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

Percentage Accurate: 41.7% → 90.3%

Time: 9.6s

Alternatives: 5

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.7% 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: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* im (sqrt (/ 0.25 re)))
   (sqrt (* (- (hypot re im) re) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = im * sqrt((0.25 / re));
	} else {
		tmp = sqrt(((hypot(re, im) - re) * 0.5));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = im * Math.sqrt((0.25 / re));
	} else {
		tmp = Math.sqrt(((Math.hypot(re, im) - re) * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = im * math.sqrt((0.25 / re))
	else:
		tmp = math.sqrt(((math.hypot(re, im) - re) * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))) <= 0.0)
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	else
		tmp = sqrt(Float64(Float64(hypot(re, im) - re) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = im * sqrt((0.25 / re));
	else
		tmp = sqrt(((hypot(re, im) - re) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.0%

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

   3. Taylor expanded in re around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*l* 98.2%

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

      4. associate-*r* 98.7%

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

   5. Simplified 98.7%

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

      1. *-commutative 98.7%

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

      2. sqrt-div 98.6%

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

      3. metadata-eval 98.6%

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

      4. un-div-inv 98.6%

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

      5. *-commutative 98.6%

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

      6. *-commutative 98.6%

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

      7. associate-*r* 98.0%

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

      8. *-commutative 98.0%

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

      9. sqrt-unprod 99.6%

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

      10. metadata-eval 99.6%

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

      11. metadata-eval 99.6%

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

      12. *-un-lft-identity 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-/l* 99.5%

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

   9. Simplified 99.5%

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

       1. *-un-lft-identity 99.5%

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

       2. *-commutative 99.5%

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

       3. add-sqr-sqrt 99.1%

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

       4. sqrt-unprod 99.5%

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

       5. frac-times 99.6%

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

       6. metadata-eval 99.6%

          \[\leadsto im \cdot \left(\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \cdot 1\right) \]

       7. add-sqr-sqrt 99.8%

          \[\leadsto im \cdot \left(\sqrt{\frac{0.25}{\color{blue}{re}}} \cdot 1\right) \]

   11. Applied egg-rr 99.8%

       \[\leadsto im \cdot \color{blue}{\left(\sqrt{\frac{0.25}{re}} \cdot 1\right)} \]
   12. Step-by-step derivation

       1. *-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

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

   1. Initial program 48.7%

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

      1. add-sqr-sqrt 48.3%

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

      2. sqrt-unprod 48.7%

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

      3. *-commutative 48.7%

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

      4. *-commutative 48.7%

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

      5. swap-sqr 48.7%

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

      6. add-sqr-sqrt 48.7%

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

      7. *-commutative 48.7%

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

      8. hypot-define 92.2%

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

      9. metadata-eval 92.2%

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

   4. Applied egg-rr 92.2%

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

      1. associate-*l* 92.2%

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

      2. metadata-eval 92.2%

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

   6. Simplified 92.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im \cdot \left(1 - \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.5e-10)
   (sqrt (- re))
   (if (<= re 1.45e-106)
     (sqrt (* 0.5 (* im (- 1.0 (/ re im)))))
     (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.5e-10) {
		tmp = sqrt(-re);
	} else if (re <= 1.45e-106) {
		tmp = sqrt((0.5 * (im * (1.0 - (re / im)))));
	} 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 <= (-7.5d-10)) then
        tmp = sqrt(-re)
    else if (re <= 1.45d-106) then
        tmp = sqrt((0.5d0 * (im * (1.0d0 - (re / im)))))
    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 <= -7.5e-10) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.45e-106) {
		tmp = Math.sqrt((0.5 * (im * (1.0 - (re / im)))));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.5e-10:
		tmp = math.sqrt(-re)
	elif re <= 1.45e-106:
		tmp = math.sqrt((0.5 * (im * (1.0 - (re / im)))))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.5e-10)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.45e-106)
		tmp = sqrt(Float64(0.5 * Float64(im * Float64(1.0 - Float64(re / im)))));
	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 <= -7.5e-10)
		tmp = sqrt(-re);
	elseif (re <= 1.45e-106)
		tmp = sqrt((0.5 * (im * (1.0 - (re / im)))));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.5e-10], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.45e-106], N[Sqrt[N[(0.5 * N[(im * N[(1.0 - N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.49999999999999995e-10

   1. Initial program 40.8%

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

      1. add-sqr-sqrt 40.5%

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

      2. sqrt-unprod 40.8%

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

      3. *-commutative 40.8%

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

      4. *-commutative 40.8%

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

      5. swap-sqr 40.8%

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

      6. add-sqr-sqrt 40.8%

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

      7. *-commutative 40.8%

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

      8. hypot-define 100.0%

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

      9. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 76.4%

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

      1. neg-mul-1 76.4%

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

   9. Simplified 76.4%

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

   if -7.49999999999999995e-10 < re < 1.45e-106

   1. Initial program 61.5%

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

      1. add-sqr-sqrt 61.1%

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

      2. sqrt-unprod 61.5%

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

      3. *-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. swap-sqr 61.5%

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

      6. add-sqr-sqrt 61.5%

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

      7. *-commutative 61.5%

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

      8. hypot-define 97.3%

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

      9. metadata-eval 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. associate-*l* 97.3%

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

      2. metadata-eval 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in im around inf 86.8%

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

      1. mul-1-neg 86.8%

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

      2. unsub-neg 86.8%

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

   9. Simplified 86.8%

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

   if 1.45e-106 < re

   1. Initial program 20.0%

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

   3. Taylor expanded in re around inf 73.7%

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

      1. *-commutative 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*l* 73.7%

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

      4. associate-*r* 74.0%

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

   5. Simplified 74.0%

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

      1. *-commutative 74.0%

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

      2. sqrt-div 74.0%

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

      3. metadata-eval 74.0%

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

      4. un-div-inv 74.0%

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

      5. *-commutative 74.0%

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

      6. *-commutative 74.0%

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

      7. associate-*r* 73.7%

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

      8. *-commutative 73.7%

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

      9. sqrt-unprod 74.7%

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

      10. metadata-eval 74.7%

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

      11. metadata-eval 74.7%

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

      12. *-un-lft-identity 74.7%

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

   7. Applied egg-rr 74.7%

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

      1. associate-/l* 74.6%

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

   9. Simplified 74.6%

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

       1. *-un-lft-identity 74.6%

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

       2. *-commutative 74.6%

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

       3. add-sqr-sqrt 74.4%

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

       4. sqrt-unprod 74.6%

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

       5. frac-times 74.7%

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

       6. metadata-eval 74.7%

          \[\leadsto im \cdot \left(\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \cdot 1\right) \]

       7. add-sqr-sqrt 74.7%

          \[\leadsto im \cdot \left(\sqrt{\frac{0.25}{\color{blue}{re}}} \cdot 1\right) \]

   11. Applied egg-rr 74.7%

       \[\leadsto im \cdot \color{blue}{\left(\sqrt{\frac{0.25}{re}} \cdot 1\right)} \]
   12. Step-by-step derivation

       1. *-rgt-identity 74.7%

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

   13. Simplified 74.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-106}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im \cdot \left(1 - \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.25e-9)
   (sqrt (- re))
   (if (<= re 8e-107) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.25e-9) {
		tmp = sqrt(-re);
	} else if (re <= 8e-107) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * sqrt((0.25 / re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.25d-9)) then
        tmp = sqrt(-re)
    else if (re <= 8d-107) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.25e-9) {
		tmp = Math.sqrt(-re);
	} else if (re <= 8e-107) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.25e-9:
		tmp = math.sqrt(-re)
	elif re <= 8e-107:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.25e-9)
		tmp = sqrt(Float64(-re));
	elseif (re <= 8e-107)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.25e-9)
		tmp = sqrt(-re);
	elseif (re <= 8e-107)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.25e-9], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 8e-107], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.25e-9

   1. Initial program 40.8%

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

      1. add-sqr-sqrt 40.5%

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

      2. sqrt-unprod 40.8%

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

      3. *-commutative 40.8%

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

      4. *-commutative 40.8%

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

      5. swap-sqr 40.8%

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

      6. add-sqr-sqrt 40.8%

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

      7. *-commutative 40.8%

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

      8. hypot-define 100.0%

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

      9. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 76.4%

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

      1. neg-mul-1 76.4%

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

   9. Simplified 76.4%

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

   if -1.25e-9 < re < 8e-107

   1. Initial program 61.5%

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

      1. add-sqr-sqrt 61.1%

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

      2. sqrt-unprod 61.5%

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

      3. *-commutative 61.5%

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

      4. *-commutative 61.5%

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

      5. swap-sqr 61.5%

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

      6. add-sqr-sqrt 61.5%

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

      7. *-commutative 61.5%

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

      8. hypot-define 97.3%

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

      9. metadata-eval 97.3%

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

   4. Applied egg-rr 97.3%

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

      1. associate-*l* 97.3%

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

      2. metadata-eval 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in re around 0 86.8%

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

      1. neg-mul-1 86.8%

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

      2. unsub-neg 86.8%

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

   9. Simplified 86.8%

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

   if 8e-107 < re

   1. Initial program 20.0%

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

   3. Taylor expanded in re around inf 73.7%

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

      1. *-commutative 73.7%

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

      2. *-commutative 73.7%

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

      3. associate-*l* 73.7%

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

      4. associate-*r* 74.0%

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

   5. Simplified 74.0%

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

      1. *-commutative 74.0%

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

      2. sqrt-div 74.0%

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

      3. metadata-eval 74.0%

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

      4. un-div-inv 74.0%

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

      5. *-commutative 74.0%

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

      6. *-commutative 74.0%

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

      7. associate-*r* 73.7%

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

      8. *-commutative 73.7%

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

      9. sqrt-unprod 74.7%

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

      10. metadata-eval 74.7%

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

      11. metadata-eval 74.7%

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

      12. *-un-lft-identity 74.7%

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

   7. Applied egg-rr 74.7%

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

      1. associate-/l* 74.6%

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

   9. Simplified 74.6%

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

       1. *-un-lft-identity 74.6%

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

       2. *-commutative 74.6%

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

       3. add-sqr-sqrt 74.4%

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

       4. sqrt-unprod 74.6%

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

       5. frac-times 74.7%

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

       6. metadata-eval 74.7%

          \[\leadsto im \cdot \left(\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}} \cdot 1\right) \]

       7. add-sqr-sqrt 74.7%

          \[\leadsto im \cdot \left(\sqrt{\frac{0.25}{\color{blue}{re}}} \cdot 1\right) \]

   11. Applied egg-rr 74.7%

       \[\leadsto im \cdot \color{blue}{\left(\sqrt{\frac{0.25}{re}} \cdot 1\right)} \]
   12. Step-by-step derivation

       1. *-rgt-identity 74.7%

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

   13. Simplified 74.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.8e-7) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.8e-7) {
		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 <= (-5.8d-7)) 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 <= -5.8e-7) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.8e-7:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.8e-7)
		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 <= -5.8e-7)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.8e-7], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.7999999999999995e-7

   1. Initial program 40.8%

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

      1. add-sqr-sqrt 40.5%

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

      2. sqrt-unprod 40.8%

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

      3. *-commutative 40.8%

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

      4. *-commutative 40.8%

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

      5. swap-sqr 40.8%

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

      6. add-sqr-sqrt 40.8%

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

      7. *-commutative 40.8%

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

      8. hypot-define 100.0%

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

      9. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-*l* 100.0%

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

      2. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 76.4%

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

      1. neg-mul-1 76.4%

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

   9. Simplified 76.4%

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

   if -5.7999999999999995e-7 < re

   1. Initial program 44.7%

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

      1. add-sqr-sqrt 44.4%

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

      2. sqrt-unprod 44.7%

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

      3. *-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. swap-sqr 44.7%

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

      6. add-sqr-sqrt 44.7%

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

      7. *-commutative 44.7%

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

      8. hypot-define 74.0%

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

      9. metadata-eval 74.0%

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

   4. Applied egg-rr 74.0%

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

      1. associate-*l* 74.0%

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

      2. metadata-eval 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in re around 0 62.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.8 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 26.8% 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 43.5%

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

   1. add-sqr-sqrt 43.2%

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

   2. sqrt-unprod 43.5%

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

   3. *-commutative 43.5%

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

   4. *-commutative 43.5%

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

   5. swap-sqr 43.5%

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

   6. add-sqr-sqrt 43.5%

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

   7. *-commutative 43.5%

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

   8. hypot-define 81.7%

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

   9. metadata-eval 81.7%

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

4. Applied egg-rr 81.7%

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

   1. associate-*l* 81.7%

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

   2. metadata-eval 81.7%

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

6. Simplified 81.7%

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

7. Taylor expanded in re around -inf 28.4%

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

   1. neg-mul-1 28.4%

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

9. Simplified 28.4%

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

10. Final simplification 28.4%

    \[\leadsto \sqrt{-re} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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)))))