math.sqrt on complex, real part

Percentage Accurate: 41.4% → 85.0%

Time: 6.9s

Alternatives: 8

Speedup: 1.2×

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: 41.4% 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: 85.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 13.3

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

   4. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-hypot.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 13.3%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 47.5

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

   9. Applied rewrites 47.5%

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

   11. Applied rewrites 56.0%

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

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

   1. Initial program 54.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift-hypot.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 94.6%

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

4. Final simplification 88.1%

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

Alternative 2: 85.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 8.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 13.3

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

   4. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-hypot.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 13.3%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 47.5

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

   9. Applied rewrites 47.5%

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

   11. Applied rewrites 56.0%

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

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

   1. Initial program 54.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 94.6

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

   4. Applied rewrites 94.6%

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

4. Final simplification 88.1%

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

Alternative 3: 59.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im} + re\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{re} \cdot im}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (+ (sqrt (+ (* re re) (* im im))) re)))
   (if (<= t_0 0.0)
     (* 0.5 (sqrt (* (/ (- im) re) im)))
     (if (<= t_0 4e+152)
       (* 0.5 (sqrt (* 2.0 (+ (sqrt (fma re re (* im im))) re))))
       (* 0.5 (sqrt (* 2.0 (+ im re))))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im))) + re;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * sqrt(((-im / re) * im));
	} else if (t_0 <= 4e+152) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) + re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im + re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) + re)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64(-im) / re) * im)));
	elseif (t_0 <= 4e+152)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) + re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im + re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+152], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 8.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 13.3

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

   4. Applied rewrites 13.3%

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

      1. lift-*.f64 N/A

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

      2. lift-hypot.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 13.3%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 47.5

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

   9. Applied rewrites 47.5%

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

   11. Applied rewrites 56.0%

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

   if 0.0 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 4.0000000000000002e152

   1. Initial program 96.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 96.1

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

   4. Applied rewrites 96.1%

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

   if 4.0000000000000002e152 < (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 3.9%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 41.1

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

   5. Applied rewrites 41.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{re \cdot re + im \cdot im} + re \leq 0:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{re} \cdot im}\\ \mathbf{elif}\;\sqrt{re \cdot re + im \cdot im} + re \leq 4 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{re} \cdot im}\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.75e+25)
   (* 0.5 (sqrt (* (/ (- im) re) im)))
   (if (<= re 2.05e+30) (* 0.5 (sqrt (* 2.0 (+ im re)))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.75e+25) {
		tmp = 0.5 * sqrt(((-im / re) * im));
	} else if (re <= 2.05e+30) {
		tmp = 0.5 * sqrt((2.0 * (im + 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 <= (-1.75d+25)) then
        tmp = 0.5d0 * sqrt(((-im / re) * im))
    else if (re <= 2.05d+30) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im + re)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.75e+25) {
		tmp = 0.5 * Math.sqrt(((-im / re) * im));
	} else if (re <= 2.05e+30) {
		tmp = 0.5 * Math.sqrt((2.0 * (im + re)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.75e+25:
		tmp = 0.5 * math.sqrt(((-im / re) * im))
	elif re <= 2.05e+30:
		tmp = 0.5 * math.sqrt((2.0 * (im + re)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.75e+25)
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64(-im) / re) * im)));
	elseif (re <= 2.05e+30)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im + re))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.75e+25)
		tmp = 0.5 * sqrt(((-im / re) * im));
	elseif (re <= 2.05e+30)
		tmp = 0.5 * sqrt((2.0 * (im + re)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.75e+25], N[(0.5 * N[Sqrt[N[(N[((-im) / re), $MachinePrecision] * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.05e+30], N[(0.5 * N[Sqrt[N[(2.0 * N[(im + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.75e25

   1. Initial program 8.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 48.3

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

   4. Applied rewrites 48.3%

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

      1. lift-*.f64 N/A

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

      2. lift-hypot.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 47.0

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

   9. Applied rewrites 47.0%

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

   11. Applied rewrites 53.9%

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

   if -1.75e25 < re < 2.05000000000000003e30

   1. Initial program 59.3%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 36.1

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

   5. Applied rewrites 36.1%

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

   if 2.05000000000000003e30 < re

   1. Initial program 52.1%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 72.8

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 72.8%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.75 \cdot 10^{+25}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-im}{re} \cdot im}\\ \mathbf{elif}\;re \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{-289}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* im im) 1e-289) (sqrt re) (* 0.5 (sqrt (* 2.0 (+ im re))))))

C

double code(double re, double im) {
	double tmp;
	if ((im * im) <= 1e-289) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (im + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im * im) <= 1d-289) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im * im) <= 1e-289) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im + re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im * im) <= 1e-289:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im + re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e-289)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im * im) <= 1e-289)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (im + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1e-289], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1e-289

   1. Initial program 45.4%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 48.8

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

   5. Applied rewrites 48.8%

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

   7. Applied rewrites 48.8%

      \[\leadsto \sqrt{re} \]

   if 1e-289 < (*.f64 im im)

   1. Initial program 47.8%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 37.0

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

   5. Applied rewrites 37.0%

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

Alternative 6: 35.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \cdot im \leq 10^{-289}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* im im) 1e-289) (sqrt re) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if ((im * im) <= 1e-289) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * 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 * im) <= 1d-289) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im * im) <= 1e-289) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im * im) <= 1e-289:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(im * im) <= 1e-289)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im * im) <= 1e-289)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(im * im), $MachinePrecision], 1e-289], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 im im) < 1e-289

   1. Initial program 45.4%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 48.8

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

   5. Applied rewrites 48.8%

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

   7. Applied rewrites 48.8%

      \[\leadsto \sqrt{re} \]

   if 1e-289 < (*.f64 im im)

   1. Initial program 47.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 85.3

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

   4. Applied rewrites 85.3%

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

   5. Taylor expanded in re around 0

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

      1. lower-*.f64 35.2

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

   7. Applied rewrites 35.2%

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

Alternative 7: 30.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -4e-310) 0.0 (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4e-310) {
		tmp = 0.0;
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4d-310)) then
        tmp = 0.0d0
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4e-310) {
		tmp = 0.0;
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4e-310:
		tmp = 0.0
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e-310)
		tmp = 0.0;
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4e-310)
		tmp = 0.0;
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4e-310], 0.0, N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.999999999999988e-310

   1. Initial program 27.2%

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

   4. Applied rewrites 9.0%

      \[\leadsto \color{blue}{0} \]

   if -3.999999999999988e-310 < re

   1. Initial program 65.5%

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

   3. Taylor expanded in re around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\sqrt{re} \cdot {\left(\sqrt{2}\right)}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 50.4%

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

Alternative 8: 6.1% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 47.1%

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

4. Applied rewrites 5.9%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Developer Target 1: 48.7% accurate, 0.6× 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 2024326 
(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)))))