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

Percentage Accurate: 40.6% → 90.1%

Time: 10.6s

Alternatives: 7

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: 40.6% 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: 90.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = 0.5 * (im / sqrt(re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - 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 = 0.5 * (im / Math.sqrt(re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - 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 = 0.5 * (im / math.sqrt(re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - 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(0.5 * Float64(im / sqrt(re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - 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 = 0.5 * (im / sqrt(re));
	else
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - 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[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 17.1%

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

   2. Taylor expanded in re around inf 62.1%

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

      1. unpow2 62.1%

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

   4. Simplified 62.1%

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

      1. add-sqr-sqrt 61.7%

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

      2. *-un-lft-identity 61.7%

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

      3. metadata-eval 61.7%

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

      4. associate-*r* 61.7%

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

      5. *-commutative 61.7%

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

      6. associate-*r* 61.7%

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

      7. metadata-eval 61.7%

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

      8. *-un-lft-identity 61.7%

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

      9. div-inv 61.8%

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

      10. associate-*l* 61.8%

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

      11. add-sqr-sqrt 61.7%

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

      12. add-sqr-sqrt 61.3%

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

      13. swap-sqr 61.5%

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

      14. *-commutative 61.5%

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

      15. *-commutative 61.5%

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

      16. swap-sqr 64.3%

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

   6. Applied egg-rr 25.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 99.2%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

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

   1. Initial program 42.8%

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

      1. hypot-def 88.8%

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

   3. Simplified 88.8%

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

4. Final simplification 90.1%

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

Alternative 2: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.6e-20)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 2.1e+37)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.6e-20) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.1e+37) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -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 <= (-8.6d-20)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 2.1d+37) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.6e-20) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.1e+37) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.6e-20:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 2.1e+37:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.6e-20)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 2.1e+37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.6e-20)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 2.1e+37)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.6e-20], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+37], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.60000000000000022e-20

   1. Initial program 50.4%

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

   2. Taylor expanded in re around -inf 84.2%

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

      1. *-commutative 84.2%

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

   4. Simplified 84.2%

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

   if -8.60000000000000022e-20 < re < 2.1000000000000001e37

   1. Initial program 49.6%

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

   2. Taylor expanded in re around 0 78.9%

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

   if 2.1000000000000001e37 < re

   1. Initial program 10.6%

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

   2. Taylor expanded in re around inf 54.4%

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

      1. unpow2 54.4%

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

   4. Simplified 54.4%

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

      1. add-sqr-sqrt 54.2%

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

      2. *-un-lft-identity 54.2%

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

      3. metadata-eval 54.2%

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

      4. associate-*r* 54.2%

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

      5. *-commutative 54.2%

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

      6. associate-*r* 54.2%

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

      7. metadata-eval 54.2%

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

      8. *-un-lft-identity 54.2%

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

      9. div-inv 54.2%

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

      10. associate-*l* 54.2%

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

      11. add-sqr-sqrt 54.1%

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

      12. add-sqr-sqrt 53.8%

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

      13. swap-sqr 54.0%

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

      14. *-commutative 54.0%

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

      15. *-commutative 54.0%

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

      16. swap-sqr 58.8%

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

   6. Applied egg-rr 76.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.6 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e-18)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 2.1e+37)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (sqrt (/ 1.0 re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-18) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.1e+37) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / 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 <= (-6.2d-18)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 2.1d+37) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.2e-18) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 2.1e+37) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.2e-18:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 2.1e+37:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.2e-18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 2.1e+37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.2e-18)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 2.1e+37)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.2e-18], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+37], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.20000000000000014e-18

   1. Initial program 50.4%

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

   2. Taylor expanded in re around -inf 84.2%

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

      1. *-commutative 84.2%

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

   4. Simplified 84.2%

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

   if -6.20000000000000014e-18 < re < 2.1000000000000001e37

   1. Initial program 49.6%

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

   2. Taylor expanded in re around 0 78.9%

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

   if 2.1000000000000001e37 < re

   1. Initial program 10.6%

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

   2. Taylor expanded in re around inf 54.4%

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

      1. unpow2 54.4%

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

   4. Simplified 54.4%

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

   5. Taylor expanded in im around 0 76.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]

Alternative 4: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+88}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.15e-18)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 4.9e+88)
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15e-18) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 4.9e+88) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * pow(re, -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 <= (-1.15d-18)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 4.9d+88) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.15e-18) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 4.9e+88) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.15e-18:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 4.9e+88:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.15e-18)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 4.9e+88)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.15e-18)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 4.9e+88)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.15e-18], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.9e+88], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.15e-18

   1. Initial program 50.4%

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

   2. Taylor expanded in re around -inf 84.2%

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

      1. *-commutative 84.2%

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

   4. Simplified 84.2%

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

   if -1.15e-18 < re < 4.9000000000000002e88

   1. Initial program 46.0%

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

      1. hypot-udef 85.1%

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

      2. add-cube-cbrt 84.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}} \]

      3. pow3 84.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   3. Applied egg-rr 84.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   4. Taylor expanded in re around 0 76.4%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot im\right)}} \]
   5. Step-by-step derivation

      1. pow-base-1 76.4%

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

      2. *-lft-identity 76.4%

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

   6. Simplified 76.4%

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

   if 4.9000000000000002e88 < re

   1. Initial program 10.4%

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

   2. Taylor expanded in re around inf 58.5%

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

      1. unpow2 58.5%

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

   4. Simplified 58.5%

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

      1. add-sqr-sqrt 58.3%

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

      2. *-un-lft-identity 58.3%

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

      3. metadata-eval 58.3%

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

      4. associate-*r* 58.3%

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

      5. *-commutative 58.3%

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

      6. associate-*r* 58.3%

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

      7. metadata-eval 58.3%

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

      8. *-un-lft-identity 58.3%

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

      9. div-inv 58.2%

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

      10. associate-*l* 58.2%

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

      11. add-sqr-sqrt 58.2%

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

      12. add-sqr-sqrt 57.8%

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

      13. swap-sqr 58.1%

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

      14. *-commutative 58.1%

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

      15. *-commutative 58.1%

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

      16. swap-sqr 64.2%

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

   6. Applied egg-rr 82.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.9 \cdot 10^{+88}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.5 \cdot 10^{+88}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.5e+88) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.5e+88) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * pow(re, -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 <= 4.5d+88) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.5e+88) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.5e+88:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.5e+88)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.5e+88)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.5e+88], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.5e88

   1. Initial program 47.2%

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

      1. hypot-udef 89.2%

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

      2. add-cube-cbrt 88.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}} \]

      3. pow3 88.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   3. Applied egg-rr 88.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   4. Taylor expanded in re around 0 61.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot im\right)}} \]
   5. Step-by-step derivation

      1. pow-base-1 61.0%

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

      2. *-lft-identity 61.0%

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

   6. Simplified 61.0%

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

   if 4.5e88 < re

   1. Initial program 10.4%

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

   2. Taylor expanded in re around inf 58.5%

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

      1. unpow2 58.5%

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

   4. Simplified 58.5%

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

      1. add-sqr-sqrt 58.3%

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

      2. *-un-lft-identity 58.3%

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

      3. metadata-eval 58.3%

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

      4. associate-*r* 58.3%

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

      5. *-commutative 58.3%

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

      6. associate-*r* 58.3%

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

      7. metadata-eval 58.3%

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

      8. *-un-lft-identity 58.3%

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

      9. div-inv 58.2%

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

      10. associate-*l* 58.2%

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

      11. add-sqr-sqrt 58.2%

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

      12. add-sqr-sqrt 57.8%

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

      13. swap-sqr 58.1%

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

      14. *-commutative 58.1%

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

      15. *-commutative 58.1%

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

      16. swap-sqr 64.2%

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

   6. Applied egg-rr 82.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.5 \cdot 10^{+88}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 6: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.7 \cdot 10^{+88}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.7e+88) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.7e+88) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / 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 <= 4.7d+88) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.7e+88) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.7e+88:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.7e+88)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.7e+88)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.7e+88], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.70000000000000007e88

   1. Initial program 47.2%

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

      1. hypot-udef 89.2%

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

      2. add-cube-cbrt 88.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}} \]

      3. pow3 88.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   3. Applied egg-rr 88.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

   4. Taylor expanded in re around 0 61.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot im\right)}} \]
   5. Step-by-step derivation

      1. pow-base-1 61.0%

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

      2. *-lft-identity 61.0%

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

   6. Simplified 61.0%

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

   if 4.70000000000000007e88 < re

   1. Initial program 10.4%

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

   2. Taylor expanded in re around inf 58.5%

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

      1. unpow2 58.5%

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

   4. Simplified 58.5%

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

      1. add-sqr-sqrt 58.3%

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

      2. *-un-lft-identity 58.3%

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

      3. metadata-eval 58.3%

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

      4. associate-*r* 58.3%

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

      5. *-commutative 58.3%

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

      6. associate-*r* 58.3%

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

      7. metadata-eval 58.3%

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

      8. *-un-lft-identity 58.3%

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

      9. div-inv 58.2%

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

      10. associate-*l* 58.2%

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

      11. add-sqr-sqrt 58.2%

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

      12. add-sqr-sqrt 57.8%

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

      13. swap-sqr 58.1%

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

      14. *-commutative 58.1%

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

      15. *-commutative 58.1%

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

      16. swap-sqr 64.2%

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

   6. Applied egg-rr 40.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 81.8%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{im}{\sqrt{re}}\right)\right)} \]

      2. expm1-log1p 82.5%

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

   8. Simplified 82.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.7 \cdot 10^{+88}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 7: 53.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

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

   1. hypot-udef 79.8%

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

   2. add-cube-cbrt 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re} \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}} \]

   3. pow3 78.9%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

3. Applied egg-rr 78.9%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(re, im\right) - re}\right)}^{3}}} \]

4. Taylor expanded in re around 0 52.9%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot im\right)}} \]
5. Step-by-step derivation

   1. pow-base-1 52.9%

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

   2. *-lft-identity 52.9%

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

6. Simplified 52.9%

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

7. Final simplification 52.9%

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

Reproduce

herbie shell --seed 2023187 
(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)))))