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

Percentage Accurate: 41.4% → 89.9%

Time: 12.1s

Alternatives: 7

Speedup: 2.2×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (* im (sqrt (/ 1.0 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((1.0 / 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((1.0 / 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((1.0 / 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(Float64(1.0 / 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((1.0 / 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[N[(1.0 / re), $MachinePrecision]], $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 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 5.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{{im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{{im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{{im}^{2}}{{re}^{3}} \cdot \frac{-1}{4}} + \frac{1}{re}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{im}^{2}}{{re}^{3}}, \frac{-1}{4}, \frac{1}{re}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{im \cdot im}}{{re}^{3}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \color{blue}{\frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      10. cube-mult N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{re \cdot \left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{{re}^{2}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{re \cdot {re}^{2}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{\left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{\left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      15. lower-/.f64 55.1

          \[\leadsto 0.5 \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \color{blue}{\frac{1}{re}}\right)} \]

   5. Simplified 55.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \frac{1}{re}\right)}} \]

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 99.7

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

   8. Simplified 99.7%

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

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

   1. Initial program 43.7%

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

      1. lower-hypot.f64 91.6

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

   4. Applied egg-rr 91.6%

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

Alternative 2: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{im - re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))
   (if (<= t_0 0.0)
     (* 0.5 (* im (sqrt (/ 1.0 re))))
     (if (<= t_0 1e+77)
       (* 0.5 (sqrt (* 2.0 (- (sqrt (fma re re (* im im))) re))))
       (* (* 0.5 (sqrt 2.0)) (sqrt (- im re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	} else if (t_0 <= 1e+77) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(re, re, (im * im))) - re)));
	} else {
		tmp = (0.5 * sqrt(2.0)) * sqrt((im - re));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	elseif (t_0 <= 1e+77)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(re, re, Float64(im * im))) - re))));
	else
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * sqrt(Float64(im - re)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+77], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(re * re + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(im - re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.5%

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

   3. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{{im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{{im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{{im}^{2}}{{re}^{3}} \cdot \frac{-1}{4}} + \frac{1}{re}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{im}^{2}}{{re}^{3}}, \frac{-1}{4}, \frac{1}{re}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{im \cdot im}}{{re}^{3}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \color{blue}{\frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      10. cube-mult N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{re \cdot \left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{{re}^{2}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{re \cdot {re}^{2}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{\left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{\left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      15. lower-/.f64 55.1

          \[\leadsto 0.5 \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \color{blue}{\frac{1}{re}}\right)} \]

   5. Simplified 55.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \frac{1}{re}\right)}} \]

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 99.7

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

   8. Simplified 99.7%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 9.99999999999999983e76

   1. Initial program 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 91.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.3

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 91.3

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

   4. Applied egg-rr 91.3%

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

   if 9.99999999999999983e76 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 3.6%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 61.9

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

   5. Simplified 61.9%

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

      1. lift--.f64 N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 62.2

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

   7. Applied egg-rr 62.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 10^{+77}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(re, re, im \cdot im\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{im - re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.2 \cdot 10^{-24}:\\ \;\;\;\;\left(0.5 \cdot \sqrt{2}\right) \cdot \sqrt{im - re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+112)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 7.2e-24)
     (* (* 0.5 (sqrt 2.0)) (sqrt (- im re)))
     (* 0.5 (* im (sqrt (/ 1.0 re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+112) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 7.2e-24) {
		tmp = (0.5 * sqrt(2.0)) * sqrt((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 <= (-1.8d+112)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 7.2d-24) then
        tmp = (0.5d0 * sqrt(2.0d0)) * sqrt((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 <= -1.8e+112) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 7.2e-24) {
		tmp = (0.5 * Math.sqrt(2.0)) * Math.sqrt((im - re));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+112:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 7.2e-24:
		tmp = (0.5 * math.sqrt(2.0)) * math.sqrt((im - re))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+112)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 7.2e-24)
		tmp = Float64(Float64(0.5 * sqrt(2.0)) * sqrt(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 <= -1.8e+112)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 7.2e-24)
		tmp = (0.5 * sqrt(2.0)) * sqrt((im - re));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+112], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.2e-24], N[(N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(im - re), $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 < -1.8e112

   1. Initial program 21.5%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 81.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 81.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.8e112 < re < 7.2000000000000002e-24

   1. Initial program 53.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 77.8

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

   5. Simplified 77.8%

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

      1. lift--.f64 N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. pow1/2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow1/2 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow1/2 N/A

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

      11. lower-sqrt.f64 77.9

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

   7. Applied egg-rr 77.9%

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

   if 7.2000000000000002e-24 < re

   1. Initial program 9.7%

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

   3. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{{im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{{im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{{im}^{2}}{{re}^{3}} \cdot \frac{-1}{4}} + \frac{1}{re}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{im}^{2}}{{re}^{3}}, \frac{-1}{4}, \frac{1}{re}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{im \cdot im}}{{re}^{3}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \color{blue}{\frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      10. cube-mult N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{re \cdot \left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{{re}^{2}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{re \cdot {re}^{2}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{\left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{\left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      15. lower-/.f64 51.9

          \[\leadsto 0.5 \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \color{blue}{\frac{1}{re}}\right)} \]

   5. Simplified 51.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \frac{1}{re}\right)}} \]

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 80.0

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

   8. Simplified 80.0%

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

Alternative 4: 76.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+112)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 6.4e-24)
     (* 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 <= -1.8e+112) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 6.4e-24) {
		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 <= (-1.8d+112)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 6.4d-24) 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 <= -1.8e+112) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 6.4e-24) {
		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 <= -1.8e+112:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 6.4e-24:
		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 <= -1.8e+112)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 6.4e-24)
		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 <= -1.8e+112)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 6.4e-24)
		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, -1.8e+112], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.4e-24], 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 < -1.8e112

   1. Initial program 21.5%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 81.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 81.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.8e112 < re < 6.40000000000000025e-24

   1. Initial program 53.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 77.8

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

   5. Simplified 77.8%

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

   if 6.40000000000000025e-24 < re

   1. Initial program 9.7%

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

   3. Taylor expanded in im around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{{im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{{im}^{2} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\left(im \cdot im\right)} \cdot \left(\frac{-1}{4} \cdot \frac{{im}^{2}}{{re}^{3}} + \frac{1}{re}\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \left(\color{blue}{\frac{{im}^{2}}{{re}^{3}} \cdot \frac{-1}{4}} + \frac{1}{re}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{{im}^{2}}{{re}^{3}}, \frac{-1}{4}, \frac{1}{re}\right)}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\frac{\color{blue}{im \cdot im}}{{re}^{3}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(\color{blue}{im \cdot \frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \color{blue}{\frac{im}{{re}^{3}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      10. cube-mult N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{re \cdot \left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{{re}^{2}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{\color{blue}{re \cdot {re}^{2}}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{\left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \color{blue}{\left(re \cdot re\right)}}, \frac{-1}{4}, \frac{1}{re}\right)} \]

      15. lower-/.f64 51.9

          \[\leadsto 0.5 \cdot \sqrt{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \color{blue}{\frac{1}{re}}\right)} \]

   5. Simplified 51.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im \cdot im\right) \cdot \mathsf{fma}\left(im \cdot \frac{im}{re \cdot \left(re \cdot re\right)}, -0.25, \frac{1}{re}\right)}} \]

   6. Taylor expanded in im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 80.0

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

   8. Simplified 80.0%

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

Alternative 5: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+112}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+112)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 6.4e-24)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+112) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 6.4e-24) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d+112)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 6.4d-24) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+112) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 6.4e-24) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+112:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 6.4e-24:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+112)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 6.4e-24)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e+112)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 6.4e-24)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+112], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.4e-24], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.8e112

   1. Initial program 21.5%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 81.9

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 81.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.8e112 < re < 6.40000000000000025e-24

   1. Initial program 53.8%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 77.8

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

   5. Simplified 77.8%

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

   if 6.40000000000000025e-24 < re

   1. Initial program 9.7%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{{im}^{2}}{re}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

      3. lower-*.f64 52.1

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

   5. Simplified 52.1%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\frac{\color{blue}{im \cdot im}}{re}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{im \cdot im}{re}}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{im \cdot im}{re}} \cdot \frac{1}{2}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{im \cdot im}{re}}} \cdot \frac{1}{2} \]

      6. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{im \cdot im}{re}}} \cdot \frac{1}{2} \]

      7. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{im \cdot im}}{\sqrt{re}}} \cdot \frac{1}{2} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sqrt{im \cdot im} \cdot \frac{1}{2}}{\sqrt{re}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{im \cdot im} \cdot \frac{1}{2}}{\sqrt{re}}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{im \cdot im}} \cdot \frac{1}{2}}{\sqrt{re}} \]

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

          \[\leadsto \frac{\color{blue}{im} \cdot \frac{1}{2}}{\sqrt{re}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{im \cdot \frac{1}{2}}}{\sqrt{re}} \]

      14. lower-sqrt.f64 79.9

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

   7. Applied egg-rr 79.9%

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

Alternative 6: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{+78}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.5e+78) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.5e+78) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.5d+78)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.5e+78) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.5e+78:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.5e+78)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.5e+78)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.5e+78], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.5000000000000001e78

   1. Initial program 24.3%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{re \cdot -4}} \]

      2. lower-*.f64 79.2

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   5. Simplified 79.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -3.5000000000000001e78 < re

   1. Initial program 40.1%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{im \cdot 2}} \]

      2. lower-*.f64 60.6

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

   5. Simplified 60.6%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{+78}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.0% accurate, 2.2× 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 37.5%

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

3. Taylor expanded in re around 0

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

   1. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \sqrt{\color{blue}{im \cdot 2}} \]

   2. lower-*.f64 54.3

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

5. Simplified 54.3%

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

6. Final simplification 54.3%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))))