math.sqrt on complex, real part

Percentage Accurate: 40.5% → 91.1%

Time: 11.1s

Alternatives: 15

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 40.5% 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: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{im \cdot \frac{im \cdot -2}{re - \mathsf{hypot}\left(re, im\right)}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-240)
   (* (sqrt (* im (/ (* im -2.0) (- re (hypot re im))))) 0.5)
   (* 0.5 (sqrt (* 2.0 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-240) {
		tmp = sqrt((im * ((im * -2.0) / (re - hypot(re, im))))) * 0.5;
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e-240) {
		tmp = Math.sqrt((im * ((im * -2.0) / (re - Math.hypot(re, im))))) * 0.5;
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-240:
		tmp = math.sqrt((im * ((im * -2.0) / (re - math.hypot(re, im))))) * 0.5
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-240)
		tmp = Float64(sqrt(Float64(im * Float64(Float64(im * -2.0) / Float64(re - hypot(re, im))))) * 0.5);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + hypot(re, im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-240)
		tmp = sqrt((im * ((im * -2.0) / (re - hypot(re, im))))) * 0.5;
	else
		tmp = 0.5 * sqrt((2.0 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-240], N[(N[Sqrt[N[(im * N[(N[(im * -2.0), $MachinePrecision] / N[(re - N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.9999999999999999e-240

   1. Initial program 24.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 24.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 24.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 24.0

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

   4. Applied rewrites 24.0%

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

   6. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(\sqrt{\left(-im \cdot im\right) \cdot -2} \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lower-hypot.f64 82.6

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

   10. Applied rewrites 82.6%

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

   if -1.9999999999999999e-240 < re

   1. Initial program 57.5%

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

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{im \cdot \frac{im \cdot -2}{re - \mathsf{hypot}\left(re, im\right)}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 5.0

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 5.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 5.0

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

   4. Applied rewrites 5.0%

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

   6. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\left(\sqrt{\left(-im \cdot im\right) \cdot -2} \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 55.4%

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

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

   1. Initial program 46.9%

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

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 88.6

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

   4. Applied rewrites 88.6%

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

4. Final simplification 85.5%

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

Alternative 3: 54.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq -3.9 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{-2 \cdot \left(-im\right)} \cdot \sqrt{im}\right) \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e+150)
   (* 0.5 (* (sqrt (* -2.0 (- (* im im)))) (sqrt (/ -1.0 (- re (- re))))))
   (if (<= re -1.3e+119)
     (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
     (if (<= re -3.9e-100)
       (*
        0.5
        (*
         (* (sqrt (* -2.0 (- im))) (sqrt im))
         (sqrt (/ -1.0 (- re (sqrt (fma re re (* im im))))))))
       (if (<= re 2.65e-21)
         (* 0.5 (sqrt (* 2.0 (+ re im))))
         (if (<= re 2.6e+123)
           (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im im (* re re)))))))
           (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * (sqrt((-2.0 * -(im * im))) * sqrt((-1.0 / (re - -re))));
	} else if (re <= -1.3e+119) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= -3.9e-100) {
		tmp = 0.5 * ((sqrt((-2.0 * -im)) * sqrt(im)) * sqrt((-1.0 / (re - sqrt(fma(re, re, (im * im)))))));
	} else if (re <= 2.65e-21) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else if (re <= 2.6e+123) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im, im, (re * re))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * Float64(sqrt(Float64(-2.0 * Float64(-Float64(im * im)))) * sqrt(Float64(-1.0 / Float64(re - Float64(-re))))));
	elseif (re <= -1.3e+119)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= -3.9e-100)
		tmp = Float64(0.5 * Float64(Float64(sqrt(Float64(-2.0 * Float64(-im))) * sqrt(im)) * sqrt(Float64(-1.0 / Float64(re - sqrt(fma(re, re, Float64(im * im))))))));
	elseif (re <= 2.65e-21)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	elseif (re <= 2.6e+123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im, im, Float64(re * re)))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e+150], N[(0.5 * N[(N[Sqrt[N[(-2.0 * (-N[(im * im), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(re - (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.3e+119], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -3.9e-100], N[(0.5 * N[(N[(N[Sqrt[N[(-2.0 * (-im)), $MachinePrecision]], $MachinePrecision] * N[Sqrt[im], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(re - N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.65e-21], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+123], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 2.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 2.5

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

   4. Applied rewrites 2.5%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.5

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

   9. Applied rewrites 75.5%

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

   if -1.1e150 < re < -1.3e119

   1. Initial program 13.9%

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

   3. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if -1.3e119 < re < -3.89999999999999977e-100

   1. Initial program 30.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 30.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 30.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 30.4

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

   4. Applied rewrites 30.4%

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

   6. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\left(\sqrt{\left(-im \cdot im\right) \cdot -2} \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-*r* N/A

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

      8. sqrt-prod N/A

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

      9. unpow1/2 N/A

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

      10. exp-to-pow N/A

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

      11. lift-log.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 36.6

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

      18. lift-exp.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-log.f64 N/A

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

      21. exp-to-pow N/A

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

      22. unpow1/2 N/A

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

      23. lower-sqrt.f64 38.6

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

   8. Applied rewrites 38.6%

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

   if -3.89999999999999977e-100 < re < 2.65e-21

   1. Initial program 59.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 45.8

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

   5. Applied rewrites 45.8%

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

   if 2.65e-21 < re < 2.59999999999999985e123

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 87.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 2.59999999999999985e123 < re

   1. Initial program 10.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.4

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

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq -3.9 \cdot 10^{-100}:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{-2 \cdot \left(-im\right)} \cdot \sqrt{im}\right) \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq -1.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{0.5}{\sqrt{\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re}} \cdot \left(im \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e+150)
   (* 0.5 (* (sqrt (* -2.0 (- (* im im)))) (sqrt (/ -1.0 (- re (- re))))))
   (if (<= re -4.2e+119)
     (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
     (if (<= re -1.5e-145)
       (* (/ 0.5 (sqrt (- (sqrt (fma re re (* im im))) re))) (* im (sqrt 2.0)))
       (if (<= re 2.65e-21)
         (* 0.5 (sqrt (* 2.0 (+ re im))))
         (if (<= re 2.6e+123)
           (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im im (* re re)))))))
           (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * (sqrt((-2.0 * -(im * im))) * sqrt((-1.0 / (re - -re))));
	} else if (re <= -4.2e+119) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= -1.5e-145) {
		tmp = (0.5 / sqrt((sqrt(fma(re, re, (im * im))) - re))) * (im * sqrt(2.0));
	} else if (re <= 2.65e-21) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else if (re <= 2.6e+123) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im, im, (re * re))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * Float64(sqrt(Float64(-2.0 * Float64(-Float64(im * im)))) * sqrt(Float64(-1.0 / Float64(re - Float64(-re))))));
	elseif (re <= -4.2e+119)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= -1.5e-145)
		tmp = Float64(Float64(0.5 / sqrt(Float64(sqrt(fma(re, re, Float64(im * im))) - re))) * Float64(im * sqrt(2.0)));
	elseif (re <= 2.65e-21)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	elseif (re <= 2.6e+123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im, im, Float64(re * re)))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e+150], N[(0.5 * N[(N[Sqrt[N[(-2.0 * (-N[(im * im), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(re - (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -4.2e+119], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.5e-145], N[(N[(0.5 / N[Sqrt[N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(im * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.65e-21], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+123], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 2.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 2.5

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

   4. Applied rewrites 2.5%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.5

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

   9. Applied rewrites 75.5%

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

   if -1.1e150 < re < -4.19999999999999966e119

   1. Initial program 13.9%

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

   3. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if -4.19999999999999966e119 < re < -1.49999999999999996e-145

   1. Initial program 32.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.2

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 32.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 32.2

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

   4. Applied rewrites 32.2%

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

   6. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\left(\sqrt{\left(-im \cdot im\right) \cdot -2} \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 38.6%

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

   if -1.49999999999999996e-145 < re < 2.65e-21

   1. Initial program 59.5%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if 2.65e-21 < re < 2.59999999999999985e123

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 87.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 2.59999999999999985e123 < re

   1. Initial program 10.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.4

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

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq -4.2 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq -1.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{0.5}{\sqrt{\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re}} \cdot \left(im \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq -0.01:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im \cdot -2}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re + im}\right)\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e+150)
   (* 0.5 (* (sqrt (* -2.0 (- (* im im)))) (sqrt (/ -1.0 (- re (- re))))))
   (if (<= re -1.3e+119)
     (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
     (if (<= re -0.01)
       (*
        0.5
        (sqrt (* im (/ (* im -2.0) (- re (sqrt (fma re re (* im im))))))))
       (if (<= re 2.65e-21)
         (* 0.5 (* (sqrt 2.0) (sqrt (+ re im))))
         (if (<= re 2.6e+123)
           (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im im (* re re)))))))
           (sqrt re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * (sqrt((-2.0 * -(im * im))) * sqrt((-1.0 / (re - -re))));
	} else if (re <= -1.3e+119) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= -0.01) {
		tmp = 0.5 * sqrt((im * ((im * -2.0) / (re - sqrt(fma(re, re, (im * im)))))));
	} else if (re <= 2.65e-21) {
		tmp = 0.5 * (sqrt(2.0) * sqrt((re + im)));
	} else if (re <= 2.6e+123) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im, im, (re * re))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * Float64(sqrt(Float64(-2.0 * Float64(-Float64(im * im)))) * sqrt(Float64(-1.0 / Float64(re - Float64(-re))))));
	elseif (re <= -1.3e+119)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= -0.01)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(Float64(im * -2.0) / Float64(re - sqrt(fma(re, re, Float64(im * im))))))));
	elseif (re <= 2.65e-21)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(re + im))));
	elseif (re <= 2.6e+123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im, im, Float64(re * re)))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e+150], N[(0.5 * N[(N[Sqrt[N[(-2.0 * (-N[(im * im), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(re - (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -1.3e+119], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -0.01], N[(0.5 * N[Sqrt[N[(im * N[(N[(im * -2.0), $MachinePrecision] / N[(re - N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.65e-21], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(re + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+123], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 2.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 2.5

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

   4. Applied rewrites 2.5%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.5

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

   9. Applied rewrites 75.5%

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

   if -1.1e150 < re < -1.3e119

   1. Initial program 13.9%

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

   3. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if -1.3e119 < re < -0.0100000000000000002

   1. Initial program 31.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.1

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 31.1

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 31.1

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

   4. Applied rewrites 31.1%

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

   6. Applied rewrites 77.8%

      \[\leadsto \color{blue}{\left(\sqrt{\left(-im \cdot im\right) \cdot -2} \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

   8. Applied rewrites 75.2%

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

   if -0.0100000000000000002 < re < 2.65e-21

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.8

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 54.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 54.8

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 42.4

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

   7. Applied rewrites 42.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 42.1

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

   9. Applied rewrites 42.1%

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

   if 2.65e-21 < re < 2.59999999999999985e123

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 87.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 2.59999999999999985e123 < re

   1. Initial program 10.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.4

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

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq -1.3 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq -0.01:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im \cdot -2}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re + im}\right)\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\\ \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq -6 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq -0.01:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-2 \cdot \left(im \cdot im\right)}{re - t\_0}}\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re + im}\right)\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (fma im im (* re re)))))
   (if (<= re -1.1e+150)
     (* 0.5 (* (sqrt (* -2.0 (- (* im im)))) (sqrt (/ -1.0 (- re (- re))))))
     (if (<= re -6e+118)
       (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
       (if (<= re -0.01)
         (* 0.5 (sqrt (/ (* -2.0 (* im im)) (- re t_0))))
         (if (<= re 2.65e-21)
           (* 0.5 (* (sqrt 2.0) (sqrt (+ re im))))
           (if (<= re 2.6e+123)
             (* 0.5 (sqrt (* 2.0 (+ re t_0))))
             (sqrt re))))))))

C

double code(double re, double im) {
	double t_0 = sqrt(fma(im, im, (re * re)));
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * (sqrt((-2.0 * -(im * im))) * sqrt((-1.0 / (re - -re))));
	} else if (re <= -6e+118) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= -0.01) {
		tmp = 0.5 * sqrt(((-2.0 * (im * im)) / (re - t_0)));
	} else if (re <= 2.65e-21) {
		tmp = 0.5 * (sqrt(2.0) * sqrt((re + im)));
	} else if (re <= 2.6e+123) {
		tmp = 0.5 * sqrt((2.0 * (re + t_0)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = sqrt(fma(im, im, Float64(re * re)))
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * Float64(sqrt(Float64(-2.0 * Float64(-Float64(im * im)))) * sqrt(Float64(-1.0 / Float64(re - Float64(-re))))));
	elseif (re <= -6e+118)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= -0.01)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-2.0 * Float64(im * im)) / Float64(re - t_0))));
	elseif (re <= 2.65e-21)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(re + im))));
	elseif (re <= 2.6e+123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + t_0))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -1.1e+150], N[(0.5 * N[(N[Sqrt[N[(-2.0 * (-N[(im * im), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(re - (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -6e+118], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, -0.01], N[(0.5 * N[Sqrt[N[(N[(-2.0 * N[(im * im), $MachinePrecision]), $MachinePrecision] / N[(re - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.65e-21], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(re + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+123], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 2.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 2.5

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

   4. Applied rewrites 2.5%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.5

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

   9. Applied rewrites 75.5%

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

   if -1.1e150 < re < -6e118

   1. Initial program 13.9%

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

   3. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

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

   if -6e118 < re < -0.0100000000000000002

   1. Initial program 31.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 30.4%

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

   5. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -0.0100000000000000002 < re < 2.65e-21

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.8

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 54.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 54.8

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 42.4

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

   7. Applied rewrites 42.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 42.1

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

   9. Applied rewrites 42.1%

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

   if 2.65e-21 < re < 2.59999999999999985e123

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 87.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 2.59999999999999985e123 < re

   1. Initial program 10.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.4

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

   5. Applied rewrites 85.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq -6 \cdot 10^{+118}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq -0.01:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{-2 \cdot \left(im \cdot im\right)}{re - \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}}}\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re + im}\right)\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e+150)
   (* 0.5 (* (sqrt (* -2.0 (- (* im im)))) (sqrt (/ -1.0 (- re (- re))))))
   (if (<= re 2.65e-21)
     (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
     (if (<= re 2.6e+123)
       (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im im (* re re)))))))
       (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * (sqrt((-2.0 * -(im * im))) * sqrt((-1.0 / (re - -re))));
	} else if (re <= 2.65e-21) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= 2.6e+123) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im, im, (re * re))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * Float64(sqrt(Float64(-2.0 * Float64(-Float64(im * im)))) * sqrt(Float64(-1.0 / Float64(re - Float64(-re))))));
	elseif (re <= 2.65e-21)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= 2.6e+123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im, im, Float64(re * re)))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e+150], N[(0.5 * N[(N[Sqrt[N[(-2.0 * (-N[(im * im), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-1.0 / N[(re - (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.65e-21], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+123], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 2.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 2.5

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

   4. Applied rewrites 2.5%

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

   6. Applied rewrites 26.1%

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

   7. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.5

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

   9. Applied rewrites 75.5%

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

   if -1.1e150 < re < 2.65e-21

   1. Initial program 47.9%

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

   3. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 38.8

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

   5. Applied rewrites 38.8%

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

   if 2.65e-21 < re < 2.59999999999999985e123

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 87.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 2.59999999999999985e123 < re

   1. Initial program 10.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{-2 \cdot \left(-im \cdot im\right)} \cdot \sqrt{\frac{-1}{re - \left(-re\right)}}\right)\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e+150)
   (* 0.5 (sqrt (* im (/ im (- re)))))
   (if (<= re 2.65e-21)
     (* 0.5 (sqrt (fma 2.0 (+ re im) (/ (* re re) im))))
     (if (<= re 2.6e+123)
       (* 0.5 (sqrt (* 2.0 (+ re (sqrt (fma im im (* re re)))))))
       (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * sqrt((im * (im / -re)));
	} else if (re <= 2.65e-21) {
		tmp = 0.5 * sqrt(fma(2.0, (re + im), ((re * re) / im)));
	} else if (re <= 2.6e+123) {
		tmp = 0.5 * sqrt((2.0 * (re + sqrt(fma(im, im, (re * re))))));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / Float64(-re)))));
	elseif (re <= 2.65e-21)
		tmp = Float64(0.5 * sqrt(fma(2.0, Float64(re + im), Float64(Float64(re * re) / im))));
	elseif (re <= 2.6e+123)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + sqrt(fma(im, im, Float64(re * re)))))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e+150], N[(0.5 * N[Sqrt[N[(im * N[(im / (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.65e-21], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision] + N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.6e+123], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 2.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 2.5

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

   4. Applied rewrites 2.5%

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

   6. Applied rewrites 26.1%

      \[\leadsto \color{blue}{\left(\sqrt{\left(-im \cdot im\right) \cdot -2} \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 70.3

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

   11. Applied rewrites 70.3%

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

   if -1.1e150 < re < 2.65e-21

   1. Initial program 47.9%

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

   3. Taylor expanded in re around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-+r+ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 38.8

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

   5. Applied rewrites 38.8%

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

   if 2.65e-21 < re < 2.59999999999999985e123

   1. Initial program 87.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.4

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 87.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 2.59999999999999985e123 < re

   1. Initial program 10.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.4

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

   5. Applied rewrites 85.4%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq 2.65 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(2, re + im, \frac{re \cdot re}{im}\right)}\\ \mathbf{elif}\;re \leq 2.6 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re + im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e+150)
   (* 0.5 (sqrt (* im (/ im (- re)))))
   (if (<= re 1.12e+52) (* 0.5 (* (sqrt 2.0) (sqrt (+ re im)))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * sqrt((im * (im / -re)));
	} else if (re <= 1.12e+52) {
		tmp = 0.5 * (sqrt(2.0) * sqrt((re + im)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.1d+150)) then
        tmp = 0.5d0 * sqrt((im * (im / -re)))
    else if (re <= 1.12d+52) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt((re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * Math.sqrt((im * (im / -re)));
	} else if (re <= 1.12e+52) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt((re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.1e+150:
		tmp = 0.5 * math.sqrt((im * (im / -re)))
	elif re <= 1.12e+52:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt((re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / Float64(-re)))));
	elseif (re <= 1.12e+52)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.1e+150)
		tmp = 0.5 * sqrt((im * (im / -re)));
	elseif (re <= 1.12e+52)
		tmp = 0.5 * (sqrt(2.0) * sqrt((re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e+150], N[(0.5 * N[Sqrt[N[(im * N[(im / (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.12e+52], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(re + im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 2.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 2.5

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

   4. Applied rewrites 2.5%

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

   6. Applied rewrites 26.1%

      \[\leadsto \color{blue}{\left(\sqrt{\left(-im \cdot im\right) \cdot -2} \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 70.3

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

   11. Applied rewrites 70.3%

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

   if -1.1e150 < re < 1.12000000000000002e52

   1. Initial program 51.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.7

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 51.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 51.7

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

   4. Applied rewrites 51.7%

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

   5. Taylor expanded in re around 0

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

      1. lower-+.f64 39.4

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

   7. Applied rewrites 39.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 39.2

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

   9. Applied rewrites 39.2%

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

   if 1.12000000000000002e52 < re

   1. Initial program 41.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{re + im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e+150)
   (* 0.5 (sqrt (* im (/ im (- re)))))
   (if (<= re 1.12e+52) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * sqrt((im * (im / -re)));
	} else if (re <= 1.12e+52) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.1d+150)) then
        tmp = 0.5d0 * sqrt((im * (im / -re)))
    else if (re <= 1.12d+52) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * Math.sqrt((im * (im / -re)));
	} else if (re <= 1.12e+52) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.1e+150:
		tmp = 0.5 * math.sqrt((im * (im / -re)))
	elif re <= 1.12e+52:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(im / Float64(-re)))));
	elseif (re <= 1.12e+52)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.1e+150)
		tmp = 0.5 * sqrt((im * (im / -re)));
	elseif (re <= 1.12e+52)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e+150], N[(0.5 * N[Sqrt[N[(im * N[(im / (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.12e+52], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 2.5

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 2.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 2.5

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

   4. Applied rewrites 2.5%

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

   6. Applied rewrites 26.1%

      \[\leadsto \color{blue}{\left(\sqrt{\left(-im \cdot im\right) \cdot -2} \cdot \sqrt{\frac{-1}{re - \sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)}}}\right)} \cdot 0.5 \]
   7. Step-by-step derivation

   8. Applied rewrites 26.5%

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

   9. Taylor expanded in re around -inf

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 70.3

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

   11. Applied rewrites 70.3%

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

   if -1.1e150 < re < 1.12000000000000002e52

   1. Initial program 51.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   if 1.12000000000000002e52 < re

   1. Initial program 41.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \frac{im}{-re}}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.1e+150)
   (* 0.5 (sqrt (/ (* im im) (- re))))
   (if (<= re 1.12e+52) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * sqrt(((im * im) / -re));
	} else if (re <= 1.12e+52) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.1d+150)) then
        tmp = 0.5d0 * sqrt(((im * im) / -re))
    else if (re <= 1.12d+52) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.1e+150) {
		tmp = 0.5 * Math.sqrt(((im * im) / -re));
	} else if (re <= 1.12e+52) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.1e+150:
		tmp = 0.5 * math.sqrt(((im * im) / -re))
	elif re <= 1.12e+52:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.1e+150)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * im) / Float64(-re))));
	elseif (re <= 1.12e+52)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.1e+150)
		tmp = 0.5 * sqrt(((im * im) / -re));
	elseif (re <= 1.12e+52)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.1e+150], N[(0.5 * N[Sqrt[N[(N[(im * im), $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.12e+52], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.1e150

   1. Initial program 2.5%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   4. 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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if -1.1e150 < re < 1.12000000000000002e52

   1. Initial program 51.7%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   if 1.12000000000000002e52 < re

   1. Initial program 41.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.1 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot im}{-re}}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.55 \cdot 10^{+179}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(-re\right)\right)}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.55e+179)
   (* 0.5 (sqrt (* 2.0 (+ re (- re)))))
   (if (<= re 1.12e+52) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.55e+179) {
		tmp = 0.5 * sqrt((2.0 * (re + -re)));
	} else if (re <= 1.12e+52) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.55d+179)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + -re)))
    else if (re <= 1.12d+52) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.55e+179) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + -re)));
	} else if (re <= 1.12e+52) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.55e+179:
		tmp = 0.5 * math.sqrt((2.0 * (re + -re)))
	elif re <= 1.12e+52:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.55e+179)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + Float64(-re)))));
	elseif (re <= 1.12e+52)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.55e+179)
		tmp = 0.5 * sqrt((2.0 * (re + -re)));
	elseif (re <= 1.12e+52)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.55e+179], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + (-re)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.12e+52], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.55e179

   1. Initial program 2.5%

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

   3. Taylor expanded in re around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.9

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

   5. Applied rewrites 30.9%

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

   if -1.55e179 < re < 1.12000000000000002e52

   1. Initial program 50.1%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 38.1

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

   5. Applied rewrites 38.1%

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

   if 1.12000000000000002e52 < re

   1. Initial program 41.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.55 \cdot 10^{+179}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + \left(-re\right)\right)}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.12e+52) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.12e+52) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.12d+52) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.12e+52) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.12e+52:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.12e+52)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.12e+52)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.12e+52], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.12000000000000002e52

   1. Initial program 43.4%

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

   3. Taylor expanded in re around 0

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

      1. lower-+.f64 32.8

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

   5. Applied rewrites 32.8%

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

   if 1.12000000000000002e52 < re

   1. Initial program 41.4%

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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.12 \cdot 10^{+52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7 \cdot 10^{+38}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7e+38) (* 0.5 (sqrt (* im 2.0))) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7e+38) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 7d+38) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 7e+38) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 7e+38:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 7e+38)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 7e+38)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 7e+38], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 7.00000000000000003e38

   1. Initial program 42.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 31.6

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

   5. Applied rewrites 31.6%

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

   if 7.00000000000000003e38 < re

   1. Initial program 45.8%

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

   3. Taylor expanded in re around 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

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. lower-sqrt.f64 82.9

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

   5. Applied rewrites 82.9%

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

Alternative 15: 27.0% accurate, 4.3× 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 43.0%

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

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. *-lft-identity N/A

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

   7. lower-sqrt.f64 26.9

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

5. Applied rewrites 26.9%

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

Developer Target 1: 47.3% 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 2024233 
(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)))))