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

Percentage Accurate: 41.3% → 90.0%

Time: 32.4s

Alternatives: 6

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.3% 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: 90.0% 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 5.9%

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

   3. Taylor expanded in re around inf 47.8%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. *-commutative 47.8%

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

      2. sqrt-div 50.1%

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

      3. sqrt-pow1 88.3%

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

      4. metadata-eval 88.3%

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

      5. pow1 88.3%

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

      6. associate-*l/ 88.3%

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

   5. Applied egg-rr 88.3%

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

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

   1. Initial program 44.1%

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

      1. sub-neg 44.1%

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

      2. sqr-neg 44.1%

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

      3. sub-neg 44.1%

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

      4. sqr-neg 44.1%

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

      5. hypot-define 88.7%

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

   3. Simplified 88.7%

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

      1. *-commutative 88.7%

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

      2. hypot-define 44.1%

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

      3. *-commutative 44.1%

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

      4. add-sqr-sqrt 43.8%

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

      5. sqrt-unprod 44.1%

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

      6. *-commutative 44.1%

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

      7. *-commutative 44.1%

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

      8. swap-sqr 44.1%

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

   6. Applied egg-rr 88.7%

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

      1. associate-*l* 88.7%

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

      2. metadata-eval 88.7%

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

   8. Simplified 88.7%

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

4. Final simplification 88.6%

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

Alternative 2: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.3e-32)
   (sqrt (- re))
   (if (<= re 5.8e+49) (sqrt (* im 0.5)) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.3e-32) {
		tmp = sqrt(-re);
	} else if (re <= 5.8e+49) {
		tmp = sqrt((im * 0.5));
	} 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 <= (-1.3d-32)) then
        tmp = sqrt(-re)
    else if (re <= 5.8d+49) then
        tmp = sqrt((im * 0.5d0))
    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 <= -1.3e-32) {
		tmp = Math.sqrt(-re);
	} else if (re <= 5.8e+49) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.3e-32:
		tmp = math.sqrt(-re)
	elif re <= 5.8e+49:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.3e-32)
		tmp = sqrt(Float64(-re));
	elseif (re <= 5.8e+49)
		tmp = sqrt(Float64(im * 0.5));
	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 <= -1.3e-32)
		tmp = sqrt(-re);
	elseif (re <= 5.8e+49)
		tmp = sqrt((im * 0.5));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.3e-32], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 5.8e+49], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.2999999999999999e-32

   1. Initial program 37.1%

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

      1. sub-neg 37.1%

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

      2. sqr-neg 37.1%

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

      3. sub-neg 37.1%

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

      4. sqr-neg 37.1%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 37.1%

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

      3. *-commutative 37.1%

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

      4. add-sqr-sqrt 36.8%

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

      5. sqrt-unprod 37.1%

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

      6. *-commutative 37.1%

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

      7. *-commutative 37.1%

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

      8. swap-sqr 37.1%

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

   6. Applied egg-rr 100.0%

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

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 72.9%

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

       1. neg-mul-1 72.9%

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

   11. Simplified 72.9%

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

   if -1.2999999999999999e-32 < re < 5.8e49

   1. Initial program 54.2%

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

      1. sub-neg 54.2%

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

      2. sqr-neg 54.2%

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

      3. sub-neg 54.2%

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

      4. sqr-neg 54.2%

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

      5. hypot-define 84.3%

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

   3. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. hypot-define 54.2%

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

      3. *-commutative 54.2%

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

      4. add-sqr-sqrt 53.8%

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

      5. sqrt-unprod 54.2%

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

      6. *-commutative 54.2%

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

      7. *-commutative 54.2%

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

      8. swap-sqr 54.2%

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

   6. Applied egg-rr 84.3%

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

      1. associate-*l* 84.3%

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

      2. metadata-eval 84.3%

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

   8. Simplified 84.3%

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

   9. Taylor expanded in re around 0 77.2%

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

       1. *-commutative 77.2%

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

   11. Simplified 77.2%

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

   if 5.8e49 < re

   1. Initial program 4.8%

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

   3. Taylor expanded in re around inf 54.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. *-commutative 54.9%

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

      2. sqrt-div 66.1%

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

      3. sqrt-pow1 83.8%

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

      4. metadata-eval 83.8%

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

      5. pow1 83.8%

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

      6. associate-*l/ 83.8%

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

   5. Applied egg-rr 83.8%

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

Alternative 3: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-32)
   (sqrt (- re))
   (if (<= re 1.8e+49) (sqrt (* im 0.5)) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-32) {
		tmp = sqrt(-re);
	} else if (re <= 1.8e+49) {
		tmp = sqrt((im * 0.5));
	} 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 <= (-2d-32)) then
        tmp = sqrt(-re)
    else if (re <= 1.8d+49) then
        tmp = sqrt((im * 0.5d0))
    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 <= -2e-32) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.8e+49) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-32:
		tmp = math.sqrt(-re)
	elif re <= 1.8e+49:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-32)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.8e+49)
		tmp = sqrt(Float64(im * 0.5));
	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 <= -2e-32)
		tmp = sqrt(-re);
	elseif (re <= 1.8e+49)
		tmp = sqrt((im * 0.5));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-32], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.8e+49], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.00000000000000011e-32

   1. Initial program 37.1%

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

      1. sub-neg 37.1%

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

      2. sqr-neg 37.1%

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

      3. sub-neg 37.1%

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

      4. sqr-neg 37.1%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 37.1%

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

      3. *-commutative 37.1%

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

      4. add-sqr-sqrt 36.8%

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

      5. sqrt-unprod 37.1%

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

      6. *-commutative 37.1%

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

      7. *-commutative 37.1%

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

      8. swap-sqr 37.1%

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

   6. Applied egg-rr 100.0%

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

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 72.9%

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

       1. neg-mul-1 72.9%

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

   11. Simplified 72.9%

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

   if -2.00000000000000011e-32 < re < 1.79999999999999998e49

   1. Initial program 54.2%

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

      1. sub-neg 54.2%

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

      2. sqr-neg 54.2%

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

      3. sub-neg 54.2%

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

      4. sqr-neg 54.2%

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

      5. hypot-define 84.3%

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

   3. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. hypot-define 54.2%

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

      3. *-commutative 54.2%

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

      4. add-sqr-sqrt 53.8%

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

      5. sqrt-unprod 54.2%

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

      6. *-commutative 54.2%

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

      7. *-commutative 54.2%

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

      8. swap-sqr 54.2%

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

   6. Applied egg-rr 84.3%

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

      1. associate-*l* 84.3%

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

      2. metadata-eval 84.3%

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

   8. Simplified 84.3%

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

   9. Taylor expanded in re around 0 77.2%

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

       1. *-commutative 77.2%

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

   11. Simplified 77.2%

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

   if 1.79999999999999998e49 < re

   1. Initial program 4.8%

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

   3. Taylor expanded in re around inf 54.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]
   4. Step-by-step derivation

      1. sqrt-div 66.1%

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

      2. sqrt-pow1 83.8%

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

      3. metadata-eval 83.8%

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

      4. pow1 83.8%

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

      5. associate-*r/ 83.8%

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

   5. Applied egg-rr 83.8%

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

      1. *-commutative 83.8%

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

      2. associate-/l* 83.7%

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

   7. Simplified 83.7%

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

Alternative 4: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.8 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.8e-33) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.8e-33) {
		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 <= (-6.8d-33)) 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 <= -6.8e-33) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.8e-33:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.8e-33)
		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 <= -6.8e-33)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.8e-33], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -6.8000000000000001e-33

   1. Initial program 37.1%

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

      1. sub-neg 37.1%

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

      2. sqr-neg 37.1%

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

      3. sub-neg 37.1%

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

      4. sqr-neg 37.1%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 37.1%

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

      3. *-commutative 37.1%

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

      4. add-sqr-sqrt 36.8%

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

      5. sqrt-unprod 37.1%

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

      6. *-commutative 37.1%

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

      7. *-commutative 37.1%

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

      8. swap-sqr 37.1%

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

   6. Applied egg-rr 100.0%

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

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 72.9%

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

       1. neg-mul-1 72.9%

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

   11. Simplified 72.9%

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

   if -6.8000000000000001e-33 < re

   1. Initial program 38.5%

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

      1. sub-neg 38.5%

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

      2. sqr-neg 38.5%

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

      3. sub-neg 38.5%

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

      4. sqr-neg 38.5%

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

      5. hypot-define 69.4%

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

   3. Simplified 69.4%

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

      1. *-commutative 69.4%

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

      2. hypot-define 38.5%

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

      3. *-commutative 38.5%

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

      4. add-sqr-sqrt 38.2%

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

      5. sqrt-unprod 38.5%

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

      6. *-commutative 38.5%

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

      7. *-commutative 38.5%

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

      8. swap-sqr 38.5%

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

   6. Applied egg-rr 69.4%

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

      1. associate-*l* 69.4%

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

      2. metadata-eval 69.4%

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

   8. Simplified 69.4%

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

   9. Taylor expanded in re around 0 59.8%

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

       1. *-commutative 59.8%

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

   11. Simplified 59.8%

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

Alternative 5: 29.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if re < -3.999999999999988e-310

   1. Initial program 50.6%

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

      1. sub-neg 50.6%

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

      2. sqr-neg 50.6%

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

      3. sub-neg 50.6%

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

      4. sqr-neg 50.6%

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

      5. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. hypot-define 50.6%

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

      3. *-commutative 50.6%

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

      4. add-sqr-sqrt 50.3%

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

      5. sqrt-unprod 50.6%

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

      6. *-commutative 50.6%

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

      7. *-commutative 50.6%

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

      8. swap-sqr 50.6%

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

   6. Applied egg-rr 100.0%

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

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 51.6%

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

       1. neg-mul-1 51.6%

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

   11. Simplified 51.6%

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

   if -3.999999999999988e-310 < re

   1. Initial program 26.6%

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

      1. sub-neg 26.6%

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

      2. sqr-neg 26.6%

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

      3. sub-neg 26.6%

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

      4. sqr-neg 26.6%

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

      5. hypot-define 57.3%

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

   3. Simplified 57.3%

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

      1. *-commutative 57.3%

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

      2. hypot-define 26.6%

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

      3. *-commutative 26.6%

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

      4. add-sqr-sqrt 26.4%

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

      5. sqrt-unprod 26.6%

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

      6. *-commutative 26.6%

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

      7. *-commutative 26.6%

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

      8. swap-sqr 26.6%

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

   6. Applied egg-rr 57.3%

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

      1. associate-*l* 57.3%

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

      2. metadata-eval 57.3%

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

   8. Simplified 57.3%

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

   9. Taylor expanded in re around -inf 0.0%

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

       1. neg-mul-1 0.0%

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

   11. Simplified 0.0%

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

       1. neg-sub0 0.0%

          \[\leadsto \sqrt{\color{blue}{0 - re}} \]

       2. sub-neg 0.0%

          \[\leadsto \sqrt{\color{blue}{0 + \left(-re\right)}} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto \sqrt{0 + \color{blue}{\sqrt{-re} \cdot \sqrt{-re}}} \]

       4. sqrt-unprod 4.5%

          \[\leadsto \sqrt{0 + \color{blue}{\sqrt{\left(-re\right) \cdot \left(-re\right)}}} \]

       5. sqr-neg 4.5%

          \[\leadsto \sqrt{0 + \sqrt{\color{blue}{re \cdot re}}} \]

       6. sqrt-unprod 5.7%

          \[\leadsto \sqrt{0 + \color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       7. add-sqr-sqrt 5.7%

          \[\leadsto \sqrt{0 + \color{blue}{re}} \]

   13. Applied egg-rr 5.7%

       \[\leadsto \sqrt{\color{blue}{0 + re}} \]
   14. Step-by-step derivation

       1. +-lft-identity 5.7%

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

   15. Simplified 5.7%

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

Alternative 6: 2.9% 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(re)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. sub-neg 38.1%

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

   2. sqr-neg 38.1%

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

   3. sub-neg 38.1%

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

   4. sqr-neg 38.1%

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

   5. hypot-define 77.8%

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

3. Simplified 77.8%

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

   1. *-commutative 77.8%

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

   2. hypot-define 38.1%

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

   3. *-commutative 38.1%

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

   4. add-sqr-sqrt 37.9%

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

   5. sqrt-unprod 38.1%

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

   6. *-commutative 38.1%

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

   7. *-commutative 38.1%

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

   8. swap-sqr 38.1%

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

6. Applied egg-rr 77.8%

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

   1. associate-*l* 77.8%

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

   2. metadata-eval 77.8%

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

8. Simplified 77.8%

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

9. Taylor expanded in re around -inf 24.8%

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

    1. neg-mul-1 24.8%

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

11. Simplified 24.8%

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

    1. neg-sub0 24.8%

       \[\leadsto \sqrt{\color{blue}{0 - re}} \]

    2. sub-neg 24.8%

       \[\leadsto \sqrt{\color{blue}{0 + \left(-re\right)}} \]

    3. add-sqr-sqrt 24.8%

       \[\leadsto \sqrt{0 + \color{blue}{\sqrt{-re} \cdot \sqrt{-re}}} \]

    4. sqrt-unprod 15.7%

       \[\leadsto \sqrt{0 + \color{blue}{\sqrt{\left(-re\right) \cdot \left(-re\right)}}} \]

    5. sqr-neg 15.7%

       \[\leadsto \sqrt{0 + \sqrt{\color{blue}{re \cdot re}}} \]

    6. sqrt-unprod 2.9%

       \[\leadsto \sqrt{0 + \color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

    7. add-sqr-sqrt 2.9%

       \[\leadsto \sqrt{0 + \color{blue}{re}} \]

13. Applied egg-rr 2.9%

    \[\leadsto \sqrt{\color{blue}{0 + re}} \]
14. Step-by-step derivation

    1. +-lft-identity 2.9%

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

15. Simplified 2.9%

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

Reproduce

herbie shell --seed 2024116 
(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)))))