
   (* 0.5 (sqrt (* im 2.0)))
   (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if ((re <= 3.9e-101) || (!(re <= 3.9e-76) && (re <= 1.25e+101))) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= 3.9d-101) .or. (.not. (re <= 3.9d-76)) .and. (re <= 1.25d+101)) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= 3.9e-101) || (!(re <= 3.9e-76) && (re <= 1.25e+101))) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= 3.9e-101) or (not (re <= 3.9e-76) and (re <= 1.25e+101)):
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= 3.9e-101) || (!(re <= 3.9e-76) && (re <= 1.25e+101)))
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= 3.9e-101) || (~((re <= 3.9e-76)) && (re <= 1.25e+101)))
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, 3.9e-101], And[N[Not[LessEqual[re, 3.9e-76]], $MachinePrecision], LessEqual[re, 1.25e+101]]], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 3.90000000000000015e-101 or 3.90000000000000025e-76 < re < 1.24999999999999997e101

   1. Initial program 39.1%

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

      1. +-commutative 39.1%

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

      2. hypot-def 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in re around 0 28.0%

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

      1. *-commutative 28.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   6. Simplified 28.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   if 3.90000000000000015e-101 < re < 3.90000000000000025e-76 or 1.24999999999999997e101 < re

   1. Initial program 35.0%

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

      1. +-commutative 35.0%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 91.3%

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

      1. associate-*r* 91.3%

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

      2. unpow2 91.3%

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

      3. rem-square-sqrt 93.1%

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

      4. metadata-eval 93.1%

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

      5. *-lft-identity 93.1%

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

   6. Simplified 93.1%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 3.9 \cdot 10^{-101} \lor \neg \left(re \leq 3.9 \cdot 10^{-76}\right) \land re \leq 1.25 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 6: 26.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 38.3%

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

   1. +-commutative 38.3%

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

   2. hypot-def 76.2%

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

3. Simplified 76.2%

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

4. Taylor expanded in im around 0 26.0%

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

   1. associate-*r* 26.0%

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

   2. unpow2 26.0%

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

   3. rem-square-sqrt 26.5%

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

   4. metadata-eval 26.5%

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

   5. *-lft-identity 26.5%

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

6. Simplified 26.5%

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

7. Final simplification 26.5%

   \[\leadsto \sqrt{re} \]

Developer target: 49.0% 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 2023230 
(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)))))