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

Percentage Accurate: 40.8% → 89.4%

Time: 11.1s

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: 40.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	else:
		tmp = math.sqrt((0.5 * (math.hypot(im, re) - re)))
	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((re ^ -0.5) * Float64(im * 0.5));
	else
		tmp = sqrt(Float64(0.5 * Float64(hypot(im, re) - re)));
	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 = (re ^ -0.5) * (im * 0.5);
	else
		tmp = sqrt((0.5 * (hypot(im, re) - re)));
	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[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[im ^ 2 + re ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.8%

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

      1. sub-neg 5.8%

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

      2. sqr-neg 5.8%

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

      3. sub-neg 5.8%

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

      4. sqr-neg 5.8%

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

      5. hypot-define 5.8%

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

   3. Simplified 5.8%

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

      1. *-commutative 5.8%

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

      2. hypot-define 5.8%

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

      3. *-commutative 5.8%

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

      4. add-sqr-sqrt 5.8%

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

      5. sqrt-unprod 5.8%

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

      6. *-commutative 5.8%

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

      7. *-commutative 5.8%

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

      8. swap-sqr 5.8%

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

   6. Applied egg-rr 5.8%

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

      1. *-commutative 5.8%

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

      2. associate-*r* 5.8%

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

      3. metadata-eval 5.8%

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

      4. hypot-undefine 5.8%

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

      5. unpow2 5.8%

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

      6. unpow2 5.8%

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

      7. +-commutative 5.8%

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

      8. unpow2 5.8%

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

      9. unpow2 5.8%

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

      10. hypot-undefine 5.8%

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

   8. Simplified 5.8%

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

   9. Taylor expanded in im around 0 98.1%

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

       1. *-commutative 98.1%

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

       2. *-commutative 98.1%

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

       3. unpow2 98.1%

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

       4. rem-square-sqrt 99.6%

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

   11. Simplified 99.6%

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

       1. *-un-lft-identity 99.6%

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

       2. inv-pow 99.6%

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

       3. sqrt-pow1 99.6%

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

       4. metadata-eval 99.6%

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

   13. Applied egg-rr 99.6%

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

       1. *-lft-identity 99.6%

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

   15. Simplified 99.6%

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

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

   1. Initial program 44.5%

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

      1. sub-neg 44.5%

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

      2. sqr-neg 44.5%

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

      3. sub-neg 44.5%

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

      4. sqr-neg 44.5%

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

      5. hypot-define 90.0%

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

   3. Simplified 90.0%

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

      1. *-commutative 90.0%

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

      2. hypot-define 44.5%

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

      3. *-commutative 44.5%

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

      4. add-sqr-sqrt 44.2%

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

      5. sqrt-unprod 44.5%

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

      6. *-commutative 44.5%

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

      7. *-commutative 44.5%

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

      8. swap-sqr 44.5%

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

   6. Applied egg-rr 90.0%

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

      1. *-commutative 90.0%

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

      2. associate-*r* 90.0%

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

      3. metadata-eval 90.0%

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

      4. hypot-undefine 44.5%

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

      5. unpow2 44.5%

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

      6. unpow2 44.5%

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

      7. +-commutative 44.5%

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

      8. unpow2 44.5%

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

      9. unpow2 44.5%

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

      10. hypot-undefine 90.0%

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

   8. Simplified 90.0%

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

4. Final simplification 91.2%

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

Alternative 2: 76.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.066 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.45 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.066e+45)
   (sqrt (- re))
   (if (<= re 1.45e-52) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.066e+45) {
		tmp = sqrt(-re);
	} else if (re <= 1.45e-52) {
		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 <= (-1.066d+45)) then
        tmp = sqrt(-re)
    else if (re <= 1.45d-52) 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 <= -1.066e+45) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.45e-52) {
		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 <= -1.066e+45:
		tmp = math.sqrt(-re)
	elif re <= 1.45e-52:
		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 <= -1.066e+45)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.45e-52)
		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 <= -1.066e+45)
		tmp = sqrt(-re);
	elseif (re <= 1.45e-52)
		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, -1.066e+45], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.45e-52], 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 < -1.06600000000000005e45

   1. Initial program 37.4%

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

      1. sub-neg 37.4%

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

      2. sqr-neg 37.4%

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

      3. sub-neg 37.4%

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

      4. sqr-neg 37.4%

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

      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 37.4%

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

      3. *-commutative 37.4%

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

      4. add-sqr-sqrt 37.1%

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

      5. sqrt-unprod 37.4%

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

      6. *-commutative 37.4%

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

      7. *-commutative 37.4%

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

      8. swap-sqr 37.4%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. metadata-eval 100.0%

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

      4. hypot-undefine 37.4%

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

      5. unpow2 37.4%

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

      6. unpow2 37.4%

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

      7. +-commutative 37.4%

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

      8. unpow2 37.4%

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

      9. unpow2 37.4%

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

      10. hypot-undefine 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 87.8%

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

       1. neg-mul-1 87.8%

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

   11. Simplified 87.8%

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

   if -1.06600000000000005e45 < re < 1.4500000000000001e-52

   1. Initial program 55.6%

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

      1. sub-neg 55.6%

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

      2. sqr-neg 55.6%

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

      3. sub-neg 55.6%

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

      4. sqr-neg 55.6%

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

      5. hypot-define 91.2%

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

   3. Simplified 91.2%

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

      1. *-commutative 91.2%

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

      2. hypot-define 55.6%

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

      3. *-commutative 55.6%

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

      4. add-sqr-sqrt 55.2%

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

      5. sqrt-unprod 55.6%

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

      6. *-commutative 55.6%

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

      7. *-commutative 55.6%

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

      8. swap-sqr 55.6%

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

   6. Applied egg-rr 91.2%

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

      1. *-commutative 91.2%

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

      2. associate-*r* 91.2%

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

      3. metadata-eval 91.2%

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

      4. hypot-undefine 55.6%

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

      5. unpow2 55.6%

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

      6. unpow2 55.6%

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

      7. +-commutative 55.6%

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

      8. unpow2 55.6%

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

      9. unpow2 55.6%

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

      10. hypot-undefine 91.2%

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

   8. Simplified 91.2%

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

   9. Taylor expanded in re around 0 81.3%

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

       1. neg-mul-1 81.3%

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

       2. sub-neg 81.3%

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

   11. Simplified 81.3%

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

   if 1.4500000000000001e-52 < re

   1. Initial program 12.1%

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

      1. sub-neg 12.1%

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

      2. sqr-neg 12.1%

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

      3. sub-neg 12.1%

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

      4. sqr-neg 12.1%

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

      5. hypot-define 43.8%

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

   3. Simplified 43.8%

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

      1. *-commutative 43.8%

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

      2. hypot-define 12.1%

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

      3. *-commutative 12.1%

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

      4. add-sqr-sqrt 12.1%

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

      5. sqrt-unprod 12.1%

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

      6. *-commutative 12.1%

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

      7. *-commutative 12.1%

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

      8. swap-sqr 12.1%

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

   6. Applied egg-rr 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-*r* 43.8%

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

      3. metadata-eval 43.8%

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

      4. hypot-undefine 12.1%

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

      5. unpow2 12.1%

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

      6. unpow2 12.1%

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

      7. +-commutative 12.1%

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

      8. unpow2 12.1%

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

      9. unpow2 12.1%

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

      10. hypot-undefine 43.8%

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

   8. Simplified 43.8%

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

   9. Taylor expanded in im around 0 70.6%

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

       1. *-commutative 70.6%

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

       2. *-commutative 70.6%

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

       3. unpow2 70.6%

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

       4. rem-square-sqrt 71.5%

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

   11. Simplified 71.5%

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

       1. *-commutative 71.5%

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

       2. sqrt-div 71.4%

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

       3. metadata-eval 71.4%

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

       4. un-div-inv 71.5%

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

       5. *-commutative 71.5%

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

   13. Applied egg-rr 71.5%

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

       1. associate-*r/ 71.4%

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

   15. Simplified 71.4%

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

       1. *-un-lft-identity 71.4%

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

       2. *-commutative 71.4%

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

       3. add-sqr-sqrt 71.1%

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

       4. sqrt-unprod 71.4%

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

       5. frac-times 71.3%

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

       6. metadata-eval 71.3%

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

       7. add-sqr-sqrt 71.5%

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

   17. Applied egg-rr 71.5%

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

       1. *-rgt-identity 71.5%

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

   19. Simplified 71.5%

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

Alternative 3: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.066 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.066e+45) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.066e+45) {
		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 <= (-1.066d+45)) 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 <= -1.066e+45) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.066e+45:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.066e+45)
		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 <= -1.066e+45)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.066e+45], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.06600000000000005e45

   1. Initial program 37.4%

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

      1. sub-neg 37.4%

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

      2. sqr-neg 37.4%

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

      3. sub-neg 37.4%

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

      4. sqr-neg 37.4%

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

      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 37.4%

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

      3. *-commutative 37.4%

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

      4. add-sqr-sqrt 37.1%

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

      5. sqrt-unprod 37.4%

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

      6. *-commutative 37.4%

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

      7. *-commutative 37.4%

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

      8. swap-sqr 37.4%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. metadata-eval 100.0%

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

      4. hypot-undefine 37.4%

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

      5. unpow2 37.4%

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

      6. unpow2 37.4%

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

      7. +-commutative 37.4%

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

      8. unpow2 37.4%

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

      9. unpow2 37.4%

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

      10. hypot-undefine 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 87.8%

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

       1. neg-mul-1 87.8%

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

   11. Simplified 87.8%

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

   if -1.06600000000000005e45 < re

   1. Initial program 40.5%

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

      1. sub-neg 40.5%

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

      2. sqr-neg 40.5%

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

      3. sub-neg 40.5%

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

      4. sqr-neg 40.5%

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

      5. hypot-define 74.7%

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

   3. Simplified 74.7%

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

      1. *-commutative 74.7%

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

      2. hypot-define 40.5%

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

      3. *-commutative 40.5%

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

      4. add-sqr-sqrt 40.2%

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

      5. sqrt-unprod 40.5%

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

      6. *-commutative 40.5%

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

      7. *-commutative 40.5%

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

      8. swap-sqr 40.5%

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

   6. Applied egg-rr 74.7%

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

      1. *-commutative 74.7%

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

      2. associate-*r* 74.7%

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

      3. metadata-eval 74.7%

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

      4. hypot-undefine 40.5%

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

      5. unpow2 40.5%

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

      6. unpow2 40.5%

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

      7. +-commutative 40.5%

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

      8. unpow2 40.5%

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

      9. unpow2 40.5%

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

      10. hypot-undefine 74.7%

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

   8. Simplified 74.7%

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

   9. Taylor expanded in im around inf 63.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.066 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 29.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-310) (sqrt (- re)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-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 <= -2e-310)
		tmp = sqrt(-re);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-310], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.999999999999994e-310

   1. Initial program 54.3%

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

      1. sub-neg 54.3%

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

      2. sqr-neg 54.3%

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

      3. sub-neg 54.3%

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

      4. sqr-neg 54.3%

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

      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 54.3%

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

      3. *-commutative 54.3%

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

      4. add-sqr-sqrt 53.9%

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

      5. sqrt-unprod 54.3%

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

      6. *-commutative 54.3%

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

      7. *-commutative 54.3%

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

      8. swap-sqr 54.3%

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

   6. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. metadata-eval 100.0%

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

      4. hypot-undefine 54.3%

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

      5. unpow2 54.3%

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

      6. unpow2 54.3%

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

      7. +-commutative 54.3%

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

      8. unpow2 54.3%

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

      9. unpow2 54.3%

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

      10. hypot-undefine 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in re around -inf 48.2%

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

       1. neg-mul-1 48.2%

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

   11. Simplified 48.2%

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

   if -1.999999999999994e-310 < re

   1. Initial program 24.2%

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

      1. sub-neg 24.2%

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

      2. sqr-neg 24.2%

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

      3. sub-neg 24.2%

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

      4. sqr-neg 24.2%

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

      5. hypot-define 58.0%

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

   3. Simplified 58.0%

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

      1. *-commutative 58.0%

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

      2. hypot-define 24.2%

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

      3. *-commutative 24.2%

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

      4. add-sqr-sqrt 24.0%

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

      5. sqrt-unprod 24.2%

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

      6. *-commutative 24.2%

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

      7. *-commutative 24.2%

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

      8. swap-sqr 24.2%

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

   6. Applied egg-rr 58.0%

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

      1. *-commutative 58.0%

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

      2. associate-*r* 58.0%

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

      3. metadata-eval 58.0%

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

      4. hypot-undefine 24.2%

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

      5. unpow2 24.2%

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

      6. unpow2 24.2%

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

      7. +-commutative 24.2%

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

      8. unpow2 24.2%

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

      9. unpow2 24.2%

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

      10. hypot-undefine 58.0%

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

   8. Simplified 58.0%

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

   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 5.8%

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

       7. add-sqr-sqrt 5.8%

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

   13. Applied egg-rr 5.8%

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

       1. +-lft-identity 5.8%

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

   15. Simplified 5.8%

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

Alternative 5: 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 39.8%

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

   1. sub-neg 39.8%

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

   2. sqr-neg 39.8%

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

   3. sub-neg 39.8%

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

   4. sqr-neg 39.8%

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

   5. hypot-define 79.8%

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

3. Simplified 79.8%

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

   1. *-commutative 79.8%

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

   2. hypot-define 39.8%

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

   3. *-commutative 39.8%

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

   4. add-sqr-sqrt 39.6%

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

   5. sqrt-unprod 39.8%

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

   6. *-commutative 39.8%

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

   7. *-commutative 39.8%

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

   8. swap-sqr 39.8%

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

6. Applied egg-rr 79.8%

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

   1. *-commutative 79.8%

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

   2. associate-*r* 79.8%

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

   3. metadata-eval 79.8%

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

   4. hypot-undefine 39.8%

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

   5. unpow2 39.8%

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

   6. unpow2 39.8%

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

   7. +-commutative 39.8%

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

   8. unpow2 39.8%

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

   9. unpow2 39.8%

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

   10. hypot-undefine 79.8%

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

8. Simplified 79.8%

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

9. Taylor expanded in re around -inf 25.0%

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

    1. neg-mul-1 25.0%

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

11. Simplified 25.0%

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

    1. neg-sub0 25.0%

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

    2. sub-neg 25.0%

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

    3. add-sqr-sqrt 25.0%

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

    4. sqrt-unprod 16.2%

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

    5. sqr-neg 16.2%

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

    6. sqrt-unprod 2.8%

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

    7. add-sqr-sqrt 2.8%

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

13. Applied egg-rr 2.8%

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

    1. +-lft-identity 2.8%

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

15. Simplified 2.8%

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

Reproduce

herbie shell --seed 2024137 
(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)))))