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

Percentage Accurate: 41.4% → 75.2%

Time: 5.8s

Alternatives: 7

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: 41.4% 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: 75.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{{re}^{-1}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

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 (pow re -1.0)))
     (if (<= t_0 5e+72) (* 0.5 t_0) (* 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(pow(re, -1.0));
	} else if (t_0 <= 5e+72) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((2.0d0 * (sqrt(((re * re) + (im * im))) - re)))
    if (t_0 <= 0.0d0) then
        tmp = (im * 0.5d0) * sqrt((re ** (-1.0d0)))
    else if (t_0 <= 5d+72) then
        tmp = 0.5d0 * t_0
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (im * 0.5) * Math.sqrt(Math.pow(re, -1.0));
	} else if (t_0 <= 5e+72) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (im * 0.5) * math.sqrt(math.pow(re, -1.0))
	elif t_0 <= 5e+72:
		tmp = 0.5 * t_0
	else:
		tmp = 0.5 * math.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 ^ -1.0)));
	elseif (t_0 <= 5e+72)
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (im * 0.5) * sqrt((re ^ -1.0));
	elseif (t_0 <= 5e+72)
		tmp = 0.5 * t_0;
	else
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	end
	tmp_2 = 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[N[Power[re, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+72], N[(0.5 * t$95$0), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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 7.4%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   7. Applied rewrites 99.7%

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

   if 0.0 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 4.99999999999999992e72

   1. Initial program 97.2%

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

   if 4.99999999999999992e72 < (sqrt.f64 (*.f64 #s(literal 2 binary64) (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 6.4%

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

   3. Taylor expanded in re around 0

      \[\leadsto \frac{1}{2} \cdot \sqrt{2 \cdot \color{blue}{\left(im + -1 \cdot re\right)}} \]
   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. lower--.f64 62.8

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

   5. Applied rewrites 62.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{{re}^{-1}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 5 \cdot 10^{+72}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.085:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{{re}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.085)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 6e+67)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (* im 0.5) (sqrt (pow re -1.0))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.085) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 6e+67) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) * sqrt(pow(re, -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.085d0)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 6d+67) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (im * 0.5d0) * sqrt((re ** (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.085) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 6e+67) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (im * 0.5) * Math.sqrt(Math.pow(re, -1.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.085:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 6e+67:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (im * 0.5) * math.sqrt(math.pow(re, -1.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.085)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 6e+67)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(im * 0.5) * sqrt((re ^ -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.085)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 6e+67)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (im * 0.5) * sqrt((re ^ -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.085], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e+67], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Sqrt[N[Power[re, -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0850000000000000061

   1. Initial program 52.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -0.0850000000000000061 < re < 6.0000000000000002e67

   1. Initial program 57.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower--.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if 6.0000000000000002e67 < re

   1. Initial program 7.2%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 81.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.085:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot \sqrt{{re}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.085:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.085)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 6e+67)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.085) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 6e+67) {
		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 <= (-0.085d0)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 6d+67) 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 <= -0.085) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 6e+67) {
		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 <= -0.085:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 6e+67:
		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 <= -0.085)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 6e+67)
		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 <= -0.085)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 6e+67)
		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, -0.085], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e+67], 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 < -0.0850000000000000061

   1. Initial program 52.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -0.0850000000000000061 < re < 6.0000000000000002e67

   1. Initial program 57.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower--.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if 6.0000000000000002e67 < re

   1. Initial program 7.2%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 81.5%

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

Alternative 4: 76.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.085:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{elif}\;re \leq 6 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{re}} \cdot im\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.085)
   (* 0.5 (sqrt (* -4.0 re)))
   (if (<= re 6e+67)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* (/ 0.5 (sqrt re)) im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.085) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else if (re <= 6e+67) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 / sqrt(re)) * im;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.085d0)) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else if (re <= 6d+67) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = (0.5d0 / sqrt(re)) * im
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.085) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else if (re <= 6e+67) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = (0.5 / Math.sqrt(re)) * im;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.085:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	elif re <= 6e+67:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = (0.5 / math.sqrt(re)) * im
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.085)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	elseif (re <= 6e+67)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(Float64(0.5 / sqrt(re)) * im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.085)
		tmp = 0.5 * sqrt((-4.0 * re));
	elseif (re <= 6e+67)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = (0.5 / sqrt(re)) * im;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.085], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6e+67], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision] * im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0850000000000000061

   1. Initial program 52.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -0.0850000000000000061 < re < 6.0000000000000002e67

   1. Initial program 57.9%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower--.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if 6.0000000000000002e67 < re

   1. Initial program 7.2%

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

   3. Taylor expanded in re around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 81.5%

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

   9. Applied rewrites 81.5%

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

Alternative 5: 59.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3e-92)
   (* 0.5 (sqrt (* -4.0 re)))
   (* 0.5 (sqrt (* 2.0 (- im re))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3e-92) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3d-92) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3e-92) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3e-92:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3e-92)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3e-92)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3e-92], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.00000000000000013e-92

   1. Initial program 47.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if 3.00000000000000013e-92 < im

   1. Initial program 46.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower--.f64 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 6: 58.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 3 \cdot 10^{-92}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 3e-92) (* 0.5 (sqrt (* -4.0 re))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 3e-92) {
		tmp = 0.5 * sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 3d-92) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * re))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 3e-92) {
		tmp = 0.5 * Math.sqrt((-4.0 * re));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 3e-92:
		tmp = 0.5 * math.sqrt((-4.0 * re))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 3e-92)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * re)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 3e-92)
		tmp = 0.5 * sqrt((-4.0 * re));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 3e-92], N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 3.00000000000000013e-92

   1. Initial program 47.8%

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

   3. Taylor expanded in re around -inf

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

      1. lower-*.f64 53.6

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

   5. Applied rewrites 53.6%

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

   if 3.00000000000000013e-92 < im

   1. Initial program 46.0%

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \]
   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. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

Alternative 7: 26.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{-4 \cdot re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (sqrt (* -4.0 re))))

C

double code(double re, double im) {
	return 0.5 * sqrt((-4.0 * re));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt(((-4.0d0) * re))
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt((-4.0 * re));
}

Python

def code(re, im):
	return 0.5 * math.sqrt((-4.0 * re))

Julia

function code(re, im)
	return Float64(0.5 * sqrt(Float64(-4.0 * re)))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt((-4.0 * re));
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[N[(-4.0 * re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 -inf

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

   1. lower-*.f64 27.7

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

5. Applied rewrites 27.7%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024315 
(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)))))