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

Percentage Accurate: 40.4% → 90.1%

Time: 10.8s

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: 40.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;{\left(re \cdot re\right)}^{-0.25} \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* (pow (* re re) -0.25) (* im 0.5))
   (* 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 = pow((re * re), -0.25) * (im * 0.5);
	} 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 = Math.pow((re * re), -0.25) * (im * 0.5);
	} 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 = math.pow((re * re), -0.25) * (im * 0.5)
	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((Float64(re * re) ^ -0.25) * Float64(im * 0.5));
	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 = ((re * re) ^ -0.25) * (im * 0.5);
	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[(N[Power[N[(re * re), $MachinePrecision], -0.25], $MachinePrecision] * N[(im * 0.5), $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 12.6%

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

   3. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 52.2%

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

   5. Simplified 52.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im + \frac{-0.25 \cdot {im}^{4}}{re \cdot re}}{re}}} \]

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. /-lowering-/.f64 99.7%

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

   8. Simplified 99.7%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. pow-prod-down N/A

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

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

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

       5. *-lowering-*.f64 99.7%

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

   12. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{{\left(re \cdot re\right)}^{-0.25}} \cdot \left(0.5 \cdot im\right) \]

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

   1. Initial program 46.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 88.3%

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

   3. Simplified 88.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;{\left(re \cdot re\right)}^{-0.25} \cdot \left(im \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + \left(re \cdot re\right) \cdot \frac{0.5}{im}\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.1e-23)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.15e+49)
     (* 0.5 (sqrt (* 2.0 (- (+ im (* (* re re) (/ 0.5 im))) re))))
     (* (* im 0.5) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.1e-23) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.15e+49) {
		tmp = 0.5 * sqrt((2.0 * ((im + ((re * re) * (0.5 / im))) - re)));
	} else {
		tmp = (im * 0.5) * pow(re, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.1d-23)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.15d+49) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((im + ((re * re) * (0.5d0 / im))) - re)))
    else
        tmp = (im * 0.5d0) * (re ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.1e-23) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.15e+49) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im + ((re * re) * (0.5 / im))) - re)));
	} else {
		tmp = (im * 0.5) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.1e-23:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.15e+49:
		tmp = 0.5 * math.sqrt((2.0 * ((im + ((re * re) * (0.5 / im))) - re)))
	else:
		tmp = (im * 0.5) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.1e-23)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.15e+49)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im + Float64(Float64(re * re) * Float64(0.5 / im))) - re))));
	else
		tmp = Float64(Float64(im * 0.5) * (re ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.1e-23)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.15e+49)
		tmp = 0.5 * sqrt((2.0 * ((im + ((re * re) * (0.5 / im))) - re)));
	else
		tmp = (im * 0.5) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.1e-23], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.15e+49], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im + N[(N[(re * re), $MachinePrecision] * N[(0.5 / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.1000000000000001e-23

   1. Initial program 53.6%

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

   3. Taylor expanded in re around -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 75.5%

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

   5. Simplified 75.5%

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

   if -2.1000000000000001e-23 < re < 1.15000000000000001e49

   1. Initial program 49.3%

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

   3. Taylor expanded in re around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 79.8%

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

   5. Simplified 79.8%

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

   if 1.15000000000000001e49 < re

   1. Initial program 9.4%

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

   3. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 48.0%

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

   5. Simplified 48.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im + \frac{-0.25 \cdot {im}^{4}}{re \cdot re}}{re}}} \]

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. /-lowering-/.f64 83.1%

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

   8. Simplified 83.1%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 83.1%

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

   10. Applied egg-rr 83.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.1 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\left(im + \left(re \cdot re\right) \cdot \frac{0.5}{im}\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(-2 + \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.9e-23)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.9e+49)
     (* 0.5 (sqrt (+ (* 2.0 im) (* re (+ -2.0 (/ re im))))))
     (* (* im 0.5) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.9e-23) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.9e+49) {
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))));
	} else {
		tmp = (im * 0.5) * pow(re, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.9d-23)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.9d+49) then
        tmp = 0.5d0 * sqrt(((2.0d0 * im) + (re * ((-2.0d0) + (re / im)))))
    else
        tmp = (im * 0.5d0) * (re ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.9e-23) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.9e+49) {
		tmp = 0.5 * Math.sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))));
	} else {
		tmp = (im * 0.5) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.9e-23:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.9e+49:
		tmp = 0.5 * math.sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))))
	else:
		tmp = (im * 0.5) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.9e-23)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.9e+49)
		tmp = Float64(0.5 * sqrt(Float64(Float64(2.0 * im) + Float64(re * Float64(-2.0 + Float64(re / im))))));
	else
		tmp = Float64(Float64(im * 0.5) * (re ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.9e-23)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.9e+49)
		tmp = 0.5 * sqrt(((2.0 * im) + (re * (-2.0 + (re / im)))));
	else
		tmp = (im * 0.5) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.9e-23], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.9e+49], 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[(N[(im * 0.5), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.8999999999999998e-23

   1. Initial program 53.6%

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

   3. Taylor expanded in re around -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 75.5%

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

   5. Simplified 75.5%

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

   if -4.8999999999999998e-23 < re < 3.9000000000000001e49

   1. Initial program 49.3%

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

   3. Taylor expanded in re around 0

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot im + re \cdot \left(\frac{re}{im} - 2\right)\right)}\right)\right) \]
   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 79.8%

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

   5. Simplified 79.8%

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

   if 3.9000000000000001e49 < re

   1. Initial program 9.4%

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

   3. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 48.0%

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

   5. Simplified 48.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im + \frac{-0.25 \cdot {im}^{4}}{re \cdot re}}{re}}} \]

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. /-lowering-/.f64 83.1%

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

   8. Simplified 83.1%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 83.1%

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

   10. Applied egg-rr 83.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.9 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.9 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im + re \cdot \left(-2 + \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.8e-23)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.5e+49)
     (* 0.5 (sqrt (* 2.0 im)))
     (* (* im 0.5) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-23) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.5e+49) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = (im * 0.5) * pow(re, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.8d-23)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.5d+49) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = (im * 0.5d0) * (re ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.8e-23) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.5e+49) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = (im * 0.5) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.8e-23:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.5e+49:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = (im * 0.5) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.8e-23)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.5e+49)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(Float64(im * 0.5) * (re ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.8e-23)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.5e+49)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = (im * 0.5) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.8e-23], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.5e+49], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.79999999999999993e-23

   1. Initial program 53.6%

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

   3. Taylor expanded in re around -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 75.5%

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

   5. Simplified 75.5%

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

   if -4.79999999999999993e-23 < re < 1.5000000000000001e49

   1. Initial program 49.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 79.2%

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

   5. Simplified 79.2%

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

   if 1.5000000000000001e49 < re

   1. Initial program 9.4%

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

   3. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 48.0%

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

   5. Simplified 48.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im + \frac{-0.25 \cdot {im}^{4}}{re \cdot re}}{re}}} \]

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. /-lowering-/.f64 83.1%

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

   8. Simplified 83.1%

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

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 83.1%

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

   10. Applied egg-rr 83.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.8 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e-26)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.75e+49) (* 0.5 (sqrt (* 2.0 im))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-26) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.75e+49) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d-26)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.75d+49) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e-26) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.75e+49) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e-26:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.75e+49:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e-26)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.75e+49)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e-26)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.75e+49)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e-26], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.75e+49], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.8000000000000001e-26

   1. Initial program 53.6%

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

   3. Taylor expanded in re around -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 75.5%

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

   5. Simplified 75.5%

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

   if -1.8000000000000001e-26 < re < 1.74999999999999987e49

   1. Initial program 49.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 79.2%

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

   5. Simplified 79.2%

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

   if 1.74999999999999987e49 < re

   1. Initial program 9.4%

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

   3. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 48.0%

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

   5. Simplified 48.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im + \frac{-0.25 \cdot {im}^{4}}{re \cdot re}}{re}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. sqrt-div N/A

         \[\leadsto \frac{\sqrt{im \cdot im + \frac{\frac{-1}{4} \cdot {im}^{4}}{re \cdot re}}}{\sqrt{re}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{im \cdot im + \frac{\frac{-1}{4} \cdot {im}^{4}}{re \cdot re}} \cdot \frac{1}{2}\right), \color{blue}{\left(\sqrt{re}\right)}\right) \]

   7. Applied egg-rr 64.4%

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

   8. Taylor expanded in im around 0

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

   10. Simplified 82.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.5e-27)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.7e+49) (* 0.5 (sqrt (* 2.0 im))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.5e-27) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.7e+49) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-8.5d-27)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.7d+49) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.5e-27) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.7e+49) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.5e-27:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.7e+49:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.5e-27)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.7e+49)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.5e-27)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.7e+49)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.5e-27], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.7e+49], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.50000000000000033e-27

   1. Initial program 53.6%

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

   3. Taylor expanded in re around -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 75.5%

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

   5. Simplified 75.5%

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

   if -8.50000000000000033e-27 < re < 1.7e49

   1. Initial program 49.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 79.2%

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

   5. Simplified 79.2%

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

   if 1.7e49 < re

   1. Initial program 9.4%

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

   3. Taylor expanded in re around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 48.0%

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

   5. Simplified 48.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{im \cdot im + \frac{-0.25 \cdot {im}^{4}}{re \cdot re}}{re}}} \]

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. /-lowering-/.f64 83.1%

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

   8. Simplified 83.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. div-inv N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 82.8%

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

   10. Applied egg-rr 82.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-27}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.35e-26) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.35e-26) {
		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 <= (-1.35d-26)) 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 <= -1.35e-26) {
		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 <= -1.35e-26:
		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 <= -1.35e-26)
		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 <= -1.35e-26)
		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, -1.35e-26], 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 < -1.34999999999999991e-26

   1. Initial program 53.6%

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

   3. Taylor expanded in re around -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 75.5%

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

   5. Simplified 75.5%

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

   if -1.34999999999999991e-26 < re

   1. Initial program 37.8%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 62.9%

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

   5. Simplified 62.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.35 \cdot 10^{-26}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.3% 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 53.8%

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

5. Simplified 53.8%

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

6. Final simplification 53.8%

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

Reproduce

herbie shell --seed 2024144 
(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)))))