math.sqrt on complex, imaginary part, im greater than 0 branch

Percentage Accurate: 41.5% → 90.1%

Time: 9.1s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

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: 41.5% 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: 90.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(im * N[(N[Power[re, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $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 8.5%

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

   3. Taylor expanded in re around inf 45.0%

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

      1. sqrt-div 47.9%

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

      2. sqrt-pow1 99.4%

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

      3. metadata-eval 99.4%

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

      4. pow1 99.4%

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

      5. associate-*r/ 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-/l* 99.4%

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

   7. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

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

   1. Initial program 42.0%

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

      1. pow1 42.0%

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

   4. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 91.5%

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

      2. *-commutative 91.5%

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

      3. associate-*r* 91.9%

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

      4. metadata-eval 91.9%

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

   6. Simplified 91.9%

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

Alternative 2: 75.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.72 \cdot 10^{-71}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left({re}^{-0.5} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-38)
   (sqrt (- re))
   (if (<= re 1.72e-71)
     (sqrt (* 0.5 (- im re)))
     (* im (* (pow re -0.5) 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-38) {
		tmp = sqrt(-re);
	} else if (re <= 1.72e-71) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (pow(re, -0.5) * 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 (re <= (-2d-38)) then
        tmp = sqrt(-re)
    else if (re <= 1.72d-71) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * ((re ** (-0.5d0)) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e-38) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.72e-71) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (Math.pow(re, -0.5) * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-38:
		tmp = math.sqrt(-re)
	elif re <= 1.72e-71:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (math.pow(re, -0.5) * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-38)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.72e-71)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64((re ^ -0.5) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-38)
		tmp = sqrt(-re);
	elseif (re <= 1.72e-71)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * ((re ^ -0.5) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-38], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.72e-71], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(N[Power[re, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9999999999999999e-38

   1. Initial program 42.8%

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

      1. pow1 42.8%

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

   4. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 79.5%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 79.5%

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

   9. Simplified 79.5%

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

   if -1.9999999999999999e-38 < re < 1.72e-71

   1. Initial program 53.2%

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

      1. pow1 53.2%

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

   4. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*r* 94.7%

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

      4. metadata-eval 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in re around 0 86.2%

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

      1. neg-mul-1 86.2%

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

      2. sub-neg 86.2%

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

   9. Simplified 86.2%

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

   if 1.72e-71 < re

   1. Initial program 8.1%

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

   3. Taylor expanded in re around inf 51.3%

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

      1. sqrt-div 56.8%

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

      2. sqrt-pow1 80.4%

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

      3. metadata-eval 80.4%

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

      4. pow1 80.4%

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

      5. associate-*r/ 80.4%

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

   5. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-/l* 80.3%

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

   7. Simplified 80.3%

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

      1. clear-num 80.3%

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

      2. associate-/r/ 80.3%

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

      3. pow1/2 80.3%

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

      4. pow-flip 80.4%

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

      5. metadata-eval 80.4%

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

   9. Applied egg-rr 80.4%

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

Alternative 3: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.3 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.3e-40)
   (sqrt (- re))
   (if (<= re 2.4e-76) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.3e-40) {
		tmp = sqrt(-re);
	} else if (re <= 2.4e-76) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * sqrt((0.25 / 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-40)) then
        tmp = sqrt(-re)
    else if (re <= 2.4d-76) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.3e-40) {
		tmp = Math.sqrt(-re);
	} else if (re <= 2.4e-76) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.3e-40:
		tmp = math.sqrt(-re)
	elif re <= 2.4e-76:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.3e-40)
		tmp = sqrt(Float64(-re));
	elseif (re <= 2.4e-76)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.3e-40)
		tmp = sqrt(-re);
	elseif (re <= 2.4e-76)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.3e-40], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 2.4e-76], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.3e-40

   1. Initial program 42.8%

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

      1. pow1 42.8%

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

   4. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 79.5%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 79.5%

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

   9. Simplified 79.5%

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

   if -2.3e-40 < re < 2.40000000000000013e-76

   1. Initial program 53.2%

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

      1. pow1 53.2%

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

   4. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*r* 94.7%

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

      4. metadata-eval 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in re around 0 86.2%

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

      1. neg-mul-1 86.2%

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

      2. sub-neg 86.2%

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

   9. Simplified 86.2%

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

   if 2.40000000000000013e-76 < re

   1. Initial program 8.1%

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

   3. Taylor expanded in re around inf 51.3%

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

      1. sqrt-div 56.8%

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

      2. sqrt-pow1 80.4%

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

      3. metadata-eval 80.4%

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

      4. pow1 80.4%

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

      5. associate-*r/ 80.4%

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

   5. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-/l* 80.3%

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

   7. Simplified 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-/r/ 80.2%

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

      3. div-inv 80.2%

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

      4. associate-/r* 80.3%

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

      5. add-sqr-sqrt 80.0%

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

      6. sqrt-unprod 80.3%

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

      7. frac-times 80.3%

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

      8. metadata-eval 80.3%

         \[\leadsto \frac{\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}}}{\frac{1}{im}} \]

      9. add-sqr-sqrt 80.3%

         \[\leadsto \frac{\sqrt{\frac{0.25}{\color{blue}{re}}}}{\frac{1}{im}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{0.25}{re}}}{\frac{1}{im}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 80.4%

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{0.25}{re}}}{1} \cdot im} \]

       2. /-rgt-identity 80.4%

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

       3. *-commutative 80.4%

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

   11. Simplified 80.4%

       \[\leadsto \color{blue}{im \cdot \sqrt{\frac{0.25}{re}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 64.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.8e-54) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.8e-54) {
		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 (re <= (-9.8d-54)) 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 (re <= -9.8e-54) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.8e-54:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.8e-54)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.8e-54)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.8e-54], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -9.80000000000000042e-54

   1. Initial program 44.2%

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

      1. pow1 44.2%

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

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 78.9%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 78.9%

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

   9. Simplified 78.9%

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

   if -9.80000000000000042e-54 < re

   1. Initial program 35.3%

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

      1. pow1 35.3%

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

   4. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 73.9%

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

      2. *-commutative 73.9%

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

      3. associate-*r* 73.9%

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

      4. metadata-eval 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in re around 0 62.3%

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

      1. *-commutative 62.3%

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

   9. Simplified 62.3%

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

Alternative 5: 28.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-310) (sqrt (- re)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-310) {
		tmp = sqrt(-re);
	} 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 <= (-2d-310)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-310)
		tmp = sqrt(-re);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-310], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.999999999999994e-310

   1. Initial program 48.8%

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

      1. pow1 48.8%

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

   4. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 59.0%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 59.0%

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

   9. Simplified 59.0%

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

   if -1.999999999999994e-310 < re

   1. Initial program 26.9%

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

      1. pow1 26.9%

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

   4. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*r* 63.9%

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

      4. metadata-eval 63.9%

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

   6. Simplified 63.9%

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

   7. Taylor expanded in re around -inf 0.0%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
   8. Step-by-step derivation

      1. neg-mul-1 0.0%

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

   9. Simplified 0.0%

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

       1. neg-sub0 0.0%

          \[\leadsto \sqrt{\color{blue}{0 - re}} \]

       2. sub-neg 0.0%

          \[\leadsto \sqrt{\color{blue}{0 + \left(-re\right)}} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto \sqrt{0 + \color{blue}{\sqrt{-re} \cdot \sqrt{-re}}} \]

       4. sqrt-unprod 4.2%

          \[\leadsto \sqrt{0 + \color{blue}{\sqrt{\left(-re\right) \cdot \left(-re\right)}}} \]

       5. sqr-neg 4.2%

          \[\leadsto \sqrt{0 + \sqrt{\color{blue}{re \cdot re}}} \]

       6. sqrt-unprod 5.8%

          \[\leadsto \sqrt{0 + \color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       7. add-sqr-sqrt 5.8%

          \[\leadsto \sqrt{0 + \color{blue}{re}} \]

   11. Applied egg-rr 5.8%

       \[\leadsto \sqrt{\color{blue}{0 + re}} \]
   12. Step-by-step derivation

       1. +-lft-identity 5.8%

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

   13. Simplified 5.8%

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

Alternative 6: 2.9% 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 38.2%

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

   1. pow1 38.2%

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

4. Applied egg-rr 82.1%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
5. Step-by-step derivation

   1. unpow1 82.1%

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

   2. *-commutative 82.1%

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

   3. associate-*r* 82.5%

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

   4. metadata-eval 82.5%

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

6. Simplified 82.5%

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

7. Taylor expanded in re around -inf 30.4%

   \[\leadsto \sqrt{\color{blue}{-1 \cdot re}} \]
8. Step-by-step derivation

   1. neg-mul-1 30.4%

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

9. Simplified 30.4%

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

    1. neg-sub0 30.4%

       \[\leadsto \sqrt{\color{blue}{0 - re}} \]

    2. sub-neg 30.4%

       \[\leadsto \sqrt{\color{blue}{0 + \left(-re\right)}} \]

    3. add-sqr-sqrt 30.4%

       \[\leadsto \sqrt{0 + \color{blue}{\sqrt{-re} \cdot \sqrt{-re}}} \]

    4. sqrt-unprod 17.5%

       \[\leadsto \sqrt{0 + \color{blue}{\sqrt{\left(-re\right) \cdot \left(-re\right)}}} \]

    5. sqr-neg 17.5%

       \[\leadsto \sqrt{0 + \sqrt{\color{blue}{re \cdot re}}} \]

    6. sqrt-unprod 2.8%

       \[\leadsto \sqrt{0 + \color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

    7. add-sqr-sqrt 2.8%

       \[\leadsto \sqrt{0 + \color{blue}{re}} \]

11. Applied egg-rr 2.8%

    \[\leadsto \sqrt{\color{blue}{0 + re}} \]
12. Step-by-step derivation

    1. +-lft-identity 2.8%

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

13. Simplified 2.8%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))