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

Percentage Accurate: 42.1% → 90.8%

Time: 13.0s

Alternatives: 6

Speedup: 2.1×

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: 42.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (/ (* im 0.5) (sqrt re))
   (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) / sqrt(re);
	} 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.sqrt(re);
	} 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.sqrt(re)
	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(Float64(im * 0.5) / sqrt(re));
	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) / sqrt(re);
	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[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $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 10.4%

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

      1. sub-neg 10.4%

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

      2. sqr-neg 10.4%

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

      3. sub-neg 10.4%

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

      4. sqr-neg 10.4%

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

      5. hypot-define 17.1%

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

   3. Simplified 17.1%

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

      1. add-sqr-sqrt 17.1%

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

      2. sqrt-unprod 17.1%

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

      3. *-commutative 17.1%

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

      4. *-commutative 17.1%

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

      5. swap-sqr 17.1%

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

      6. add-sqr-sqrt 17.1%

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

      7. *-commutative 17.1%

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

      8. metadata-eval 17.1%

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

   6. Applied egg-rr 17.1%

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

      1. associate-*l* 17.1%

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

      2. metadata-eval 17.1%

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

   8. Simplified 17.1%

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

   9. Taylor expanded in re around inf 92.6%

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

       1. associate-*l* 92.3%

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

       2. unpow2 92.3%

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

       3. rem-square-sqrt 93.8%

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

   11. Simplified 93.8%

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

       1. associate-*r* 93.8%

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

       2. sqrt-div 93.8%

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

       3. metadata-eval 93.8%

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

       4. un-div-inv 94.1%

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

   13. Applied egg-rr 94.1%

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

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

   1. Initial program 47.9%

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

      1. sub-neg 47.9%

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

      2. sqr-neg 47.9%

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

      3. sub-neg 47.9%

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

      4. sqr-neg 47.9%

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

      5. hypot-define 92.1%

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

   3. Simplified 92.1%

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

      1. add-sqr-sqrt 91.5%

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

      2. sqrt-unprod 92.1%

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

      3. *-commutative 92.1%

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

      4. *-commutative 92.1%

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

      5. swap-sqr 92.1%

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

      6. add-sqr-sqrt 92.1%

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

      7. *-commutative 92.1%

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

      8. metadata-eval 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. associate-*l* 92.1%

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

      2. metadata-eval 92.1%

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

   8. Simplified 92.1%

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

4. Final simplification 92.4%

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

Alternative 2: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e+53)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5.5e-9) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+53) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.5e-9) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d+53)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.5d-9) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+53) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.5e-9) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e+53:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.5e-9:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e+53)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.5e-9)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e+53)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.5e-9)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e+53], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.5e-9], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.10000000000000019e53

   1. Initial program 31.4%

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

   3. Taylor expanded in re around -inf 78.9%

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

      1. *-commutative 78.9%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 78.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -3.10000000000000019e53 < re < 5.4999999999999996e-9

   1. Initial program 60.3%

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

      1. sub-neg 60.3%

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

      2. sqr-neg 60.3%

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

      3. sub-neg 60.3%

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

      4. sqr-neg 60.3%

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

      5. hypot-define 93.1%

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

   3. Simplified 93.1%

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

      1. add-sqr-sqrt 92.4%

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

      2. sqrt-unprod 93.1%

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

      3. *-commutative 93.1%

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

      4. *-commutative 93.1%

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

      5. swap-sqr 93.1%

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

      6. add-sqr-sqrt 93.1%

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

      7. *-commutative 93.1%

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

      8. metadata-eval 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. associate-*l* 93.1%

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

      2. metadata-eval 93.1%

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

   8. Simplified 93.1%

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

   9. Taylor expanded in re around 0 83.3%

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

       1. neg-mul-1 83.3%

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

       2. unsub-neg 83.3%

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

   11. Simplified 83.3%

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

   if 5.4999999999999996e-9 < re

   1. Initial program 15.2%

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

      1. sub-neg 15.2%

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

      2. sqr-neg 15.2%

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

      3. sub-neg 15.2%

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

      4. sqr-neg 15.2%

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

      5. hypot-define 41.7%

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

   3. Simplified 41.7%

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

      1. add-sqr-sqrt 41.6%

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

      2. sqrt-unprod 41.7%

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

      3. *-commutative 41.7%

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

      4. *-commutative 41.7%

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

      5. swap-sqr 41.7%

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

      6. add-sqr-sqrt 41.7%

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

      7. *-commutative 41.7%

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

      8. metadata-eval 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. associate-*l* 41.7%

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

      2. metadata-eval 41.7%

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

   8. Simplified 41.7%

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

   9. Taylor expanded in re around inf 76.1%

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

       1. associate-*l* 76.0%

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

       2. unpow2 76.0%

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

       3. rem-square-sqrt 76.8%

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

   11. Simplified 76.8%

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

       1. associate-*r* 76.8%

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

       2. sqrt-div 76.8%

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

       3. metadata-eval 76.8%

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

       4. un-div-inv 76.9%

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

   13. Applied egg-rr 76.9%

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

       1. associate-*r/ 76.8%

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

   15. Simplified 76.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.6e+53)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5.8e-9) (sqrt (* 0.5 (- im re))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+53) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.8e-9) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.6d+53)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.8d-9) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.6e+53) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.8e-9) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.6e+53:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.8e-9:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.6e+53)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.8e-9)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.6e+53)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.8e-9)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.6e+53], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.8e-9], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.6e53

   1. Initial program 31.4%

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

   3. Taylor expanded in re around -inf 78.9%

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

      1. *-commutative 78.9%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 78.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -3.6e53 < re < 5.79999999999999982e-9

   1. Initial program 60.3%

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

      1. sub-neg 60.3%

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

      2. sqr-neg 60.3%

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

      3. sub-neg 60.3%

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

      4. sqr-neg 60.3%

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

      5. hypot-define 93.1%

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

   3. Simplified 93.1%

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

      1. add-sqr-sqrt 92.4%

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

      2. sqrt-unprod 93.1%

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

      3. *-commutative 93.1%

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

      4. *-commutative 93.1%

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

      5. swap-sqr 93.1%

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

      6. add-sqr-sqrt 93.1%

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

      7. *-commutative 93.1%

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

      8. metadata-eval 93.1%

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

   6. Applied egg-rr 93.1%

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

      1. associate-*l* 93.1%

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

      2. metadata-eval 93.1%

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

   8. Simplified 93.1%

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

   9. Taylor expanded in re around 0 83.3%

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

       1. neg-mul-1 83.3%

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

       2. unsub-neg 83.3%

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

   11. Simplified 83.3%

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

   if 5.79999999999999982e-9 < re

   1. Initial program 15.2%

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

      1. sub-neg 15.2%

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

      2. sqr-neg 15.2%

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

      3. sub-neg 15.2%

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

      4. sqr-neg 15.2%

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

      5. hypot-define 41.7%

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

   3. Simplified 41.7%

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

      1. add-sqr-sqrt 41.6%

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

      2. sqrt-unprod 41.7%

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

      3. *-commutative 41.7%

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

      4. *-commutative 41.7%

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

      5. swap-sqr 41.7%

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

      6. add-sqr-sqrt 41.7%

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

      7. *-commutative 41.7%

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

      8. metadata-eval 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. associate-*l* 41.7%

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

      2. metadata-eval 41.7%

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

   8. Simplified 41.7%

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

   9. Taylor expanded in re around inf 76.1%

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

       1. associate-*l* 76.0%

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

       2. unpow2 76.0%

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

       3. rem-square-sqrt 76.8%

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

   11. Simplified 76.8%

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

       1. associate-*r* 76.8%

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

       2. sqrt-div 76.8%

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

       3. metadata-eval 76.8%

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

       4. un-div-inv 76.9%

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

   13. Applied egg-rr 76.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.6 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.2e+54) (* 0.5 (sqrt (* re -4.0))) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.2e+54) {
		tmp = 0.5 * sqrt((re * -4.0));
	} 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 <= (-3.2d+54)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    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 <= -3.2e+54) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.2e+54:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.2e+54)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.2e+54)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.2e+54], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.2e54

   1. Initial program 31.4%

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

   3. Taylor expanded in re around -inf 78.9%

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

      1. *-commutative 78.9%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 78.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -3.2e54 < re

   1. Initial program 47.0%

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

      1. sub-neg 47.0%

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

      2. sqr-neg 47.0%

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

      3. sub-neg 47.0%

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

      4. sqr-neg 47.0%

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

      5. hypot-define 77.9%

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

   3. Simplified 77.9%

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

      1. add-sqr-sqrt 77.3%

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

      2. sqrt-unprod 77.9%

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

      3. *-commutative 77.9%

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

      4. *-commutative 77.9%

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

      5. swap-sqr 77.9%

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

      6. add-sqr-sqrt 77.9%

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

      7. *-commutative 77.9%

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

      8. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. associate-*l* 77.9%

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

      2. metadata-eval 77.9%

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

   8. Simplified 77.9%

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

   9. Taylor expanded in re around 0 65.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.2 \cdot 10^{+54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

   1. sub-neg 43.3%

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

   2. sqr-neg 43.3%

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

   3. sub-neg 43.3%

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

   4. sqr-neg 43.3%

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

   5. hypot-define 83.1%

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

3. Simplified 83.1%

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

   1. add-sqr-sqrt 82.5%

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

   2. sqrt-unprod 83.1%

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

   3. *-commutative 83.1%

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

   4. *-commutative 83.1%

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

   5. swap-sqr 83.1%

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

   6. add-sqr-sqrt 83.1%

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

   7. *-commutative 83.1%

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

   8. metadata-eval 83.1%

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

6. Applied egg-rr 83.1%

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

   1. associate-*l* 83.1%

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

   2. metadata-eval 83.1%

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

8. Simplified 83.1%

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

9. Taylor expanded in re around 0 58.7%

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

    1. neg-mul-1 58.7%

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

    2. unsub-neg 58.7%

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

11. Simplified 58.7%

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

12. Final simplification 58.7%

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

Alternative 6: 53.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{im \cdot 0.5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.3%

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

   1. sub-neg 43.3%

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

   2. sqr-neg 43.3%

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

   3. sub-neg 43.3%

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

   4. sqr-neg 43.3%

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

   5. hypot-define 83.1%

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

3. Simplified 83.1%

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

   1. add-sqr-sqrt 82.5%

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

   2. sqrt-unprod 83.1%

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

   3. *-commutative 83.1%

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

   4. *-commutative 83.1%

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

   5. swap-sqr 83.1%

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

   6. add-sqr-sqrt 83.1%

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

   7. *-commutative 83.1%

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

   8. metadata-eval 83.1%

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

6. Applied egg-rr 83.1%

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

   1. associate-*l* 83.1%

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

   2. metadata-eval 83.1%

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

8. Simplified 83.1%

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

9. Taylor expanded in re around 0 56.5%

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

10. Final simplification 56.5%

    \[\leadsto \sqrt{im \cdot 0.5} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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)))))