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

Percentage Accurate: 40.8% → 76.1%

Time: 7.7s

Alternatives: 5

Speedup: 2.2×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.8% 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: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e-17)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5.3e+24) (* 0.5 (sqrt (* im 2.0))) (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-17) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.3e+24) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = (0.5 * im) / 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 <= (-3.1d-17)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.3d+24) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-17) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.3e+24) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e-17:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.3e+24:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e-17)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.3e+24)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e-17)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.3e+24)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e-17], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.3e+24], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.0999999999999998e-17

   1. Initial program 44.7%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   if -3.0999999999999998e-17 < re < 5.2999999999999999e24

   1. Initial program 50.1%

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

   3. Taylor expanded in re around 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. lower-*.f64 77.3

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

   5. Simplified 77.3%

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

   if 5.2999999999999999e24 < re

   1. Initial program 7.1%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.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. lower-*.f64 42.3

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

   5. Simplified 42.3%

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

      1. associate-/l* N/A

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

      2. sqrt-prod N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate-*r/ N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 79.8

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

   7. Applied egg-rr 79.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 5.3 \cdot 10^{+24}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e-17)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 5.3e+24) (* 0.5 (sqrt (* im 2.0))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-17) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 5.3e+24) {
		tmp = 0.5 * sqrt((im * 2.0));
	} 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 <= (-3.1d-17)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 5.3d+24) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    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 <= -3.1e-17) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 5.3e+24) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e-17:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 5.3e+24:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e-17)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 5.3e+24)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	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 <= -3.1e-17)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 5.3e+24)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e-17], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 5.3e+24], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.0999999999999998e-17

   1. Initial program 44.7%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   if -3.0999999999999998e-17 < re < 5.2999999999999999e24

   1. Initial program 50.1%

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

   3. Taylor expanded in re around 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. lower-*.f64 77.3

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

   5. Simplified 77.3%

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

   if 5.2999999999999999e24 < re

   1. Initial program 7.1%

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

   3. Taylor expanded in re around inf

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

      1. lower-/.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. lower-*.f64 42.3

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

   5. Simplified 42.3%

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

      1. associate-/l* N/A

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

      2. sqrt-prod N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-div N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 79.6

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

   7. Applied egg-rr 79.6%

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

Alternative 3: 64.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e-17) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-17) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.1d-17)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e-17) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e-17:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e-17)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e-17)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e-17], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3.0999999999999998e-17

   1. Initial program 44.7%

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

   3. Taylor expanded in re around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 79.2

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

   5. Simplified 79.2%

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

   if -3.0999999999999998e-17 < re

   1. Initial program 36.1%

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

   3. Taylor expanded in re around 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. lower-*.f64 60.5

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

   5. Simplified 60.5%

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

Alternative 4: 52.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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. lower-*.f64 51.2

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

5. Simplified 51.2%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{im \cdot 2}} \]
6. Add Preprocessing

Alternative 5: 13.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.5%

   \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
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. lower-*.f64 51.2

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

5. Simplified 51.2%

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

6. Taylor expanded in im around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 0.0

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

8. Simplified 0.0%

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

   1. lift-neg.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot {\left(\mathsf{neg}\left(im\right)\right)}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. pow-prod-down N/A

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

   7. lift-neg.f64 N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. lift-*.f64 N/A

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

   12. remove-double-neg N/A

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

   13. lift-*.f64 N/A

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

   14. pow-prod-down N/A

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

   15. pow-sqr N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{1}{2} \cdot {im}^{\left(2 \cdot \color{blue}{\frac{1}{4}}\right)} \]

   17. metadata-eval N/A

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

   18. pow1/2 N/A

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

   19. lift-sqrt.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 13.1

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

10. Applied egg-rr 13.1%

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

11. Final simplification 13.1%

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

Reproduce

herbie shell --seed 2024214 
(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)))))