math.sqrt on complex, real part

Percentage Accurate: 40.2% → 84.1%

Time: 9.3s

Alternatives: 7

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: 40.2% 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: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;\sqrt{-0.25 \cdot \frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (+ re (sqrt (+ (* re re) (* im im)))))) 0.0)
   (sqrt (* -0.25 (/ (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 = sqrt((-0.25 * (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 = Math.sqrt((-0.25 * (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 = math.sqrt((-0.25 * (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 = sqrt(Float64(-0.25 * 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 = sqrt((-0.25 * ((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[Sqrt[N[(-0.25 * 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 #s(literal 2 binary64) (+.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. sqr-neg 10.7%

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

      2. +-commutative 10.7%

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

      3. sqr-neg 10.7%

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

      4. +-commutative 10.7%

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

      5. distribute-rgt-in 10.7%

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

      6. cancel-sign-sub 10.7%

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

      7. distribute-rgt-out-- 10.7%

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

      8. sub-neg 10.7%

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

      9. remove-double-neg 10.7%

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

      10. +-commutative 10.7%

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

   3. Simplified 10.7%

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

      1. hypot-define 10.7%

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

      2. +-commutative 10.7%

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

      3. add-sqr-sqrt 10.7%

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

      4. sqrt-unprod 10.7%

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

      5. *-commutative 10.7%

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

      6. *-commutative 10.7%

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

      7. swap-sqr 10.7%

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

   6. Applied egg-rr 10.7%

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

      1. *-commutative 10.7%

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

      2. associate-*r* 10.7%

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

      3. metadata-eval 10.7%

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

   8. Simplified 10.7%

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

   9. Taylor expanded in re around -inf 58.0%

      \[\leadsto \sqrt{\color{blue}{-0.25 \cdot \frac{{im}^{2}}{re}}} \]

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 46.1%

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

      1. sqr-neg 46.1%

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

      2. +-commutative 46.1%

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

      3. sqr-neg 46.1%

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

      4. +-commutative 46.1%

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

      5. distribute-rgt-in 46.1%

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

      6. cancel-sign-sub 46.1%

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

      7. distribute-rgt-out-- 46.1%

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

      8. sub-neg 46.1%

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

      9. remove-double-neg 46.1%

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

      10. +-commutative 46.1%

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

   3. Simplified 91.9%

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

      1. hypot-define 46.1%

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

      2. +-commutative 46.1%

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

      3. add-sqr-sqrt 45.7%

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

      4. sqrt-unprod 46.1%

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

      5. *-commutative 46.1%

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

      6. *-commutative 46.1%

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

      7. swap-sqr 46.1%

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

   6. Applied egg-rr 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-*r* 91.9%

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

      3. metadata-eval 91.9%

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

   8. Simplified 91.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(re + \sqrt{re \cdot re + im \cdot im}\right)} \leq 0:\\ \;\;\;\;\sqrt{-0.25 \cdot \frac{{im}^{2}}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{-0.25 \cdot \frac{{im}^{2}}{re}}\\ \mathbf{elif}\;re \leq 8.8 \cdot 10^{+64}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im}{0.5 \cdot \frac{im}{re + im}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9e+92)
   (sqrt (* -0.25 (/ (pow im 2.0) re)))
   (if (<= re 8.8e+64)
     (* 0.5 (sqrt (/ im (* 0.5 (/ im (+ re im))))))
     (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9e+92) {
		tmp = sqrt((-0.25 * (pow(im, 2.0) / re)));
	} else if (re <= 8.8e+64) {
		tmp = 0.5 * sqrt((im / (0.5 * (im / (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 <= (-9d+92)) then
        tmp = sqrt(((-0.25d0) * ((im ** 2.0d0) / re)))
    else if (re <= 8.8d+64) then
        tmp = 0.5d0 * sqrt((im / (0.5d0 * (im / (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 <= -9e+92) {
		tmp = Math.sqrt((-0.25 * (Math.pow(im, 2.0) / re)));
	} else if (re <= 8.8e+64) {
		tmp = 0.5 * Math.sqrt((im / (0.5 * (im / (re + im)))));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9e+92:
		tmp = math.sqrt((-0.25 * (math.pow(im, 2.0) / re)))
	elif re <= 8.8e+64:
		tmp = 0.5 * math.sqrt((im / (0.5 * (im / (re + im)))))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9e+92)
		tmp = sqrt(Float64(-0.25 * Float64((im ^ 2.0) / re)));
	elseif (re <= 8.8e+64)
		tmp = Float64(0.5 * sqrt(Float64(im / Float64(0.5 * Float64(im / Float64(re + im))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9e+92)
		tmp = sqrt((-0.25 * ((im ^ 2.0) / re)));
	elseif (re <= 8.8e+64)
		tmp = 0.5 * sqrt((im / (0.5 * (im / (re + im)))));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9e+92], N[Sqrt[N[(-0.25 * N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[re, 8.8e+64], N[(0.5 * N[Sqrt[N[(im / N[(0.5 * N[(im / N[(re + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.9999999999999998e92

   1. Initial program 8.4%

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

      1. sqr-neg 8.4%

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

      2. +-commutative 8.4%

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

      3. sqr-neg 8.4%

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

      4. +-commutative 8.4%

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

      5. distribute-rgt-in 8.4%

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

      6. cancel-sign-sub 8.4%

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

      7. distribute-rgt-out-- 8.4%

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

      8. sub-neg 8.4%

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

      9. remove-double-neg 8.4%

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

      10. +-commutative 8.4%

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

   3. Simplified 46.2%

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

      1. hypot-define 8.4%

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

      2. +-commutative 8.4%

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

      3. add-sqr-sqrt 8.4%

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

      4. sqrt-unprod 8.4%

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

      5. *-commutative 8.4%

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

      6. *-commutative 8.4%

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

      7. swap-sqr 8.4%

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

   6. Applied egg-rr 46.2%

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

      1. *-commutative 46.2%

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

      2. associate-*r* 46.2%

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

      3. metadata-eval 46.2%

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

   8. Simplified 46.2%

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

   9. Taylor expanded in re around -inf 56.9%

      \[\leadsto \sqrt{\color{blue}{-0.25 \cdot \frac{{im}^{2}}{re}}} \]

   if -8.9999999999999998e92 < re < 8.80000000000000007e64

   1. Initial program 53.1%

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

      1. sqr-neg 53.1%

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

      2. +-commutative 53.1%

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

      3. sqr-neg 53.1%

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

      4. +-commutative 53.1%

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

      5. distribute-rgt-in 53.1%

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

      6. cancel-sign-sub 53.1%

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

      7. distribute-rgt-out-- 53.1%

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

      8. sub-neg 53.1%

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

      9. remove-double-neg 53.1%

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

      10. +-commutative 53.1%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in im around inf 40.7%

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

   6. Taylor expanded in im around 0 40.7%

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

      1. distribute-lft-out 40.7%

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

   8. Simplified 40.7%

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

      1. clear-num 40.7%

         \[\leadsto 0.5 \cdot \sqrt{im \cdot \color{blue}{\frac{1}{\frac{im}{2 \cdot \left(im + re\right)}}}} \]

      2. un-div-inv 40.7%

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

      3. *-un-lft-identity 40.7%

         \[\leadsto 0.5 \cdot \sqrt{\frac{im}{\frac{\color{blue}{1 \cdot im}}{2 \cdot \left(im + re\right)}}} \]

      4. times-frac 40.7%

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

      5. metadata-eval 40.7%

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

      6. +-commutative 40.7%

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

   10. Applied egg-rr 40.7%

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

   if 8.80000000000000007e64 < re

   1. Initial program 31.6%

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

      1. sqr-neg 31.6%

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

      2. +-commutative 31.6%

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

      3. sqr-neg 31.6%

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

      4. +-commutative 31.6%

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

      5. distribute-rgt-in 31.6%

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

      6. cancel-sign-sub 31.6%

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

      7. distribute-rgt-out-- 31.6%

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

      8. sub-neg 31.6%

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

      9. remove-double-neg 31.6%

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

      10. +-commutative 31.6%

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

   3. Simplified 100.0%

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

      1. hypot-define 31.6%

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

      2. +-commutative 31.6%

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

      3. add-sqr-sqrt 31.4%

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

      4. sqrt-unprod 31.6%

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

      5. *-commutative 31.6%

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

      6. *-commutative 31.6%

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

      7. swap-sqr 31.6%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around inf 83.3%

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

Alternative 3: 42.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{im \cdot \left(0.5 + 0.5 \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.15e+195)
   (sqrt (* 0.5 (- re re)))
   (if (<= re 1.4e+65) (sqrt (* im (+ 0.5 (* 0.5 (/ re im))))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.15e+195) {
		tmp = sqrt((0.5 * (re - re)));
	} else if (re <= 1.4e+65) {
		tmp = sqrt((im * (0.5 + (0.5 * (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 <= (-2.15d+195)) then
        tmp = sqrt((0.5d0 * (re - re)))
    else if (re <= 1.4d+65) then
        tmp = sqrt((im * (0.5d0 + (0.5d0 * (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 <= -2.15e+195) {
		tmp = Math.sqrt((0.5 * (re - re)));
	} else if (re <= 1.4e+65) {
		tmp = Math.sqrt((im * (0.5 + (0.5 * (re / im)))));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.15e+195:
		tmp = math.sqrt((0.5 * (re - re)))
	elif re <= 1.4e+65:
		tmp = math.sqrt((im * (0.5 + (0.5 * (re / im)))))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.15e+195)
		tmp = sqrt(Float64(0.5 * Float64(re - re)));
	elseif (re <= 1.4e+65)
		tmp = sqrt(Float64(im * Float64(0.5 + Float64(0.5 * Float64(re / im)))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.15e+195)
		tmp = sqrt((0.5 * (re - re)));
	elseif (re <= 1.4e+65)
		tmp = sqrt((im * (0.5 + (0.5 * (re / im)))));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.15e+195], N[Sqrt[N[(0.5 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[re, 1.4e+65], N[Sqrt[N[(im * N[(0.5 + N[(0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.14999999999999991e195

   1. Initial program 2.0%

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

      1. sqr-neg 2.0%

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

      2. +-commutative 2.0%

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

      3. sqr-neg 2.0%

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

      4. +-commutative 2.0%

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

      5. distribute-rgt-in 2.0%

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

      6. cancel-sign-sub 2.0%

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

      7. distribute-rgt-out-- 2.0%

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

      8. sub-neg 2.0%

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

      9. remove-double-neg 2.0%

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

      10. +-commutative 2.0%

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

   3. Simplified 60.0%

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

      1. hypot-define 2.0%

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

      2. +-commutative 2.0%

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

      3. add-sqr-sqrt 2.0%

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

      4. sqrt-unprod 2.0%

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

      5. *-commutative 2.0%

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

      6. *-commutative 2.0%

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

      7. swap-sqr 2.0%

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

   6. Applied egg-rr 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-*r* 60.0%

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

      3. metadata-eval 60.0%

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

   8. Simplified 60.0%

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

   9. Taylor expanded in re around -inf 50.2%

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

       1. neg-mul-1 50.2%

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

   11. Simplified 50.2%

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

   if -2.14999999999999991e195 < re < 1.3999999999999999e65

   1. Initial program 48.7%

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

      1. sqr-neg 48.7%

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

      2. +-commutative 48.7%

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

      3. sqr-neg 48.7%

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

      4. +-commutative 48.7%

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

      5. distribute-rgt-in 48.7%

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

      6. cancel-sign-sub 48.7%

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

      7. distribute-rgt-out-- 48.7%

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

      8. sub-neg 48.7%

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

      9. remove-double-neg 48.7%

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

      10. +-commutative 48.7%

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

   3. Simplified 79.8%

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

      1. hypot-define 48.7%

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

      2. +-commutative 48.7%

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

      3. add-sqr-sqrt 48.4%

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

      4. sqrt-unprod 48.7%

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

      5. *-commutative 48.7%

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

      6. *-commutative 48.7%

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

      7. swap-sqr 48.7%

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

   6. Applied egg-rr 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*r* 79.8%

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

      3. metadata-eval 79.8%

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

   8. Simplified 79.8%

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

   9. Taylor expanded in im around inf 36.7%

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

   if 1.3999999999999999e65 < re

   1. Initial program 31.6%

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

      1. sqr-neg 31.6%

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

      2. +-commutative 31.6%

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

      3. sqr-neg 31.6%

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

      4. +-commutative 31.6%

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

      5. distribute-rgt-in 31.6%

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

      6. cancel-sign-sub 31.6%

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

      7. distribute-rgt-out-- 31.6%

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

      8. sub-neg 31.6%

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

      9. remove-double-neg 31.6%

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

      10. +-commutative 31.6%

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

   3. Simplified 100.0%

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

      1. hypot-define 31.6%

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

      2. +-commutative 31.6%

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

      3. add-sqr-sqrt 31.4%

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

      4. sqrt-unprod 31.6%

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

      5. *-commutative 31.6%

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

      6. *-commutative 31.6%

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

      7. swap-sqr 31.6%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around inf 83.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.15 \cdot 10^{+195}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{im \cdot \left(0.5 + 0.5 \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e+200)
   (sqrt (* 0.5 (- re re)))
   (if (<= re 4.9e+67) (sqrt (* 0.5 (+ re im))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+200) {
		tmp = sqrt((0.5 * (re - re)));
	} else if (re <= 4.9e+67) {
		tmp = sqrt((0.5 * (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 <= (-3.6d+200)) then
        tmp = sqrt((0.5d0 * (re - re)))
    else if (re <= 4.9d+67) then
        tmp = sqrt((0.5d0 * (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 <= -3.6e+200) {
		tmp = Math.sqrt((0.5 * (re - re)));
	} else if (re <= 4.9e+67) {
		tmp = Math.sqrt((0.5 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e+200:
		tmp = math.sqrt((0.5 * (re - re)))
	elif re <= 4.9e+67:
		tmp = math.sqrt((0.5 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e+200)
		tmp = sqrt(Float64(0.5 * Float64(re - re)));
	elseif (re <= 4.9e+67)
		tmp = sqrt(Float64(0.5 * Float64(re + im)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.6e+200)
		tmp = sqrt((0.5 * (re - re)));
	elseif (re <= 4.9e+67)
		tmp = sqrt((0.5 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e+200], N[Sqrt[N[(0.5 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[re, 4.9e+67], N[Sqrt[N[(0.5 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.5999999999999998e200

   1. Initial program 2.0%

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

      1. sqr-neg 2.0%

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

      2. +-commutative 2.0%

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

      3. sqr-neg 2.0%

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

      4. +-commutative 2.0%

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

      5. distribute-rgt-in 2.0%

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

      6. cancel-sign-sub 2.0%

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

      7. distribute-rgt-out-- 2.0%

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

      8. sub-neg 2.0%

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

      9. remove-double-neg 2.0%

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

      10. +-commutative 2.0%

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

   3. Simplified 60.0%

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

      1. hypot-define 2.0%

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

      2. +-commutative 2.0%

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

      3. add-sqr-sqrt 2.0%

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

      4. sqrt-unprod 2.0%

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

      5. *-commutative 2.0%

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

      6. *-commutative 2.0%

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

      7. swap-sqr 2.0%

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

   6. Applied egg-rr 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-*r* 60.0%

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

      3. metadata-eval 60.0%

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

   8. Simplified 60.0%

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

   9. Taylor expanded in re around -inf 50.2%

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

       1. neg-mul-1 50.2%

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

   11. Simplified 50.2%

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

   if -3.5999999999999998e200 < re < 4.8999999999999999e67

   1. Initial program 48.7%

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

      1. sqr-neg 48.7%

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

      2. +-commutative 48.7%

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

      3. sqr-neg 48.7%

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

      4. +-commutative 48.7%

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

      5. distribute-rgt-in 48.7%

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

      6. cancel-sign-sub 48.7%

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

      7. distribute-rgt-out-- 48.7%

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

      8. sub-neg 48.7%

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

      9. remove-double-neg 48.7%

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

      10. +-commutative 48.7%

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

   3. Simplified 79.8%

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

      1. hypot-define 48.7%

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

      2. +-commutative 48.7%

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

      3. add-sqr-sqrt 48.4%

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

      4. sqrt-unprod 48.7%

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

      5. *-commutative 48.7%

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

      6. *-commutative 48.7%

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

      7. swap-sqr 48.7%

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

   6. Applied egg-rr 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*r* 79.8%

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

      3. metadata-eval 79.8%

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

   8. Simplified 79.8%

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

   9. Taylor expanded in re around 0 36.7%

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

   if 4.8999999999999999e67 < re

   1. Initial program 31.6%

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

      1. sqr-neg 31.6%

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

      2. +-commutative 31.6%

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

      3. sqr-neg 31.6%

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

      4. +-commutative 31.6%

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

      5. distribute-rgt-in 31.6%

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

      6. cancel-sign-sub 31.6%

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

      7. distribute-rgt-out-- 31.6%

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

      8. sub-neg 31.6%

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

      9. remove-double-neg 31.6%

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

      10. +-commutative 31.6%

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

   3. Simplified 100.0%

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

      1. hypot-define 31.6%

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

      2. +-commutative 31.6%

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

      3. add-sqr-sqrt 31.4%

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

      4. sqrt-unprod 31.6%

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

      5. *-commutative 31.6%

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

      6. *-commutative 31.6%

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

      7. swap-sqr 31.6%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around inf 83.3%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+200}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re - re\right)}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3.3e-91) (sqrt re) (sqrt (* 0.5 (+ re im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3.3e-91) {
		tmp = sqrt(re);
	} else {
		tmp = sqrt((0.5 * (re + im)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3.3d-91) then
        tmp = sqrt(re)
    else
        tmp = sqrt((0.5d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3.3e-91) {
		tmp = Math.sqrt(re);
	} else {
		tmp = Math.sqrt((0.5 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3.3e-91:
		tmp = math.sqrt(re)
	else:
		tmp = math.sqrt((0.5 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3.3e-91)
		tmp = sqrt(re);
	else
		tmp = sqrt(Float64(0.5 * Float64(re + im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3.3e-91)
		tmp = sqrt(re);
	else
		tmp = sqrt((0.5 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3.3e-91], N[Sqrt[re], $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.30000000000000011e-91

   1. Initial program 42.9%

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

      1. sqr-neg 42.9%

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

      2. +-commutative 42.9%

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

      3. sqr-neg 42.9%

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

      4. +-commutative 42.9%

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

      5. distribute-rgt-in 42.9%

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

      6. cancel-sign-sub 42.9%

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

      7. distribute-rgt-out-- 42.9%

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

      8. sub-neg 42.9%

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

      9. remove-double-neg 42.9%

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

      10. +-commutative 42.9%

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

   3. Simplified 78.7%

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

      1. hypot-define 42.9%

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

      2. +-commutative 42.9%

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

      3. add-sqr-sqrt 42.5%

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

      4. sqrt-unprod 42.9%

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

      5. *-commutative 42.9%

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

      6. *-commutative 42.9%

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

      7. swap-sqr 42.9%

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

   6. Applied egg-rr 78.7%

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

      1. *-commutative 78.7%

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

      2. associate-*r* 78.7%

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

      3. metadata-eval 78.7%

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

   8. Simplified 78.7%

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

   9. Taylor expanded in re around inf 32.6%

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

   if 3.30000000000000011e-91 < im

   1. Initial program 39.7%

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

      1. sqr-neg 39.7%

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

      2. +-commutative 39.7%

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

      3. sqr-neg 39.7%

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

      4. +-commutative 39.7%

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

      5. distribute-rgt-in 39.7%

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

      6. cancel-sign-sub 39.7%

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

      7. distribute-rgt-out-- 39.7%

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

      8. sub-neg 39.7%

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

      9. remove-double-neg 39.7%

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

      10. +-commutative 39.7%

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

   3. Simplified 88.8%

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

      1. hypot-define 39.7%

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

      2. +-commutative 39.7%

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

      3. add-sqr-sqrt 39.5%

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

      4. sqrt-unprod 39.7%

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

      5. *-commutative 39.7%

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

      6. *-commutative 39.7%

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

      7. swap-sqr 39.7%

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

   6. Applied egg-rr 88.8%

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

      1. *-commutative 88.8%

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

      2. associate-*r* 88.8%

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

      3. metadata-eval 88.8%

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

   8. Simplified 88.8%

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

   9. Taylor expanded in re around 0 74.3%

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

Alternative 6: 43.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 6.5 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 6.5e-85) (sqrt re) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 6.5e-85) {
		tmp = sqrt(re);
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 6.5d-85) then
        tmp = sqrt(re)
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 6.5e-85) {
		tmp = Math.sqrt(re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 6.5e-85:
		tmp = math.sqrt(re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 6.5e-85)
		tmp = sqrt(re);
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 6.5e-85)
		tmp = sqrt(re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 6.5e-85], N[Sqrt[re], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 6.5e-85

   1. Initial program 43.0%

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

      1. sqr-neg 43.0%

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

      2. +-commutative 43.0%

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

      3. sqr-neg 43.0%

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

      4. +-commutative 43.0%

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

      5. distribute-rgt-in 43.0%

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

      6. cancel-sign-sub 43.0%

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

      7. distribute-rgt-out-- 43.0%

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

      8. sub-neg 43.0%

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

      9. remove-double-neg 43.0%

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

      10. +-commutative 43.0%

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

   3. Simplified 78.9%

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

      1. hypot-define 43.0%

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

      2. +-commutative 43.0%

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

      3. add-sqr-sqrt 42.6%

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

      4. sqrt-unprod 43.0%

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

      5. *-commutative 43.0%

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

      6. *-commutative 43.0%

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

      7. swap-sqr 43.0%

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

   6. Applied egg-rr 78.9%

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

      1. *-commutative 78.9%

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

      2. associate-*r* 78.9%

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

      3. metadata-eval 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in re around inf 32.8%

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

   if 6.5e-85 < im

   1. Initial program 39.4%

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

      1. sqr-neg 39.4%

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

      2. +-commutative 39.4%

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

      3. sqr-neg 39.4%

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

      4. +-commutative 39.4%

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

      5. distribute-rgt-in 39.4%

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

      6. cancel-sign-sub 39.4%

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

      7. distribute-rgt-out-- 39.4%

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

      8. sub-neg 39.4%

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

      9. remove-double-neg 39.4%

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

      10. +-commutative 39.4%

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

   3. Simplified 88.5%

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

      1. hypot-define 39.4%

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

      2. +-commutative 39.4%

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

      3. add-sqr-sqrt 39.2%

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

      4. sqrt-unprod 39.4%

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

      5. *-commutative 39.4%

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

      6. *-commutative 39.4%

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

      7. swap-sqr 39.4%

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

   6. Applied egg-rr 88.5%

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

      1. *-commutative 88.5%

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

      2. associate-*r* 88.5%

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

      3. metadata-eval 88.5%

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

   8. Simplified 88.5%

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

   9. Taylor expanded in re around 0 72.5%

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

       1. *-commutative 72.5%

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

   11. Simplified 72.5%

       \[\leadsto \sqrt{\color{blue}{im \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 26.9% 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 41.8%

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

   1. sqr-neg 41.8%

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

   2. +-commutative 41.8%

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

   3. sqr-neg 41.8%

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

   4. +-commutative 41.8%

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

   5. distribute-rgt-in 41.8%

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

   6. cancel-sign-sub 41.8%

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

   7. distribute-rgt-out-- 41.8%

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

   8. sub-neg 41.8%

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

   9. remove-double-neg 41.8%

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

   10. +-commutative 41.8%

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

3. Simplified 82.1%

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

   1. hypot-define 41.8%

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

   2. +-commutative 41.8%

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

   3. add-sqr-sqrt 41.5%

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

   4. sqrt-unprod 41.8%

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

   5. *-commutative 41.8%

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

   6. *-commutative 41.8%

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

   7. swap-sqr 41.8%

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

6. Applied egg-rr 82.1%

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

   1. *-commutative 82.1%

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

   2. associate-*r* 82.1%

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

   3. metadata-eval 82.1%

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

8. Simplified 82.1%

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

9. Taylor expanded in re around inf 27.9%

   \[\leadsto \sqrt{\color{blue}{re}} \]
10. Add Preprocessing

Developer Target 1: 47.5% 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 2024185 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< re 0) (* 1/2 (* (sqrt 2) (sqrt (/ (* im im) (- (modulus re im) re))))) (* 1/2 (sqrt (* 2 (+ (modulus re im) re))))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))