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

Percentage Accurate: 42.1% → 90.4%

Time: 10.0s

Alternatives: 8

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: 42.1% 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: 90.4% accurate, 0.5× speedup

Math

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

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

   1. Initial program 11.4%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 11.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 11.4%

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

   5. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \color{blue}{\sqrt{2}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 98.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

   7. Simplified 98.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right)}\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{2} \cdot \frac{1}{re}}\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{2 \cdot \left(\frac{1}{2} \cdot \frac{1}{re}\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\left(2 \cdot \frac{1}{2}\right) \cdot \frac{1}{re}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{1 \cdot \frac{1}{re}}\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{re}}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{1}{\sqrt{\color{blue}{re}}}\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{im}{\color{blue}{\sqrt{re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \color{blue}{\left(\sqrt{re}\right)}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \mathsf{sqrt.f64}\left(re\right)\right)\right) \]

   9. Applied egg-rr 99.7%

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

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

   1. Initial program 46.4%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{re \cdot re + im \cdot im} - re\right)\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right)\right)\right)\right) \]

      5. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right)\right)\right)\right) \]

      6. hypot-lowering-hypot.f64 88.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right)\right)\right)\right) \]

   3. Simplified 88.8%

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

Alternative 2: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0102:\\ \;\;\;\;\sqrt{re \cdot -2} \cdot \left(0.5 \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(-2 + \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0102)
   (* (sqrt (* re -2.0)) (* 0.5 (sqrt 2.0)))
   (if (<= re 1.75e+83)
     (* 0.5 (sqrt (+ (* 2.0 im) (* re (+ -2.0 (/ re im))))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0102) {
		tmp = sqrt((re * -2.0)) * (0.5 * sqrt(2.0));
	} else if (re <= 1.75e+83) {
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (-2.0 + (re / 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 <= (-0.0102d0)) then
        tmp = sqrt((re * (-2.0d0))) * (0.5d0 * sqrt(2.0d0))
    else if (re <= 1.75d+83) then
        tmp = 0.5d0 * sqrt(((2.0d0 * im) + (re * ((-2.0d0) + (re / 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 <= -0.0102) {
		tmp = Math.sqrt((re * -2.0)) * (0.5 * Math.sqrt(2.0));
	} else if (re <= 1.75e+83) {
		tmp = 0.5 * Math.sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0102:
		tmp = math.sqrt((re * -2.0)) * (0.5 * math.sqrt(2.0))
	elif re <= 1.75e+83:
		tmp = 0.5 * math.sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0102)
		tmp = Float64(sqrt(Float64(re * -2.0)) * Float64(0.5 * sqrt(2.0)));
	elseif (re <= 1.75e+83)
		tmp = Float64(0.5 * sqrt(Float64(Float64(2.0 * im) + Float64(re * Float64(-2.0 + Float64(re / 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 <= -0.0102)
		tmp = sqrt((re * -2.0)) * (0.5 * sqrt(2.0));
	elseif (re <= 1.75e+83)
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0102], N[(N[Sqrt[N[(re * -2.0), $MachinePrecision]], $MachinePrecision] * N[(0.5 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.75e+83], N[(0.5 * N[Sqrt[N[(N[(2.0 * im), $MachinePrecision] + N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.010200000000000001

   1. Initial program 39.7%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 99.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot re\right)}, \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(re \cdot -2\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(re, -2\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   7. Simplified 77.2%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{re \cdot -2}\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(re \cdot -2\right)\right), \mathsf{*.f64}\left(\color{blue}{\frac{1}{2}}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

      3. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -2\right)\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   9. Applied egg-rr 77.2%

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

   if -0.010200000000000001 < re < 1.74999999999999989e83

   1. Initial program 52.2%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im + re \cdot \left(\frac{re}{im} - 2\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot im\right), \left(re \cdot \left(\frac{re}{im} - 2\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(im \cdot 2\right), \left(re \cdot \left(\frac{re}{im} - 2\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \left(re \cdot \left(\frac{re}{im} - 2\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \left(\frac{re}{im} - 2\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \left(\frac{re}{im} + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \left(\frac{re}{im} + -2\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \left(-2 + \frac{re}{im}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-2, \left(\frac{re}{im}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 77.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(re, im\right)\right)\right)\right)\right)\right) \]

   5. Simplified 77.4%

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

   if 1.74999999999999989e83 < re

   1. Initial program 11.5%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 38.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \color{blue}{\sqrt{2}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 87.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

   7. Simplified 87.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right)}\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{2} \cdot \frac{1}{re}}\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{2 \cdot \left(\frac{1}{2} \cdot \frac{1}{re}\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\left(2 \cdot \frac{1}{2}\right) \cdot \frac{1}{re}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{1 \cdot \frac{1}{re}}\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{re}}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{1}{\sqrt{\color{blue}{re}}}\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{im}{\color{blue}{\sqrt{re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \color{blue}{\left(\sqrt{re}\right)}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 88.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \mathsf{sqrt.f64}\left(re\right)\right)\right) \]

   9. Applied egg-rr 88.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0102:\\ \;\;\;\;\sqrt{re \cdot -2} \cdot \left(0.5 \cdot \sqrt{2}\right)\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(-2 + \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.00102:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.72 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(-2 + \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.00102)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.72e+83)
     (* 0.5 (sqrt (+ (* 2.0 im) (* re (+ -2.0 (/ re im))))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.00102) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.72e+83) {
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (-2.0 + (re / 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 <= (-0.00102d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.72d+83) then
        tmp = 0.5d0 * sqrt(((2.0d0 * im) + (re * ((-2.0d0) + (re / 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 <= -0.00102) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.72e+83) {
		tmp = 0.5 * Math.sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.00102:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.72e+83:
		tmp = 0.5 * math.sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.00102)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.72e+83)
		tmp = Float64(0.5 * sqrt(Float64(Float64(2.0 * im) + Float64(re * Float64(-2.0 + Float64(re / 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 <= -0.00102)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.72e+83)
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.00102], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.72e+83], N[(0.5 * N[Sqrt[N[(N[(2.0 * im), $MachinePrecision] + N[(re * N[(-2.0 + N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00102

   1. Initial program 39.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 76.0%

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

   if -0.00102 < re < 1.72000000000000006e83

   1. Initial program 52.2%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im + re \cdot \left(\frac{re}{im} - 2\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot im\right), \left(re \cdot \left(\frac{re}{im} - 2\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(im \cdot 2\right), \left(re \cdot \left(\frac{re}{im} - 2\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \left(re \cdot \left(\frac{re}{im} - 2\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \left(\frac{re}{im} - 2\right)\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \left(\frac{re}{im} + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \left(\frac{re}{im} + -2\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \left(-2 + \frac{re}{im}\right)\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-2, \left(\frac{re}{im}\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 77.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(im, 2\right), \mathsf{*.f64}\left(re, \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(re, im\right)\right)\right)\right)\right)\right) \]

   5. Simplified 77.4%

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

   if 1.72000000000000006e83 < re

   1. Initial program 11.5%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 38.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \color{blue}{\sqrt{2}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 87.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

   7. Simplified 87.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right)}\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{2} \cdot \frac{1}{re}}\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{2 \cdot \left(\frac{1}{2} \cdot \frac{1}{re}\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\left(2 \cdot \frac{1}{2}\right) \cdot \frac{1}{re}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{1 \cdot \frac{1}{re}}\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{re}}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{1}{\sqrt{\color{blue}{re}}}\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{im}{\color{blue}{\sqrt{re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \color{blue}{\left(\sqrt{re}\right)}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 88.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \mathsf{sqrt.f64}\left(re\right)\right)\right) \]

   9. Applied egg-rr 88.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.00102:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.72 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(-2 + \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(2 + \frac{re \cdot -2}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4e-5)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 7.5e+84)
     (* 0.5 (sqrt (* im (+ 2.0 (/ (* re -2.0) im)))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4e-5) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 7.5e+84) {
		tmp = 0.5 * sqrt((im * (2.0 + ((re * -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 <= (-4d-5)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 7.5d+84) then
        tmp = 0.5d0 * sqrt((im * (2.0d0 + ((re * (-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 <= -4e-5) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 7.5e+84) {
		tmp = 0.5 * Math.sqrt((im * (2.0 + ((re * -2.0) / im))));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4e-5:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 7.5e+84:
		tmp = 0.5 * math.sqrt((im * (2.0 + ((re * -2.0) / im))))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4e-5)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 7.5e+84)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(2.0 + Float64(Float64(re * -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 <= -4e-5)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 7.5e+84)
		tmp = 0.5 * sqrt((im * (2.0 + ((re * -2.0) / im))));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4e-5], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.5e+84], N[(0.5 * N[Sqrt[N[(im * N[(2.0 + N[(N[(re * -2.0), $MachinePrecision] / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.00000000000000033e-5

   1. Initial program 39.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 76.0%

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

   if -4.00000000000000033e-5 < re < 7.5000000000000001e84

   1. Initial program 52.2%

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

   3. Taylor expanded in im around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(im \cdot \left(2 + -2 \cdot \frac{re}{im}\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \left(2 + -2 \cdot \frac{re}{im}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \left(-2 \cdot \frac{re}{im}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \left(\frac{-2 \cdot re}{im}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(-2 \cdot re\right), im\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\left(re \cdot -2\right), im\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 77.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(re, -2\right), im\right)\right)\right)\right)\right) \]

   5. Simplified 77.4%

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

   if 7.5000000000000001e84 < re

   1. Initial program 11.5%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 38.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \color{blue}{\sqrt{2}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 87.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

   7. Simplified 87.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right)}\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{2} \cdot \frac{1}{re}}\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{2 \cdot \left(\frac{1}{2} \cdot \frac{1}{re}\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\left(2 \cdot \frac{1}{2}\right) \cdot \frac{1}{re}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{1 \cdot \frac{1}{re}}\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{re}}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{1}{\sqrt{\color{blue}{re}}}\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{im}{\color{blue}{\sqrt{re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \color{blue}{\left(\sqrt{re}\right)}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 88.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \mathsf{sqrt.f64}\left(re\right)\right)\right) \]

   9. Applied egg-rr 88.6%

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

Alternative 5: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0061:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.7 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0061)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.7e+83)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0061) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.7e+83) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} 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 <= (-0.0061d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.7d+83) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    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 <= -0.0061) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.7e+83) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0061:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.7e+83:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0061)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.7e+83)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	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 <= -0.0061)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.7e+83)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0061], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.7e+83], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00610000000000000039

   1. Initial program 39.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 76.0%

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

   if -0.00610000000000000039 < re < 3.7000000000000002e83

   1. Initial program 52.2%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{\_.f64}\left(\color{blue}{im}, re\right)\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 77.4%

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

   if 3.7000000000000002e83 < re

   1. Initial program 11.5%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 38.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \color{blue}{\sqrt{2}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 87.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

   7. Simplified 87.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right)}\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{2} \cdot \frac{1}{re}}\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{2 \cdot \left(\frac{1}{2} \cdot \frac{1}{re}\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\left(2 \cdot \frac{1}{2}\right) \cdot \frac{1}{re}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{1 \cdot \frac{1}{re}}\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{re}}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{1}{\sqrt{\color{blue}{re}}}\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{im}{\color{blue}{\sqrt{re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \color{blue}{\left(\sqrt{re}\right)}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 88.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \mathsf{sqrt.f64}\left(re\right)\right)\right) \]

   9. Applied egg-rr 88.6%

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

Alternative 6: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.7 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e-5)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.7e+83) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-5) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.7e+83) {
		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 <= (-3.4d-5)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.7d+83) 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 <= -3.4e-5) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.7e+83) {
		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 <= -3.4e-5:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.7e+83:
		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 <= -3.4e-5)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.7e+83)
		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 <= -3.4e-5)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.7e+83)
		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, -3.4e-5], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.7e+83], 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 3 regimes

2. if re < -3.4e-5

   1. Initial program 39.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 76.0%

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

   if -3.4e-5 < re < 3.7000000000000002e83

   1. Initial program 52.2%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 77.2%

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

   if 3.7000000000000002e83 < re

   1. Initial program 11.5%

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

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{re \cdot re + im \cdot im} - re\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)}\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{re \cdot re + im \cdot im} - re\right), \frac{1}{2}\right), \left(\color{blue}{\frac{1}{2}} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{re \cdot re + im \cdot im}\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{hypot}\left(re, im\right)\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      9. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \left(\frac{1}{2} \cdot {2}^{\frac{1}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({2}^{\frac{1}{2}}\right)}\right)\right) \]

      11. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{2}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 38.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(re, im\right), re\right), \frac{1}{2}\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(2\right)\right)\right) \]

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in re around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\left(im \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{\frac{1}{re}} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \color{blue}{\left(\sqrt{\frac{1}{re}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \color{blue}{\sqrt{2}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{re}} \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\frac{1}{re}}\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{re}\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \left(\sqrt{2}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 87.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(im, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, re\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

   7. Simplified 87.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{re}}\right)}\right)\right)\right) \]

      2. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{2} \cdot \frac{1}{re}}\right)\right)\right) \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{2 \cdot \left(\frac{1}{2} \cdot \frac{1}{re}\right)}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\left(2 \cdot \frac{1}{2}\right) \cdot \frac{1}{re}}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{1 \cdot \frac{1}{re}}\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \sqrt{\frac{1}{re}}\right)\right) \]

      7. sqrt-div N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{re}}}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(im \cdot \frac{1}{\sqrt{\color{blue}{re}}}\right)\right) \]

      9. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\frac{im}{\color{blue}{\sqrt{re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \color{blue}{\left(\sqrt{re}\right)}\right)\right) \]

      11. sqrt-lowering-sqrt.f64 88.6%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(im, \mathsf{sqrt.f64}\left(re\right)\right)\right) \]

   9. Applied egg-rr 88.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.7 \cdot 10^{+83}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.000155:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.000155) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.000155) {
		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 <= (-0.000155d0)) 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 <= -0.000155) {
		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 <= -0.000155:
		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 <= -0.000155)
		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 <= -0.000155)
		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, -0.000155], 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 < -1.55e-4

   1. Initial program 39.7%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(re \cdot -4\right)\right)\right) \]

      2. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(re, -4\right)\right)\right) \]

   5. Simplified 76.0%

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

   if -1.55e-4 < re

   1. Initial program 42.3%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

      2. *-lowering-*.f64 62.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

   5. Simplified 62.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.000155:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.0% 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 41.8%

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

3. Taylor expanded in re around 0

   \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im\right)}\right)\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(im \cdot 2\right)\right)\right) \]

   2. *-lowering-*.f64 55.4%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(im, 2\right)\right)\right) \]

5. Simplified 55.4%

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

6. Final simplification 55.4%

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

Reproduce

herbie shell --seed 2024161 
(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)))))