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

Percentage Accurate: 41.6% → 90.3%

Time: 9.5s

Alternatives: 8

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: 41.6% 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: 90.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (- (sqrt (+ (* re re) (* im im))) re) 0.0)
   (* (/ im (sqrt re)) 0.5)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))))

C

double code(double re, double im) {
	double tmp;
	if ((sqrt(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = (im / sqrt(re)) * 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(((re * re) + (im * im))) - re) <= 0.0) {
		tmp = (im / Math.sqrt(re)) * 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(((re * re) + (im * im))) - re) <= 0.0:
		tmp = (im / math.sqrt(re)) * 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 (Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re) <= 0.0)
		tmp = Float64(Float64(im / sqrt(re)) * 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(((re * re) + (im * im))) - re) <= 0.0)
		tmp = (im / sqrt(re)) * 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[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision], 0.0], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $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 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 4.2%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 35.9

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

   5. Simplified 35.9%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 92.7

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

   7. Applied egg-rr 92.7%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)

   1. Initial program 49.9%

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

      1. accelerator-lowering-hypot.f64 89.2

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

   4. Applied egg-rr 89.2%

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

Alternative 2: 76.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e+43)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 3e-29)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (sqrt (/ 1.0 re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e+43) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 3e-29) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * 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 <= (-2d+43)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 3d-29) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e+43) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 3e-29) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e+43:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 3e-29:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e+43)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 3e-29)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e+43)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 3e-29)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e+43], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3e-29], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.00000000000000003e43

   1. Initial program 28.6%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.7

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

   5. Simplified 86.7%

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

   if -2.00000000000000003e43 < re < 3.0000000000000003e-29

   1. Initial program 60.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 74.1

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

   5. Simplified 74.1%

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

   if 3.0000000000000003e-29 < re

   1. Initial program 16.2%

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

   3. Taylor expanded in re around inf

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 76.1

          \[\leadsto 0.5 \cdot \left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot im\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{re}}}\right) \]

   5. Simplified 76.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sqrt-unprod N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

      10. /-lowering-/.f64 76.8

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

   7. Applied egg-rr 76.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 3 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(im \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.4e+43)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.8e-29)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (sqrt (/ 1.0 re)) (* im 0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.4e+43) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.8e-29) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = sqrt((1.0 / re)) * (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 <= (-2.4d+43)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.8d-29) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = sqrt((1.0d0 / re)) * (im * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.4e+43) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.8e-29) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = Math.sqrt((1.0 / re)) * (im * 0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.4e+43:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.8e-29:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = math.sqrt((1.0 / re)) * (im * 0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.4e+43)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.8e-29)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(sqrt(Float64(1.0 / re)) * Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.4e+43)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.8e-29)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = sqrt((1.0 / re)) * (im * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.4e+43], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.8e-29], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.40000000000000023e43

   1. Initial program 28.6%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.7

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

   5. Simplified 86.7%

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

   if -2.40000000000000023e43 < re < 2.8000000000000002e-29

   1. Initial program 60.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 74.1

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

   5. Simplified 74.1%

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

   if 2.8000000000000002e-29 < re

   1. Initial program 16.2%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

   6. Taylor expanded in im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. /-lowering-/.f64 76.8

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

   8. Simplified 76.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.4 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{re}} \cdot \left(im \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.1 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im}{\sqrt{re}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+43)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.1e-29)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (/ im (sqrt re)) 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+43) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.1e-29) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / sqrt(re)) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d+43)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.1d-29) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im / sqrt(re)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+43) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.1e-29) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im / Math.sqrt(re)) * 0.5;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+43:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.1e-29:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im / math.sqrt(re)) * 0.5
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+43)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.1e-29)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im / sqrt(re)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.8e+43)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.1e-29)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im / sqrt(re)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+43], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.1e-29], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.80000000000000005e43

   1. Initial program 28.6%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.7

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

   5. Simplified 86.7%

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

   if -1.80000000000000005e43 < re < 4.0999999999999998e-29

   1. Initial program 60.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 74.1

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

   5. Simplified 74.1%

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

   if 4.0999999999999998e-29 < re

   1. Initial program 16.2%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

      1. *-commutative N/A

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

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

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

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

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

      7. sqrt-lowering-sqrt.f64 76.8

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

   7. Applied egg-rr 76.8%

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

Alternative 5: 76.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.95e+43)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.55e-29)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.95e+43) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.55e-29) {
		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 <= (-1.95d+43)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.55d-29) 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 <= -1.95e+43) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.55e-29) {
		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 <= -1.95e+43:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.55e-29:
		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 <= -1.95e+43)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.55e-29)
		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 <= -1.95e+43)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.55e-29)
		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, -1.95e+43], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.55e-29], 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 < -1.95e43

   1. Initial program 28.6%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.7

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

   5. Simplified 86.7%

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

   if -1.95e43 < re < 1.55000000000000013e-29

   1. Initial program 60.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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 74.1

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

   5. Simplified 74.1%

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

   if 1.55000000000000013e-29 < re

   1. Initial program 16.2%

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

   3. Taylor expanded in re around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 45.3

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

   5. Simplified 45.3%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-*l/ N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. sqrt-lowering-sqrt.f64 76.7

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

   7. Applied egg-rr 76.7%

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

Alternative 6: 65.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1600:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.2 \cdot 10^{+201}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1600.0)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.2e+201) (* 0.5 (sqrt (* im 2.0))) 0.0)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1600.0) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.2e+201) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= (-1600.0d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.2d+201) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1600.0) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.2e+201) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1600.0:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.2e+201:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1600.0)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.2e+201)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1600.0)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.2e+201)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1600.0], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.2e+201], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if re < -1600

   1. Initial program 35.6%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 79.9

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

   5. Simplified 79.9%

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

   if -1600 < re < 1.19999999999999996e201

   1. Initial program 50.7%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 65.7

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

   5. Simplified 65.7%

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

   if 1.19999999999999996e201 < re

   1. Initial program 2.3%

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

   3. Taylor expanded in re around inf

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

   5. Simplified 27.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} - re\right)} \]
   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 27.2

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

   7. Applied egg-rr 27.2%

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

Alternative 7: 52.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6 \cdot 10^{+198}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6e+198) (* 0.5 (sqrt (* im 2.0))) 0.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6e+198) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= 6d+198) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 6e+198) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6e+198:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6e+198)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 6e+198)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6e+198], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if re < 6.00000000000000037e198

   1. Initial program 46.6%

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

   3. Taylor expanded in re around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 54.7

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

   5. Simplified 54.7%

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

   if 6.00000000000000037e198 < re

   1. Initial program 2.3%

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

   3. Taylor expanded in re around inf

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

   5. Simplified 27.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} - re\right)} \]
   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 27.2

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

   7. Applied egg-rr 27.2%

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

Alternative 8: 6.1% 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 42.9%

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

3. Taylor expanded in re around inf

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

5. Simplified 5.3%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} - re\right)} \]
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 5.3

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

7. Applied egg-rr 5.3%

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

Reproduce

herbie shell --seed 2024194 
(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)))))