math.sqrt on complex, real part

Percentage Accurate: 41.6% → 83.7%

Time: 8.8s

Alternatives: 5

Speedup: 2.0×

Specification

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: 41.6% 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: 83.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 10.0%

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

      1. sqr-neg 10.0%

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

      2. +-commutative 10.0%

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

      3. sqr-neg 10.0%

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

      4. +-commutative 10.0%

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

      5. distribute-rgt-in 10.0%

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

      6. cancel-sign-sub 10.0%

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

      7. distribute-rgt-out-- 10.0%

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

      8. sub-neg 10.0%

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

      9. remove-double-neg 10.0%

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

      10. +-commutative 10.0%

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

      11. hypot-define 17.0%

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

   3. Simplified 17.0%

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

      1. hypot-define 10.0%

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

      2. distribute-lft-in 10.0%

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

      3. +-commutative 10.0%

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

      4. *-commutative 10.0%

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

      5. add-sqr-sqrt 5.5%

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

      6. fma-define 2.8%

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

      7. hypot-define 2.8%

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

      8. hypot-define 9.8%

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

   6. Applied egg-rr 9.8%

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

   7. Taylor expanded in re around -inf 5.0%

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

      1. mul-1-neg 5.0%

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

      2. *-commutative 5.0%

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

      3. distribute-rgt-neg-in 5.0%

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

      4. unpow2 5.0%

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

      5. rem-square-sqrt 10.0%

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

      6. associate-+r- 40.6%

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

      7. unpow2 40.6%

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

      8. rem-square-sqrt 40.9%

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

      9. associate-*r/ 40.9%

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

      10. metadata-eval 40.9%

          \[\leadsto 0.5 \cdot \sqrt{\left(0.5 \cdot \left({im}^{2} \cdot \frac{2}{{re}^{2}}\right) + \color{blue}{0}\right) \cdot \left(-re\right)} \]

   9. Simplified 40.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(0.5 \cdot \left({im}^{2} \cdot \frac{2}{{re}^{2}}\right) + 0\right) \cdot \left(-re\right)}} \]
   10. Step-by-step derivation

       1. +-rgt-identity 40.9%

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

       2. distribute-rgt-neg-out 40.9%

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

       3. neg-sub0 40.9%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - \left(0.5 \cdot \left({im}^{2} \cdot \frac{2}{{re}^{2}}\right)\right) \cdot re}} \]

       4. *-commutative 40.9%

          \[\leadsto 0.5 \cdot \sqrt{0 - \color{blue}{\left(\left({im}^{2} \cdot \frac{2}{{re}^{2}}\right) \cdot 0.5\right)} \cdot re} \]

       5. associate-*l* 41.0%

          \[\leadsto 0.5 \cdot \sqrt{0 - \color{blue}{\left({im}^{2} \cdot \frac{2}{{re}^{2}}\right) \cdot \left(0.5 \cdot re\right)}} \]

       6. add-sqr-sqrt 40.9%

          \[\leadsto 0.5 \cdot \sqrt{0 - \color{blue}{\left(\sqrt{{im}^{2} \cdot \frac{2}{{re}^{2}}} \cdot \sqrt{{im}^{2} \cdot \frac{2}{{re}^{2}}}\right)} \cdot \left(0.5 \cdot re\right)} \]

       7. pow2 40.9%

          \[\leadsto 0.5 \cdot \sqrt{0 - \color{blue}{{\left(\sqrt{{im}^{2} \cdot \frac{2}{{re}^{2}}}\right)}^{2}} \cdot \left(0.5 \cdot re\right)} \]

       8. sqrt-prod 40.9%

          \[\leadsto 0.5 \cdot \sqrt{0 - {\color{blue}{\left(\sqrt{{im}^{2}} \cdot \sqrt{\frac{2}{{re}^{2}}}\right)}}^{2} \cdot \left(0.5 \cdot re\right)} \]

       9. sqrt-pow1 47.1%

          \[\leadsto 0.5 \cdot \sqrt{0 - {\left(\color{blue}{{im}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{2}{{re}^{2}}}\right)}^{2} \cdot \left(0.5 \cdot re\right)} \]

       10. metadata-eval 47.1%

           \[\leadsto 0.5 \cdot \sqrt{0 - {\left({im}^{\color{blue}{1}} \cdot \sqrt{\frac{2}{{re}^{2}}}\right)}^{2} \cdot \left(0.5 \cdot re\right)} \]

       11. pow1 47.1%

           \[\leadsto 0.5 \cdot \sqrt{0 - {\left(\color{blue}{im} \cdot \sqrt{\frac{2}{{re}^{2}}}\right)}^{2} \cdot \left(0.5 \cdot re\right)} \]

       12. sqrt-div 47.0%

           \[\leadsto 0.5 \cdot \sqrt{0 - {\left(im \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{{re}^{2}}}}\right)}^{2} \cdot \left(0.5 \cdot re\right)} \]

       13. sqrt-pow1 55.9%

           \[\leadsto 0.5 \cdot \sqrt{0 - {\left(im \cdot \frac{\sqrt{2}}{\color{blue}{{re}^{\left(\frac{2}{2}\right)}}}\right)}^{2} \cdot \left(0.5 \cdot re\right)} \]

       14. metadata-eval 55.9%

           \[\leadsto 0.5 \cdot \sqrt{0 - {\left(im \cdot \frac{\sqrt{2}}{{re}^{\color{blue}{1}}}\right)}^{2} \cdot \left(0.5 \cdot re\right)} \]

       15. pow1 55.9%

           \[\leadsto 0.5 \cdot \sqrt{0 - {\left(im \cdot \frac{\sqrt{2}}{\color{blue}{re}}\right)}^{2} \cdot \left(0.5 \cdot re\right)} \]

   11. Applied egg-rr 55.9%

       \[\leadsto 0.5 \cdot \sqrt{\color{blue}{0 - {\left(im \cdot \frac{\sqrt{2}}{re}\right)}^{2} \cdot \left(0.5 \cdot re\right)}} \]
   12. Step-by-step derivation

       1. neg-sub0 55.9%

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

       2. distribute-rgt-neg-in 55.9%

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

       3. distribute-lft-neg-in 55.9%

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

       4. metadata-eval 55.9%

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

   13. Simplified 55.9%

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

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

   1. Initial program 43.1%

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

      1. sqr-neg 43.1%

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

      2. +-commutative 43.1%

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

      3. sqr-neg 43.1%

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

      4. +-commutative 43.1%

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

      5. distribute-rgt-in 43.1%

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

      6. cancel-sign-sub 43.1%

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

      7. distribute-rgt-out-- 43.1%

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

      8. sub-neg 43.1%

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

      9. remove-double-neg 43.1%

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

      10. +-commutative 43.1%

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

      11. hypot-define 91.2%

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

   3. Simplified 91.2%

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

      1. *-commutative 91.2%

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

      2. hypot-define 43.1%

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

      3. +-commutative 43.1%

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

      4. *-commutative 43.1%

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

      5. add-sqr-sqrt 42.8%

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

      6. sqrt-unprod 43.1%

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

      7. *-commutative 43.1%

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

      8. *-commutative 43.1%

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

      9. swap-sqr 43.1%

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

   6. Applied egg-rr 91.2%

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

      1. *-commutative 91.2%

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

      2. associate-*r* 91.2%

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

      3. metadata-eval 91.2%

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

   8. Simplified 91.2%

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

4. Final simplification 84.9%

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

Alternative 2: 84.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(N[Power[im, 2.0], $MachinePrecision] / (-re)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $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 12.1%

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

      1. sqr-neg 12.1%

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

      2. +-commutative 12.1%

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

      3. sqr-neg 12.1%

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

      4. +-commutative 12.1%

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

      5. distribute-rgt-in 12.1%

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

      6. cancel-sign-sub 12.1%

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

      7. distribute-rgt-out-- 12.1%

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

      8. sub-neg 12.1%

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

      9. remove-double-neg 12.1%

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

      10. +-commutative 12.1%

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

      11. hypot-define 12.1%

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

   3. Simplified 12.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in re around -inf 51.9%

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

      1. mul-1-neg 51.9%

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

      2. distribute-neg-frac2 51.9%

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

   7. Simplified 51.9%

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

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

   1. Initial program 41.5%

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

      1. sqr-neg 41.5%

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

      2. +-commutative 41.5%

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

      3. sqr-neg 41.5%

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

      4. +-commutative 41.5%

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

      5. distribute-rgt-in 41.5%

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

      6. cancel-sign-sub 41.5%

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

      7. distribute-rgt-out-- 41.5%

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

      8. sub-neg 41.5%

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

      9. remove-double-neg 41.5%

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

      10. +-commutative 41.5%

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

      11. hypot-define 89.4%

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

   3. Simplified 89.4%

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

      1. *-commutative 89.4%

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

      2. hypot-define 41.5%

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

      3. +-commutative 41.5%

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

      4. *-commutative 41.5%

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

      5. add-sqr-sqrt 41.2%

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

      6. sqrt-unprod 41.5%

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

      7. *-commutative 41.5%

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

      8. *-commutative 41.5%

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

      9. swap-sqr 41.5%

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

   6. Applied egg-rr 89.4%

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

      1. *-commutative 89.4%

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

      2. associate-*r* 89.4%

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

      3. metadata-eval 89.4%

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

   8. Simplified 89.4%

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

4. Final simplification 83.8%

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

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. sqr-neg 37.1%

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

   2. +-commutative 37.1%

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

   3. sqr-neg 37.1%

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

   4. +-commutative 37.1%

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

   5. distribute-rgt-in 37.1%

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

   6. cancel-sign-sub 37.1%

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

   7. distribute-rgt-out-- 37.1%

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

   8. sub-neg 37.1%

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

   9. remove-double-neg 37.1%

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

   10. +-commutative 37.1%

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

   11. hypot-define 77.9%

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

3. Simplified 77.9%

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

   1. *-commutative 77.9%

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

   2. hypot-define 37.1%

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

   3. +-commutative 37.1%

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

   4. *-commutative 37.1%

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

   5. add-sqr-sqrt 36.9%

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

   6. sqrt-unprod 37.1%

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

   7. *-commutative 37.1%

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

   8. *-commutative 37.1%

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

   9. 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 77.9%

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

   1. *-commutative 77.9%

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

   2. associate-*r* 77.9%

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

   3. metadata-eval 77.9%

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

8. Simplified 77.9%

   \[\leadsto \color{blue}{\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
9. Add Preprocessing

Alternative 4: 42.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.4e-13) (sqrt (* im 0.5)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.4e-13) {
		tmp = sqrt((im * 0.5));
	} 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 <= 1.4d-13) then
        tmp = sqrt((im * 0.5d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.4e-13) {
		tmp = Math.sqrt((im * 0.5));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.4e-13:
		tmp = math.sqrt((im * 0.5))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.4e-13)
		tmp = sqrt(Float64(im * 0.5));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.4e-13)
		tmp = sqrt((im * 0.5));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.4e-13], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.4000000000000001e-13

   1. Initial program 37.0%

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

      1. sqr-neg 37.0%

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

      2. +-commutative 37.0%

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

      3. sqr-neg 37.0%

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

      4. +-commutative 37.0%

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

      5. distribute-rgt-in 37.0%

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

      6. cancel-sign-sub 37.0%

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

      7. distribute-rgt-out-- 37.0%

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

      8. sub-neg 37.0%

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

      9. remove-double-neg 37.0%

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

      10. +-commutative 37.0%

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

      11. hypot-define 70.2%

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

   3. Simplified 70.2%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in re around 0 29.0%

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

      1. associate-*r* 29.0%

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

   7. Simplified 29.0%

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

      1. add-sqr-sqrt 28.8%

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

      2. pow2 28.8%

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

      3. associate-*l* 28.8%

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

      4. sqrt-unprod 28.9%

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

   9. Applied egg-rr 28.9%

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

       1. unpow2 28.9%

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

       2. sqrt-unprod 29.2%

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

       3. *-commutative 29.2%

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

       4. *-commutative 29.2%

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

       5. swap-sqr 29.2%

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

       6. add-sqr-sqrt 29.2%

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

       7. metadata-eval 29.2%

          \[\leadsto \sqrt{\left(im \cdot 2\right) \cdot \color{blue}{0.25}} \]

   11. Applied egg-rr 29.2%

       \[\leadsto \color{blue}{\sqrt{\left(im \cdot 2\right) \cdot 0.25}} \]
   12. Step-by-step derivation

       1. associate-*l* 29.2%

          \[\leadsto \sqrt{\color{blue}{im \cdot \left(2 \cdot 0.25\right)}} \]

       2. metadata-eval 29.2%

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

   13. Simplified 29.2%

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

   if 1.4000000000000001e-13 < re

   1. Initial program 37.6%

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

      1. sqr-neg 37.6%

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

      2. +-commutative 37.6%

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

      3. sqr-neg 37.6%

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

      4. +-commutative 37.6%

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

      5. distribute-rgt-in 37.6%

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

      6. cancel-sign-sub 37.6%

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

      7. distribute-rgt-out-- 37.6%

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

      8. sub-neg 37.6%

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

      9. remove-double-neg 37.6%

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

      10. +-commutative 37.6%

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

      11. hypot-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in re around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. unpow2 74.8%

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

      3. rem-square-sqrt 76.3%

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

      4. associate-*r* 76.3%

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

      5. metadata-eval 76.3%

         \[\leadsto \color{blue}{1} \cdot \sqrt{re} \]

   7. Simplified 76.3%

      \[\leadsto \color{blue}{1 \cdot \sqrt{re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 26.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. sqr-neg 37.1%

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

   2. +-commutative 37.1%

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

   3. sqr-neg 37.1%

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

   4. +-commutative 37.1%

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

   5. distribute-rgt-in 37.1%

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

   6. cancel-sign-sub 37.1%

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

   7. distribute-rgt-out-- 37.1%

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

   8. sub-neg 37.1%

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

   9. remove-double-neg 37.1%

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

   10. +-commutative 37.1%

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

   11. hypot-define 77.9%

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

3. Simplified 77.9%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in re around 0 25.2%

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

   1. associate-*r* 25.2%

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

7. Simplified 25.2%

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

   1. add-sqr-sqrt 25.1%

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

   2. pow2 25.1%

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

   3. associate-*l* 25.1%

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

   4. sqrt-unprod 25.2%

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

9. Applied egg-rr 25.2%

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

    1. unpow2 25.2%

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

    2. sqrt-unprod 25.4%

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

    3. *-commutative 25.4%

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

    4. *-commutative 25.4%

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

    5. swap-sqr 25.4%

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

    6. add-sqr-sqrt 25.4%

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

    7. metadata-eval 25.4%

       \[\leadsto \sqrt{\left(im \cdot 2\right) \cdot \color{blue}{0.25}} \]

11. Applied egg-rr 25.4%

    \[\leadsto \color{blue}{\sqrt{\left(im \cdot 2\right) \cdot 0.25}} \]
12. Step-by-step derivation

    1. associate-*l* 25.4%

       \[\leadsto \sqrt{\color{blue}{im \cdot \left(2 \cdot 0.25\right)}} \]

    2. metadata-eval 25.4%

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

13. Simplified 25.4%

    \[\leadsto \color{blue}{\sqrt{im \cdot 0.5}} \]
14. Add Preprocessing

Developer target: 48.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t\_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t\_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024101 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))