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

Percentage Accurate: 41.5% → 90.0%

Time: 10.5s

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: 41.5% 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: 90.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* (pow re -0.5) (* 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, -0.5) * (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, -0.5) * (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, -0.5) * (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((re ^ -0.5) * 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 ^ -0.5) * (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[re, -0.5], $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 8.5%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-hypot.f64 N/A

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

      9. metadata-eval 8.5%

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

   4. Applied egg-rr 8.5%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in re around inf

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

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

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

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

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

      3. /-lowering-/.f64 99.4%

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

   7. Simplified 99.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
   8. 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. *-commutative N/A

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

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

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

      4. pow1/2 N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 99.6%

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

   9. Applied egg-rr 99.6%

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

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

   1. Initial program 42.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. accelerator-lowering-hypot.f64 91.5%

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

   3. Simplified 91.5%

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

Alternative 2: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.72 \cdot 10^{-71}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-38)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.72e-71)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (pow re -0.5) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-38) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.72e-71) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = pow(re, -0.5) * (im * 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 <= (-2d-38)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.72d-71) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (re ** (-0.5d0)) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e-38) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.72e-71) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = Math.pow(re, -0.5) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-38:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.72e-71:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-38)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.72e-71)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-38)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.72e-71)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (re ^ -0.5) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-38], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.72e-71], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.9999999999999999e-38

   1. Initial program 42.8%

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

   3. Taylor expanded in re around -inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\color{blue}{\left(-4 \cdot re\right)}\right)\right) \]
   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 79.5%

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

   5. Simplified 79.5%

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

   if -1.9999999999999999e-38 < re < 1.72e-71

   1. Initial program 53.2%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 86.2%

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

   if 1.72e-71 < re

   1. Initial program 8.1%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-hypot.f64 N/A

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

      9. metadata-eval 40.5%

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

   4. Applied egg-rr 40.5%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in re around inf

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

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

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

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

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

      3. /-lowering-/.f64 80.4%

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

   7. Simplified 80.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
   8. 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. *-commutative N/A

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

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

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

      4. pow1/2 N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 80.4%

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

   9. Applied egg-rr 80.4%

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

Alternative 3: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.25 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.7e-54)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3.25e-72)
     (* 0.5 (sqrt (* 2.0 im)))
     (* (pow re -0.5) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.7e-54) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3.25e-72) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = pow(re, -0.5) * (im * 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 <= (-1.7d-54)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3.25d-72) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = (re ** (-0.5d0)) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.7e-54) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3.25e-72) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = Math.pow(re, -0.5) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.7e-54:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3.25e-72:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = math.pow(re, -0.5) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.7e-54)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3.25e-72)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64((re ^ -0.5) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.7e-54)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3.25e-72)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = (re ^ -0.5) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.7e-54], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.25e-72], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[re, -0.5], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.69999999999999994e-54

   1. Initial program 44.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.9%

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

   5. Simplified 78.9%

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

   if -1.69999999999999994e-54 < re < 3.2499999999999998e-72

   1. Initial program 52.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 85.8%

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

   5. Simplified 85.8%

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

   if 3.2499999999999998e-72 < re

   1. Initial program 8.1%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-hypot.f64 N/A

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

      9. metadata-eval 40.5%

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

   4. Applied egg-rr 40.5%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in re around inf

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

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

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

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

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

      3. /-lowering-/.f64 80.4%

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

   7. Simplified 80.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
   8. 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. *-commutative N/A

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

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

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

      4. pow1/2 N/A

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

      5. inv-pow N/A

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

      6. pow-pow N/A

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 80.4%

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

   9. Applied egg-rr 80.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3.25 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;{re}^{-0.5} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.45e-53)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4e-68) (* 0.5 (sqrt (* 2.0 im))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.45e-53) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4e-68) {
		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 <= (-1.45d-53)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4d-68) 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 <= -1.45e-53) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4e-68) {
		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 <= -1.45e-53:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4e-68:
		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 <= -1.45e-53)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4e-68)
		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 <= -1.45e-53)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4e-68)
		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, -1.45e-53], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4e-68], 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 < -1.4499999999999999e-53

   1. Initial program 44.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.9%

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

   5. Simplified 78.9%

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

   if -1.4499999999999999e-53 < re < 4.00000000000000027e-68

   1. Initial program 52.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 85.8%

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

   5. Simplified 85.8%

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

   if 4.00000000000000027e-68 < re

   1. Initial program 8.1%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-hypot.f64 N/A

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

      9. metadata-eval 40.5%

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

   4. Applied egg-rr 40.5%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in re around inf

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

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

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

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

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

      3. /-lowering-/.f64 80.4%

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

   7. Simplified 80.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
   8. 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. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. un-div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. sqrt-lowering-sqrt.f64 80.4%

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

   9. Applied egg-rr 80.4%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.45 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4 \cdot 10^{-68}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.8e-54)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.5e-72) (* 0.5 (sqrt (* 2.0 im))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.8e-54) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.5e-72) {
		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 <= (-9.8d-54)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.5d-72) 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 <= -9.8e-54) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.5e-72) {
		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 <= -9.8e-54:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.5e-72:
		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 <= -9.8e-54)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.5e-72)
		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 <= -9.8e-54)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.5e-72)
		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, -9.8e-54], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e-72], 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 < -9.80000000000000042e-54

   1. Initial program 44.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.9%

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

   5. Simplified 78.9%

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

   if -9.80000000000000042e-54 < re < 4.5e-72

   1. Initial program 52.3%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 85.8%

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

   5. Simplified 85.8%

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

   if 4.5e-72 < re

   1. Initial program 8.1%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

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

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

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

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

      8. accelerator-lowering-hypot.f64 N/A

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

      9. metadata-eval 40.5%

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

   4. Applied egg-rr 40.5%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right)}^{0.25}\right)}^{2}} \]

   5. Taylor expanded in re around inf

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

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

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

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

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

      3. /-lowering-/.f64 80.4%

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

   7. Simplified 80.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
   8. 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. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. sqrt-lowering-sqrt.f64 80.3%

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

   9. Applied egg-rr 80.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.7e-53) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.7e-53) {
		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 <= (-1.7d-53)) 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 <= -1.7e-53) {
		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 <= -1.7e-53:
		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 <= -1.7e-53)
		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 <= -1.7e-53)
		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, -1.7e-53], 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 < -1.7e-53

   1. Initial program 44.2%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 78.9%

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

   5. Simplified 78.9%

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

   if -1.7e-53 < re

   1. Initial program 35.3%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 62.3%

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

   5. Simplified 62.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.5% 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 38.2%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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 50.2%

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

5. Simplified 50.2%

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

6. Final simplification 50.2%

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

Reproduce

herbie shell --seed 2024191 
(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)))))