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

Percentage Accurate: 41.3% → 89.9%

Time: 8.3s

Alternatives: 9

Speedup: 1.7×

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: 41.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 0.3× speedup

Math

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

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 5.1%

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

   3. Taylor expanded in re around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 45.5

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

   5. Simplified 45.5%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 56.9

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

   7. Applied egg-rr 56.9%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

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

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

      11. sqrt-lowering-sqrt.f64 99.6

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

   9. Applied egg-rr 99.6%

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

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

   1. Initial program 44.3%

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

      1. accelerator-lowering-hypot.f64 91.2

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

   4. Applied egg-rr 91.2%

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

Alternative 2: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))
   (if (<= t_0 0.0)
     (/ (* im 0.5) (sqrt re))
     (if (<= t_0 4e-82)
       (* 0.5 (sqrt (* re -4.0)))
       (if (<= t_0 4e+75)
         (* 0.5 (sqrt (* 2.0 (- (sqrt (fma im im (* re re))) re))))
         (* 0.5 (sqrt (* 2.0 (- im re)))))))))

C

double code(double re, double im) {
	double t_0 = sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im * 0.5) / sqrt(re);
	} else if (t_0 <= 4e-82) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (t_0 <= 4e+75) {
		tmp = 0.5 * sqrt((2.0 * (sqrt(fma(im, im, (re * re))) - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	elseif (t_0 <= 4e-82)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (t_0 <= 4e+75)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(sqrt(fma(im, im, Float64(re * re))) - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-82], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+75], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(im * im + N[(re * re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 5.1%

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

   3. Taylor expanded in re around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 45.5

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

   5. Simplified 45.5%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 56.9

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

   7. Applied egg-rr 56.9%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

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

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

      11. sqrt-lowering-sqrt.f64 99.6

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

   9. Applied egg-rr 99.6%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 4e-82

   1. Initial program 16.3%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-lowering-*.f64 16.3

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

   4. Applied egg-rr 16.3%

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

   5. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 67.0

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

   7. Simplified 67.0%

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

   if 4e-82 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 3.99999999999999971e75

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-lowering-*.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   if 3.99999999999999971e75 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 3.6%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 65.5

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

   5. Simplified 65.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 4 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\mathsf{fma}\left(im, im, re \cdot re\right)} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.25e+107)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.25e+125)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* im 0.5) (sqrt (/ 1.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.25e+107) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.25e+125) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) * sqrt((1.0 / re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.25d+107)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.25d+125) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im * 0.5d0) * sqrt((1.0d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.25e+107) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.25e+125) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) * Math.sqrt((1.0 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.25e+107:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.25e+125:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) * math.sqrt((1.0 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.25e+107)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.25e+125)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im * 0.5) * sqrt(Float64(1.0 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.25e+107)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.25e+125)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) * sqrt((1.0 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.25e+107], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.25e+125], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.25e107

   1. Initial program 31.2%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-lowering-*.f64 31.2

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

   4. Applied egg-rr 31.2%

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

   5. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 85.0

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

   7. Simplified 85.0%

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

   if -1.25e107 < re < 2.25e125

   1. Initial program 48.9%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.4

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

   5. Simplified 75.4%

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

   if 2.25e125 < re

   1. Initial program 3.1%

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

   3. Taylor expanded in re around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 52.8

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

   5. Simplified 52.8%

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

   6. Taylor expanded in im around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 83.3

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

   8. Simplified 83.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.25 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{+125}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{\frac{1}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.42 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.42e+107)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.8e+124)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.42e+107) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.8e+124) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.42d+107)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.8d+124) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.42e+107) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.8e+124) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.42e+107:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.8e+124:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.42e+107)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.8e+124)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.42e+107)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.8e+124)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.42e+107], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.8e+124], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.42000000000000006e107

   1. Initial program 31.2%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-lowering-*.f64 31.2

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

   4. Applied egg-rr 31.2%

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

   5. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 85.0

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

   7. Simplified 85.0%

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

   if -1.42000000000000006e107 < re < 4.80000000000000013e124

   1. Initial program 48.9%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.4

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

   5. Simplified 75.4%

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

   if 4.80000000000000013e124 < re

   1. Initial program 3.1%

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

   3. Taylor expanded in re around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 52.8

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

   5. Simplified 52.8%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 68.0

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

   7. Applied egg-rr 68.0%

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

      1. associate-*l/ N/A

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

      2. sqrt-div N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

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

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

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

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

      11. sqrt-lowering-sqrt.f64 83.2

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

   9. Applied egg-rr 83.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.42 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.4e+105)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 6e+124)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.4e+105) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 6e+124) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.4d+105)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 6d+124) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.4e+105) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 6e+124) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.4e+105:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 6e+124:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.4e+105)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 6e+124)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	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 <= -4.4e+105)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 6e+124)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.4e+105], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e+124], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.40000000000000014e105

   1. Initial program 31.2%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-lowering-*.f64 31.2

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

   4. Applied egg-rr 31.2%

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

   5. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 85.0

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

   7. Simplified 85.0%

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

   if -4.40000000000000014e105 < re < 5.9999999999999999e124

   1. Initial program 48.9%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 75.4

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

   5. Simplified 75.4%

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

   if 5.9999999999999999e124 < re

   1. Initial program 3.1%

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

   3. Taylor expanded in re around inf

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 52.8

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

   5. Simplified 52.8%

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. /-lowering-/.f64 68.0

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

   7. Applied egg-rr 68.0%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. sqrt-prod N/A

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

      5. sqrt-prod N/A

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

      6. rem-square-sqrt N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 83.2

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

   9. Applied egg-rr 83.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+124}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+184}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.15e+106)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.1e+184) (* 0.5 (sqrt (* 2.0 (- im re)))) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+106) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.1e+184) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.15d+106)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.1d+184) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.15e+106) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.1e+184) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.15e+106:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.1e+184:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.15e+106)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.1e+184)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.15e+106)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.1e+184)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.15e+106], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.1e+184], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -1.1500000000000001e106

   1. Initial program 31.2%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-lowering-*.f64 31.2

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

   4. Applied egg-rr 31.2%

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

   5. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 85.0

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

   7. Simplified 85.0%

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

   if -1.1500000000000001e106 < re < 2.1e184

   1. Initial program 46.7%

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

   3. Taylor expanded in re around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 73.7

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

   5. Simplified 73.7%

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

   if 2.1e184 < re

   1. Initial program 2.7%

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

   3. Taylor expanded in re around inf

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

   5. Simplified 31.8%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. +-inverses N/A

         \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

      6. metadata-eval 31.8

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.15 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.1 \cdot 10^{+184}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e-37)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.5e+192) (* 0.5 (sqrt (* 2.0 im))) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-37) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.5e+192) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.3d-37)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.5d+192) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-37) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.5e+192) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.3e-37:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.5e+192:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.3e-37)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.5e+192)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.3e-37)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.5e+192)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e-37], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.5e+192], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -3.29999999999999982e-37

   1. Initial program 47.5%

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

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

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

      9. *-lowering-*.f64 47.5

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

   4. Applied egg-rr 47.5%

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

   5. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 71.2

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

   7. Simplified 71.2%

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

   if -3.29999999999999982e-37 < re < 4.5e192

   1. Initial program 43.2%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 76.2%

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

   if 4.5e192 < re

   1. Initial program 2.6%

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

   3. Taylor expanded in re around inf

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

   5. Simplified 33.0%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. +-inverses N/A

         \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

      6. metadata-eval 33.0

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.5 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.1 \cdot 10^{+193}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.1e+193) (* 0.5 (sqrt (* 2.0 im))) 0.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.1e+193) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 1.1d+193) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.1e+193) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.1e+193:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.1e+193)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.1e+193)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.1e+193], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if re < 1.09999999999999993e193

   1. Initial program 44.4%

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

   3. Taylor expanded in re around 0

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

   5. Simplified 64.8%

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

   if 1.09999999999999993e193 < re

   1. Initial program 2.6%

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

   3. Taylor expanded in re around inf

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

   5. Simplified 33.0%

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

      1. pow1/2 N/A

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

      2. *-commutative N/A

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

      3. +-inverses N/A

         \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

      6. metadata-eval 33.0

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 33.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 5.9% accurate, 47.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in re around inf

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

5. Simplified 5.7%

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

   1. pow1/2 N/A

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

   2. *-commutative N/A

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

   3. +-inverses N/A

      \[\leadsto \frac{1}{2} \cdot {\left(\color{blue}{0} \cdot 2\right)}^{\frac{1}{2}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot {\color{blue}{0}}^{\frac{1}{2}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{0} \]

   6. metadata-eval 5.7

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 5.7%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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)))))