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

Percentage Accurate: 42.0% → 89.8%

Time: 12.9s

Alternatives: 7

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (/ (* im 0.5) (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 = (im * 0.5) / 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 = (im * 0.5) / 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 = (im * 0.5) / 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(Float64(im * 0.5) / 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 = (im * 0.5) / 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[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $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 8.7%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 49.4%

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

   5. Simplified 49.4%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-unprod N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 99.6%

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

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

   1. Initial program 42.5%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 91.9%

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

   3. Simplified 91.9%

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

4. Final simplification 92.7%

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

Alternative 2: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+104}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.8e+104)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4e-44)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* im 0.5) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.8e+104) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4e-44) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-3.8d+104)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4d-44) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -3.8e+104) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4e-44) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.8e+104:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4e-44:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.8e+104)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4e-44)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(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 <= -3.8e+104)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4e-44)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.8e+104], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4e-44], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - 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 < -3.79999999999999969e104

   1. Initial program 26.2%

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

   3. Taylor expanded in re around -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 82.8%

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

   5. Simplified 82.8%

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

   if -3.79999999999999969e104 < re < 3.99999999999999981e-44

   1. Initial program 55.6%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 77.4%

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

   5. Simplified 77.4%

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

   if 3.99999999999999981e-44 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 39.8%

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

   5. Simplified 39.8%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sqrt-unprod N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

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

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

      18. pow1/2 N/A

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

      19. inv-pow N/A

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

      20. pow-pow N/A

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

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

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

      22. metadata-eval N/A

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

      23. *-commutative N/A

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

   7. Applied egg-rr 74.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.8 \cdot 10^{+104}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.36e+63)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.75e-19)
     (* 0.5 (sqrt (* 2.0 im)))
     (* (* im 0.5) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.36e+63) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.75e-19) {
		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 <= (-1.36d+63)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.75d-19) 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 <= -1.36e+63) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.75e-19) {
		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 <= -1.36e+63:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.75e-19:
		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 <= -1.36e+63)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.75e-19)
		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 <= -1.36e+63)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.75e-19)
		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, -1.36e+63], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.75e-19], 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 < -1.36000000000000006e63

   1. Initial program 29.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.6%

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

   5. Simplified 78.6%

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

   if -1.36000000000000006e63 < re < 1.75000000000000008e-19

   1. Initial program 54.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 74.7%

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

   5. Simplified 74.7%

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

   if 1.75000000000000008e-19 < re

   1. Initial program 8.8%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 41.2%

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

   5. Simplified 41.2%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. sqrt-unprod N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. *-commutative N/A

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

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

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

      18. pow1/2 N/A

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

      19. inv-pow N/A

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

      20. pow-pow N/A

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

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

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

      22. metadata-eval N/A

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

      23. *-commutative N/A

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

   7. Applied egg-rr 77.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.36 \cdot 10^{+63}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 10^{-14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.2e+62)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1e-14) (* 0.5 (sqrt (* 2.0 im))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.2e+62) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1e-14) {
		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 <= (-6.2d+62)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1d-14) 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 <= -6.2e+62) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1e-14) {
		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 <= -6.2e+62:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1e-14:
		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 <= -6.2e+62)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1e-14)
		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 <= -6.2e+62)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1e-14)
		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, -6.2e+62], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e-14], 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 < -6.20000000000000029e62

   1. Initial program 29.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.6%

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

   5. Simplified 78.6%

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

   if -6.20000000000000029e62 < re < 9.99999999999999999e-15

   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 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 74.4%

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

   5. Simplified 74.4%

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

   if 9.99999999999999999e-15 < re

   1. Initial program 8.9%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 42.5%

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

   5. Simplified 42.5%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

      6. metadata-eval N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sqrt-unprod N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 77.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.2 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 10^{-14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.4e+68)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.1e-15) (* 0.5 (sqrt (* 2.0 im))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.4e+68) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.1e-15) {
		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 <= (-6.4d+68)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.1d-15) 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 <= -6.4e+68) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.1e-15) {
		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 <= -6.4e+68:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.1e-15:
		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 <= -6.4e+68)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.1e-15)
		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 <= -6.4e+68)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.1e-15)
		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, -6.4e+68], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e-15], 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 < -6.39999999999999989e68

   1. Initial program 29.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.6%

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

   5. Simplified 78.6%

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

   if -6.39999999999999989e68 < re < 2.09999999999999981e-15

   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 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 74.4%

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

   5. Simplified 74.4%

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

   if 2.09999999999999981e-15 < re

   1. Initial program 8.9%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 42.5%

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

   5. Simplified 42.5%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. sqrt-prod N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. un-div-inv N/A

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

      12. /-lowering-/.f64 N/A

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

      13. sqrt-lowering-sqrt.f64 77.8%

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

   7. Applied egg-rr 77.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e+68) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e+68) {
		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 <= (-2.4d+68)) 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 <= -2.4e+68) {
		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 <= -2.4e+68:
		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 <= -2.4e+68)
		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 <= -2.4e+68)
		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, -2.4e+68], 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 < -2.40000000000000008e68

   1. Initial program 29.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.6%

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

   5. Simplified 78.6%

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

   if -2.40000000000000008e68 < re

   1. Initial program 41.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 61.6%

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

   5. Simplified 61.6%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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 54.1%

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

5. Simplified 54.1%

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

6. Final simplification 54.1%

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

Reproduce

herbie shell --seed 2024145 
(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)))))