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

Percentage Accurate: 40.8% → 89.9%

Time: 10.8s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.8% 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: 89.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 11.5%

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

      1. add-sqr-sqrt 11.5%

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

      2. sqrt-unprod 11.5%

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

      3. *-commutative 11.5%

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

      4. *-commutative 11.5%

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

      5. swap-sqr 11.5%

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

      6. add-sqr-sqrt 11.5%

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

      7. *-commutative 11.5%

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

      8. hypot-define 18.3%

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

      9. metadata-eval 18.3%

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

   4. Applied egg-rr 18.3%

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

      1. associate-*l* 18.3%

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

      2. metadata-eval 18.3%

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

   6. Simplified 18.3%

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

   7. Taylor expanded in re around inf 92.8%

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

      1. associate-*l* 92.8%

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

      2. unpow2 92.8%

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

      3. rem-square-sqrt 93.9%

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

      4. rem-exp-log 88.7%

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

      5. exp-neg 88.7%

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

      6. unpow1/2 88.7%

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

      7. exp-prod 88.7%

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

      8. distribute-lft-neg-out 88.7%

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

      9. distribute-rgt-neg-in 88.7%

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

      10. metadata-eval 88.7%

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

      11. exp-to-pow 94.1%

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

   9. Simplified 94.1%

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

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

   1. Initial program 45.5%

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

      1. add-sqr-sqrt 45.2%

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

      2. sqrt-unprod 45.5%

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

      3. *-commutative 45.5%

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

      4. *-commutative 45.5%

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

      5. swap-sqr 45.5%

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

      6. add-sqr-sqrt 45.5%

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

      7. *-commutative 45.5%

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

      8. hypot-define 91.3%

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

      9. metadata-eval 91.3%

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

   4. Applied egg-rr 91.3%

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

      1. associate-*l* 91.3%

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

      2. metadata-eval 91.3%

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

   6. Simplified 91.3%

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

4. Final simplification 91.6%

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

Alternative 2: 76.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 1.66 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.25e-22)
   (sqrt (- re))
   (if (<= re 4.1e-10)
     (sqrt (* 0.5 (- im re)))
     (if (<= re 4.9e+62)
       (* 0.5 (/ im (sqrt re)))
       (if (<= re 1.66e+106)
         (sqrt (* im 0.5))
         (* im (* 0.5 (pow re -0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.25e-22) {
		tmp = sqrt(-re);
	} else if (re <= 4.1e-10) {
		tmp = sqrt((0.5 * (im - re)));
	} else if (re <= 4.9e+62) {
		tmp = 0.5 * (im / sqrt(re));
	} else if (re <= 1.66e+106) {
		tmp = sqrt((im * 0.5));
	} else {
		tmp = im * (0.5 * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.25d-22)) then
        tmp = sqrt(-re)
    else if (re <= 4.1d-10) then
        tmp = sqrt((0.5d0 * (im - re)))
    else if (re <= 4.9d+62) then
        tmp = 0.5d0 * (im / sqrt(re))
    else if (re <= 1.66d+106) then
        tmp = sqrt((im * 0.5d0))
    else
        tmp = im * (0.5d0 * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.25e-22) {
		tmp = Math.sqrt(-re);
	} else if (re <= 4.1e-10) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else if (re <= 4.9e+62) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else if (re <= 1.66e+106) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = im * (0.5 * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.25e-22:
		tmp = math.sqrt(-re)
	elif re <= 4.1e-10:
		tmp = math.sqrt((0.5 * (im - re)))
	elif re <= 4.9e+62:
		tmp = 0.5 * (im / math.sqrt(re))
	elif re <= 1.66e+106:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = im * (0.5 * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.25e-22)
		tmp = sqrt(Float64(-re));
	elseif (re <= 4.1e-10)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	elseif (re <= 4.9e+62)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	elseif (re <= 1.66e+106)
		tmp = sqrt(Float64(im * 0.5));
	else
		tmp = Float64(im * Float64(0.5 * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.25e-22)
		tmp = sqrt(-re);
	elseif (re <= 4.1e-10)
		tmp = sqrt((0.5 * (im - re)));
	elseif (re <= 4.9e+62)
		tmp = 0.5 * (im / sqrt(re));
	elseif (re <= 1.66e+106)
		tmp = sqrt((im * 0.5));
	else
		tmp = im * (0.5 * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.25e-22], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 4.1e-10], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[re, 4.9e+62], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.66e+106], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1.24999999999999988e-22

   1. Initial program 43.7%

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

      1. add-sqr-sqrt 43.3%

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

      2. sqrt-unprod 43.7%

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

      3. *-commutative 43.7%

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

      4. *-commutative 43.7%

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

      5. swap-sqr 43.7%

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

      6. add-sqr-sqrt 43.7%

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

      7. *-commutative 43.7%

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

      8. hypot-define 98.6%

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

      9. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. associate-*l* 98.6%

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

      2. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in re around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   9. Simplified 86.5%

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

   if -1.24999999999999988e-22 < re < 4.0999999999999998e-10

   1. Initial program 55.1%

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

      1. add-sqr-sqrt 54.7%

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

      2. sqrt-unprod 55.1%

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

      3. *-commutative 55.1%

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

      4. *-commutative 55.1%

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

      5. swap-sqr 55.1%

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

      6. add-sqr-sqrt 55.1%

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

      7. *-commutative 55.1%

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

      8. hypot-define 93.7%

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

      9. metadata-eval 93.7%

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

   4. Applied egg-rr 93.7%

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

      1. associate-*l* 93.7%

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

      2. metadata-eval 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in re around 0 86.1%

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

      1. neg-mul-1 86.1%

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

      2. unsub-neg 86.1%

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

   9. Simplified 86.1%

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

   if 4.0999999999999998e-10 < re < 4.8999999999999997e62

   1. Initial program 11.8%

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

   3. Taylor expanded in re around inf 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-*l* 77.2%

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

      3. *-commutative 77.2%

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

   5. Simplified 77.2%

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

      1. pow1 77.2%

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

      2. sqrt-unprod 78.2%

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

      3. metadata-eval 78.2%

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

      4. metadata-eval 78.2%

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

      5. *-un-lft-identity 78.2%

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

      6. sqrt-div 78.1%

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

      7. metadata-eval 78.1%

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

      8. un-div-inv 78.5%

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

   7. Applied egg-rr 78.5%

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

      1. unpow1 78.5%

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

   9. Simplified 78.5%

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

   if 4.8999999999999997e62 < re < 1.66e106

   1. Initial program 33.1%

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

      1. add-sqr-sqrt 33.0%

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

      2. sqrt-unprod 33.1%

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

      3. *-commutative 33.1%

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

      4. *-commutative 33.1%

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

      5. swap-sqr 33.1%

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

      6. add-sqr-sqrt 33.1%

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

      7. *-commutative 33.1%

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

      8. hypot-define 81.0%

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

      9. metadata-eval 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. associate-*l* 81.0%

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

      2. metadata-eval 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in re around 0 81.1%

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

   if 1.66e106 < re

   1. Initial program 10.9%

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

      1. add-sqr-sqrt 10.9%

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

      2. sqrt-unprod 10.9%

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

      3. *-commutative 10.9%

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

      4. *-commutative 10.9%

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

      5. swap-sqr 10.9%

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

      6. add-sqr-sqrt 10.9%

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

      7. *-commutative 10.9%

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

      8. hypot-define 43.1%

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

      9. metadata-eval 43.1%

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

   4. Applied egg-rr 43.1%

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

      1. associate-*l* 43.1%

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

      2. metadata-eval 43.1%

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

   6. Simplified 43.1%

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

   7. Taylor expanded in re around inf 81.0%

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

      1. associate-*l* 81.0%

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

      2. unpow2 81.0%

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

      3. rem-square-sqrt 81.8%

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

      4. rem-exp-log 77.1%

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

      5. exp-neg 77.1%

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

      6. unpow1/2 77.1%

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

      7. exp-prod 77.1%

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

      8. distribute-lft-neg-out 77.1%

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

      9. distribute-rgt-neg-in 77.1%

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

      10. metadata-eval 77.1%

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

      11. exp-to-pow 81.9%

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

   9. Simplified 81.9%

      \[\leadsto \color{blue}{im \cdot \left(0.5 \cdot {re}^{-0.5}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 1.66 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+62} \lor \neg \left(re \leq 1.2 \cdot 10^{+106}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.9e-24)
   (sqrt (- re))
   (if (<= re 2.1e-5)
     (sqrt (* 0.5 (- im re)))
     (if (or (<= re 1.05e+62) (not (<= re 1.2e+106)))
       (* 0.5 (/ im (sqrt re)))
       (sqrt (* im 0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.9e-24) {
		tmp = sqrt(-re);
	} else if (re <= 2.1e-5) {
		tmp = sqrt((0.5 * (im - re)));
	} else if ((re <= 1.05e+62) || !(re <= 1.2e+106)) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.9d-24)) then
        tmp = sqrt(-re)
    else if (re <= 2.1d-5) then
        tmp = sqrt((0.5d0 * (im - re)))
    else if ((re <= 1.05d+62) .or. (.not. (re <= 1.2d+106))) then
        tmp = 0.5d0 * (im / sqrt(re))
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.9e-24) {
		tmp = Math.sqrt(-re);
	} else if (re <= 2.1e-5) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else if ((re <= 1.05e+62) || !(re <= 1.2e+106)) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.9e-24:
		tmp = math.sqrt(-re)
	elif re <= 2.1e-5:
		tmp = math.sqrt((0.5 * (im - re)))
	elif (re <= 1.05e+62) or not (re <= 1.2e+106):
		tmp = 0.5 * (im / math.sqrt(re))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.9e-24)
		tmp = sqrt(Float64(-re));
	elseif (re <= 2.1e-5)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	elseif ((re <= 1.05e+62) || !(re <= 1.2e+106))
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.9e-24)
		tmp = sqrt(-re);
	elseif (re <= 2.1e-5)
		tmp = sqrt((0.5 * (im - re)));
	elseif ((re <= 1.05e+62) || ~((re <= 1.2e+106)))
		tmp = 0.5 * (im / sqrt(re));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.9e-24], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 2.1e-5], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[re, 1.05e+62], N[Not[LessEqual[re, 1.2e+106]], $MachinePrecision]], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -3.9e-24

   1. Initial program 43.7%

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

      1. add-sqr-sqrt 43.3%

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

      2. sqrt-unprod 43.7%

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

      3. *-commutative 43.7%

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

      4. *-commutative 43.7%

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

      5. swap-sqr 43.7%

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

      6. add-sqr-sqrt 43.7%

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

      7. *-commutative 43.7%

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

      8. hypot-define 98.6%

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

      9. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. associate-*l* 98.6%

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

      2. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in re around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   9. Simplified 86.5%

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

   if -3.9e-24 < re < 2.09999999999999988e-5

   1. Initial program 55.1%

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

      1. add-sqr-sqrt 54.7%

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

      2. sqrt-unprod 55.1%

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

      3. *-commutative 55.1%

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

      4. *-commutative 55.1%

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

      5. swap-sqr 55.1%

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

      6. add-sqr-sqrt 55.1%

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

      7. *-commutative 55.1%

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

      8. hypot-define 93.7%

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

      9. metadata-eval 93.7%

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

   4. Applied egg-rr 93.7%

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

      1. associate-*l* 93.7%

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

      2. metadata-eval 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in re around 0 86.1%

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

      1. neg-mul-1 86.1%

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

      2. unsub-neg 86.1%

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

   9. Simplified 86.1%

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

   if 2.09999999999999988e-5 < re < 1.05e62 or 1.2e106 < re

   1. Initial program 11.1%

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

   3. Taylor expanded in re around inf 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*l* 80.1%

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

      3. *-commutative 80.1%

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

   5. Simplified 80.1%

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

      1. pow1 80.1%

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

      2. sqrt-unprod 80.9%

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

      3. metadata-eval 80.9%

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

      4. metadata-eval 80.9%

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

      5. *-un-lft-identity 80.9%

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

      6. sqrt-div 80.9%

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

      7. metadata-eval 80.9%

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

      8. un-div-inv 81.0%

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

   7. Applied egg-rr 81.0%

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

      1. unpow1 81.0%

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

   9. Simplified 81.0%

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

   if 1.05e62 < re < 1.2e106

   1. Initial program 33.1%

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

      1. add-sqr-sqrt 33.0%

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

      2. sqrt-unprod 33.1%

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

      3. *-commutative 33.1%

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

      4. *-commutative 33.1%

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

      5. swap-sqr 33.1%

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

      6. add-sqr-sqrt 33.1%

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

      7. *-commutative 33.1%

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

      8. hypot-define 81.0%

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

      9. metadata-eval 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. associate-*l* 81.0%

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

      2. metadata-eval 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in re around 0 81.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.9 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.05 \cdot 10^{+62} \lor \neg \left(re \leq 1.2 \cdot 10^{+106}\right):\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4e-23)
   (sqrt (- re))
   (if (<= re 5e-8)
     (sqrt (* 0.5 (- im re)))
     (if (<= re 3.2e+62)
       (* 0.5 (/ im (sqrt re)))
       (if (<= re 6.6e+104) (sqrt (* im 0.5)) (* im (sqrt (/ 0.25 re))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4e-23) {
		tmp = sqrt(-re);
	} else if (re <= 5e-8) {
		tmp = sqrt((0.5 * (im - re)));
	} else if (re <= 3.2e+62) {
		tmp = 0.5 * (im / sqrt(re));
	} else if (re <= 6.6e+104) {
		tmp = sqrt((im * 0.5));
	} else {
		tmp = im * sqrt((0.25 / re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4d-23)) then
        tmp = sqrt(-re)
    else if (re <= 5d-8) then
        tmp = sqrt((0.5d0 * (im - re)))
    else if (re <= 3.2d+62) then
        tmp = 0.5d0 * (im / sqrt(re))
    else if (re <= 6.6d+104) then
        tmp = sqrt((im * 0.5d0))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4e-23) {
		tmp = Math.sqrt(-re);
	} else if (re <= 5e-8) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else if (re <= 3.2e+62) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else if (re <= 6.6e+104) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4e-23:
		tmp = math.sqrt(-re)
	elif re <= 5e-8:
		tmp = math.sqrt((0.5 * (im - re)))
	elif re <= 3.2e+62:
		tmp = 0.5 * (im / math.sqrt(re))
	elif re <= 6.6e+104:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e-23)
		tmp = sqrt(Float64(-re));
	elseif (re <= 5e-8)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	elseif (re <= 3.2e+62)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	elseif (re <= 6.6e+104)
		tmp = sqrt(Float64(im * 0.5));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4e-23)
		tmp = sqrt(-re);
	elseif (re <= 5e-8)
		tmp = sqrt((0.5 * (im - re)));
	elseif (re <= 3.2e+62)
		tmp = 0.5 * (im / sqrt(re));
	elseif (re <= 6.6e+104)
		tmp = sqrt((im * 0.5));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4e-23], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 5e-8], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[re, 3.2e+62], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.6e+104], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -3.99999999999999984e-23

   1. Initial program 43.7%

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

      1. add-sqr-sqrt 43.3%

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

      2. sqrt-unprod 43.7%

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

      3. *-commutative 43.7%

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

      4. *-commutative 43.7%

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

      5. swap-sqr 43.7%

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

      6. add-sqr-sqrt 43.7%

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

      7. *-commutative 43.7%

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

      8. hypot-define 98.6%

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

      9. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. associate-*l* 98.6%

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

      2. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in re around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   9. Simplified 86.5%

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

   if -3.99999999999999984e-23 < re < 4.9999999999999998e-8

   1. Initial program 55.1%

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

      1. add-sqr-sqrt 54.7%

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

      2. sqrt-unprod 55.1%

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

      3. *-commutative 55.1%

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

      4. *-commutative 55.1%

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

      5. swap-sqr 55.1%

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

      6. add-sqr-sqrt 55.1%

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

      7. *-commutative 55.1%

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

      8. hypot-define 93.7%

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

      9. metadata-eval 93.7%

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

   4. Applied egg-rr 93.7%

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

      1. associate-*l* 93.7%

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

      2. metadata-eval 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in re around 0 86.1%

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

      1. neg-mul-1 86.1%

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

      2. unsub-neg 86.1%

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

   9. Simplified 86.1%

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

   if 4.9999999999999998e-8 < re < 3.19999999999999984e62

   1. Initial program 11.8%

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

   3. Taylor expanded in re around inf 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-*l* 77.2%

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

      3. *-commutative 77.2%

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

   5. Simplified 77.2%

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

      1. pow1 77.2%

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

      2. sqrt-unprod 78.2%

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

      3. metadata-eval 78.2%

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

      4. metadata-eval 78.2%

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

      5. *-un-lft-identity 78.2%

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

      6. sqrt-div 78.1%

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

      7. metadata-eval 78.1%

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

      8. un-div-inv 78.5%

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

   7. Applied egg-rr 78.5%

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

      1. unpow1 78.5%

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

   9. Simplified 78.5%

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

   if 3.19999999999999984e62 < re < 6.59999999999999969e104

   1. Initial program 33.1%

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

      1. add-sqr-sqrt 33.0%

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

      2. sqrt-unprod 33.1%

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

      3. *-commutative 33.1%

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

      4. *-commutative 33.1%

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

      5. swap-sqr 33.1%

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

      6. add-sqr-sqrt 33.1%

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

      7. *-commutative 33.1%

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

      8. hypot-define 81.0%

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

      9. metadata-eval 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. associate-*l* 81.0%

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

      2. metadata-eval 81.0%

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

   6. Simplified 81.0%

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

   7. Taylor expanded in re around 0 81.1%

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

   if 6.59999999999999969e104 < re

   1. Initial program 10.9%

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

   3. Taylor expanded in re around inf 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-*l* 81.0%

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

      3. *-commutative 81.0%

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

   5. Simplified 81.0%

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

      1. pow1 81.0%

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

      2. sqrt-unprod 81.7%

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

      3. metadata-eval 81.7%

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

      4. metadata-eval 81.7%

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

      5. *-un-lft-identity 81.7%

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

      6. sqrt-div 81.7%

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

      7. metadata-eval 81.7%

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

      8. un-div-inv 81.7%

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

   7. Applied egg-rr 81.7%

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

      1. unpow1 81.7%

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

      2. associate-*r/ 81.7%

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

      3. *-commutative 81.7%

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

      4. associate-/l* 81.8%

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

   9. Simplified 81.8%

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

       1. add-sqr-sqrt 81.5%

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

       2. sqrt-unprod 81.8%

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

       3. frac-times 81.7%

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

       4. metadata-eval 81.7%

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

       5. add-sqr-sqrt 81.8%

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

   11. Applied egg-rr 81.8%

       \[\leadsto im \cdot \color{blue}{\sqrt{\frac{0.25}{re}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 6.6 \cdot 10^{+104}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.2e-23) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.2e-23) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-9.2d-23)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9.2e-23) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.2e-23:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.2e-23)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.2e-23)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.2e-23], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -9.2000000000000004e-23

   1. Initial program 43.7%

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

      1. add-sqr-sqrt 43.3%

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

      2. sqrt-unprod 43.7%

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

      3. *-commutative 43.7%

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

      4. *-commutative 43.7%

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

      5. swap-sqr 43.7%

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

      6. add-sqr-sqrt 43.7%

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

      7. *-commutative 43.7%

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

      8. hypot-define 98.6%

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

      9. metadata-eval 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. associate-*l* 98.6%

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

      2. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in re around -inf 86.5%

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

      1. neg-mul-1 86.5%

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

   9. Simplified 86.5%

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

   if -9.2000000000000004e-23 < re

   1. Initial program 40.4%

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

      1. add-sqr-sqrt 40.1%

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

      2. sqrt-unprod 40.4%

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

      3. *-commutative 40.4%

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

      4. *-commutative 40.4%

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

      5. swap-sqr 40.4%

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

      6. add-sqr-sqrt 40.4%

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

      7. *-commutative 40.4%

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

      8. hypot-define 76.2%

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

      9. metadata-eval 76.2%

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

   4. Applied egg-rr 76.2%

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

      1. associate-*l* 76.2%

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

      2. metadata-eval 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in re around 0 66.2%

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

Alternative 6: 25.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(Float64(-re))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.3%

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

   1. add-sqr-sqrt 41.0%

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

   2. sqrt-unprod 41.3%

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

   3. *-commutative 41.3%

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

   4. *-commutative 41.3%

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

   5. swap-sqr 41.3%

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

   6. add-sqr-sqrt 41.3%

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

   7. *-commutative 41.3%

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

   8. hypot-define 82.2%

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

   9. metadata-eval 82.2%

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

4. Applied egg-rr 82.2%

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

   1. associate-*l* 82.2%

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

   2. metadata-eval 82.2%

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

6. Simplified 82.2%

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

7. Taylor expanded in re around -inf 27.6%

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

   1. neg-mul-1 27.6%

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

9. Simplified 27.6%

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

Reproduce

herbie shell --seed 2024100 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  :pre (> im 0.0)
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))