math.sqrt on complex, real part

Percentage Accurate: 41.1% → 84.1%

Time: 7.1s

Alternatives: 9

Speedup: 2.0×

Specification

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: 41.1% 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: 84.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double re, double im) {
	double tmp;
	if ((re + sqrt(((re * re) + (im * im)))) <= 0.0) {
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

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

Python

def code(re, im):
	tmp = 0
	if (re + math.sqrt(((re * re) + (im * im)))) <= 0.0:
		tmp = 0.5 * math.sqrt((2.0 * ((im / (re / im)) * -0.5)))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im * im)))) <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im / Float64(re / im)) * -0.5))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re + sqrt(((re * re) + (im * im)))) <= 0.0)
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $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 3.2%

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

   2. Taylor expanded in re around -inf 31.0%

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

      1. *-commutative 31.0%

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

      2. unpow2 31.0%

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

      3. associate-/l* 56.5%

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

   4. Simplified 56.5%

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

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

   1. Initial program 47.5%

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

      1. add-sqr-sqrt 47.2%

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

      2. sqrt-unprod 47.5%

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

      3. *-commutative 47.5%

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

      4. *-commutative 47.5%

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

      5. swap-sqr 47.5%

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

      6. add-sqr-sqrt 47.5%

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

      7. *-commutative 47.5%

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

      8. +-commutative 47.5%

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

      9. hypot-def 92.4%

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

      10. metadata-eval 92.4%

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

   3. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{\sqrt{\left(\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2\right) \cdot 0.25}} \]
   4. Step-by-step derivation

      1. associate-*l* 92.4%

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

      2. metadata-eval 92.4%

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

   5. Simplified 92.4%

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

4. Final simplification 87.8%

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

Alternative 2: 60.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -3.2 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -7 \cdot 10^{-278}:\\ \;\;\;\;0.5 \cdot \frac{-im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.35e-117)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im -3.2e-234)
     (sqrt re)
     (if (<= im -7e-278)
       (* 0.5 (/ (- im) (sqrt (- re))))
       (if (<= im 2.5e-305)
         (sqrt re)
         (if (<= im 2.6e-190)
           (* 0.5 (* im (sqrt (/ -1.0 re))))
           (if (<= im 5e-112) (sqrt re) (* 0.5 (sqrt (* 2.0 (+ re im)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.35e-117) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= -3.2e-234) {
		tmp = sqrt(re);
	} else if (im <= -7e-278) {
		tmp = 0.5 * (-im / sqrt(-re));
	} else if (im <= 2.5e-305) {
		tmp = sqrt(re);
	} else if (im <= 2.6e-190) {
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	} else if (im <= 5e-112) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (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 (im <= (-1.35d-117)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= (-3.2d-234)) then
        tmp = sqrt(re)
    else if (im <= (-7d-278)) then
        tmp = 0.5d0 * (-im / sqrt(-re))
    else if (im <= 2.5d-305) then
        tmp = sqrt(re)
    else if (im <= 2.6d-190) then
        tmp = 0.5d0 * (im * sqrt(((-1.0d0) / re)))
    else if (im <= 5d-112) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.35e-117) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= -3.2e-234) {
		tmp = Math.sqrt(re);
	} else if (im <= -7e-278) {
		tmp = 0.5 * (-im / Math.sqrt(-re));
	} else if (im <= 2.5e-305) {
		tmp = Math.sqrt(re);
	} else if (im <= 2.6e-190) {
		tmp = 0.5 * (im * Math.sqrt((-1.0 / re)));
	} else if (im <= 5e-112) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.35e-117:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= -3.2e-234:
		tmp = math.sqrt(re)
	elif im <= -7e-278:
		tmp = 0.5 * (-im / math.sqrt(-re))
	elif im <= 2.5e-305:
		tmp = math.sqrt(re)
	elif im <= 2.6e-190:
		tmp = 0.5 * (im * math.sqrt((-1.0 / re)))
	elif im <= 5e-112:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.35e-117)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= -3.2e-234)
		tmp = sqrt(re);
	elseif (im <= -7e-278)
		tmp = Float64(0.5 * Float64(Float64(-im) / sqrt(Float64(-re))));
	elseif (im <= 2.5e-305)
		tmp = sqrt(re);
	elseif (im <= 2.6e-190)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(-1.0 / re))));
	elseif (im <= 5e-112)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.35e-117)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= -3.2e-234)
		tmp = sqrt(re);
	elseif (im <= -7e-278)
		tmp = 0.5 * (-im / sqrt(-re));
	elseif (im <= 2.5e-305)
		tmp = sqrt(re);
	elseif (im <= 2.6e-190)
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	elseif (im <= 5e-112)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.35e-117], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -3.2e-234], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, -7e-278], N[(0.5 * N[((-im) / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.5e-305], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 2.6e-190], N[(0.5 * N[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5e-112], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -1.35000000000000001e-117

   1. Initial program 43.2%

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

   2. Taylor expanded in im around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. sub-neg 77.5%

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

   4. Simplified 77.5%

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

   if -1.35000000000000001e-117 < im < -3.1999999999999999e-234 or -6.99999999999999941e-278 < im < 2.49999999999999993e-305 or 2.5999999999999998e-190 < im < 5.00000000000000044e-112

   1. Initial program 47.6%

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

   2. Taylor expanded in im around 0 60.2%

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

      1. associate-*r* 60.2%

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

      2. unpow2 60.2%

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

      3. rem-square-sqrt 61.4%

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

      4. metadata-eval 61.4%

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

      5. *-lft-identity 61.4%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   4. Simplified 61.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if -3.1999999999999999e-234 < im < -6.99999999999999941e-278

   1. Initial program 14.2%

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

   2. Taylor expanded in re around -inf 26.1%

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

      1. *-commutative 26.1%

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

      2. associate-*l/ 26.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot -0.5}{re}}} \]

      3. unpow2 26.1%

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

   4. Simplified 26.1%

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

      1. sqrt-prod 26.1%

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

      2. associate-/l* 26.1%

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

      3. sqrt-div 25.3%

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

      4. sqrt-prod 0.0%

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

      5. add-sqr-sqrt 24.4%

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

      6. div-inv 24.4%

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

      7. metadata-eval 24.4%

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

   6. Applied egg-rr 24.4%

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

      1. *-commutative 24.4%

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

      2. associate-/r/ 24.4%

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

   8. Simplified 24.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}} \]
   9. Step-by-step derivation

      1. frac-2neg 24.4%

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

      2. neg-sub0 24.4%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{0 - im}}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}} \]

      3. div-sub 24.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{0}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}} - \frac{im}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right)} \]

      4. sqrt-undiv 24.4%

         \[\leadsto 0.5 \cdot \left(\frac{0}{-\color{blue}{\sqrt{\frac{re \cdot -2}{2}}}} - \frac{im}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right) \]

      5. div-inv 24.4%

         \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\color{blue}{\left(re \cdot -2\right) \cdot \frac{1}{2}}}} - \frac{im}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right) \]

      6. metadata-eval 24.4%

         \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot \color{blue}{0.5}}} - \frac{im}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right) \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{\color{blue}{\sqrt{im} \cdot \sqrt{im}}}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right) \]

      8. sqrt-prod 25.3%

         \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{\color{blue}{\sqrt{im \cdot im}}}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right) \]

      9. sqr-neg 25.3%

         \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{\sqrt{\color{blue}{\left(-im\right) \cdot \left(-im\right)}}}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right) \]

      10. sqrt-unprod 77.3%

          \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right) \]

      11. add-sqr-sqrt 77.4%

          \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{\color{blue}{-im}}{-\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}\right) \]

      12. frac-2neg 77.4%

          \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \color{blue}{\frac{im}{\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}}\right) \]

      13. sqrt-undiv 77.4%

          \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{im}{\color{blue}{\sqrt{\frac{re \cdot -2}{2}}}}\right) \]

      14. div-inv 77.4%

          \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{im}{\sqrt{\color{blue}{\left(re \cdot -2\right) \cdot \frac{1}{2}}}}\right) \]

      15. metadata-eval 77.4%

          \[\leadsto 0.5 \cdot \left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{im}{\sqrt{\left(re \cdot -2\right) \cdot \color{blue}{0.5}}}\right) \]

   10. Applied egg-rr 77.4%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{0}{-\sqrt{\left(re \cdot -2\right) \cdot 0.5}} - \frac{im}{\sqrt{\left(re \cdot -2\right) \cdot 0.5}}\right)} \]
   11. Step-by-step derivation

       1. div0 77.4%

          \[\leadsto 0.5 \cdot \left(\color{blue}{0} - \frac{im}{\sqrt{\left(re \cdot -2\right) \cdot 0.5}}\right) \]

       2. neg-sub0 77.4%

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

       3. distribute-neg-frac 77.4%

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

       4. associate-*l* 77.4%

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

       5. metadata-eval 77.4%

          \[\leadsto 0.5 \cdot \frac{-im}{\sqrt{re \cdot \color{blue}{-1}}} \]

       6. *-commutative 77.4%

          \[\leadsto 0.5 \cdot \frac{-im}{\sqrt{\color{blue}{-1 \cdot re}}} \]

       7. neg-mul-1 77.4%

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

   12. Simplified 77.4%

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

   if 2.49999999999999993e-305 < im < 2.5999999999999998e-190

   1. Initial program 12.2%

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

   2. Taylor expanded in re around -inf 26.6%

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

      1. *-commutative 26.6%

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

      2. associate-*l/ 26.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot -0.5}{re}}} \]

      3. unpow2 26.6%

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

   4. Simplified 26.6%

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

      1. sqrt-prod 26.6%

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

      2. associate-/l* 26.6%

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

      3. sqrt-div 25.8%

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

      4. sqrt-prod 68.6%

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

      5. add-sqr-sqrt 68.7%

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

      6. div-inv 68.7%

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

      7. metadata-eval 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-/r/ 68.7%

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

   8. Simplified 68.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}} \]
   9. Step-by-step derivation

      1. div-inv 68.8%

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

      2. clear-num 68.8%

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

      3. sqrt-undiv 69.0%

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

   10. Applied egg-rr 69.0%

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

       1. *-commutative 69.0%

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

       2. associate-/r* 69.0%

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

       3. metadata-eval 69.0%

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

   12. Simplified 69.0%

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

   if 5.00000000000000044e-112 < im

   1. Initial program 47.7%

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

   2. Taylor expanded in re around 0 73.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + im\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.35 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -3.2 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -7 \cdot 10^{-278}:\\ \;\;\;\;0.5 \cdot \frac{-im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 2.5 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.6 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 3: 60.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -5.8 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -2.8 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -1.35 \cdot 10^{-277}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \frac{1}{-\sqrt{0.5 \cdot \left(re \cdot -2\right)}}\right)\\ \mathbf{elif}\;im \leq 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-197}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -5.8e-118)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im -2.8e-237)
     (sqrt re)
     (if (<= im -1.35e-277)
       (* 0.5 (* im (/ 1.0 (- (sqrt (* 0.5 (* re -2.0)))))))
       (if (<= im 1e-305)
         (sqrt re)
         (if (<= im 2.55e-197)
           (* 0.5 (* im (sqrt (/ -1.0 re))))
           (if (<= im 1.6e-112)
             (sqrt re)
             (* 0.5 (sqrt (* 2.0 (+ re im)))))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -5.8e-118) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= -2.8e-237) {
		tmp = sqrt(re);
	} else if (im <= -1.35e-277) {
		tmp = 0.5 * (im * (1.0 / -sqrt((0.5 * (re * -2.0)))));
	} else if (im <= 1e-305) {
		tmp = sqrt(re);
	} else if (im <= 2.55e-197) {
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	} else if (im <= 1.6e-112) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (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 (im <= (-5.8d-118)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= (-2.8d-237)) then
        tmp = sqrt(re)
    else if (im <= (-1.35d-277)) then
        tmp = 0.5d0 * (im * (1.0d0 / -sqrt((0.5d0 * (re * (-2.0d0))))))
    else if (im <= 1d-305) then
        tmp = sqrt(re)
    else if (im <= 2.55d-197) then
        tmp = 0.5d0 * (im * sqrt(((-1.0d0) / re)))
    else if (im <= 1.6d-112) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -5.8e-118) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= -2.8e-237) {
		tmp = Math.sqrt(re);
	} else if (im <= -1.35e-277) {
		tmp = 0.5 * (im * (1.0 / -Math.sqrt((0.5 * (re * -2.0)))));
	} else if (im <= 1e-305) {
		tmp = Math.sqrt(re);
	} else if (im <= 2.55e-197) {
		tmp = 0.5 * (im * Math.sqrt((-1.0 / re)));
	} else if (im <= 1.6e-112) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -5.8e-118:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= -2.8e-237:
		tmp = math.sqrt(re)
	elif im <= -1.35e-277:
		tmp = 0.5 * (im * (1.0 / -math.sqrt((0.5 * (re * -2.0)))))
	elif im <= 1e-305:
		tmp = math.sqrt(re)
	elif im <= 2.55e-197:
		tmp = 0.5 * (im * math.sqrt((-1.0 / re)))
	elif im <= 1.6e-112:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -5.8e-118)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= -2.8e-237)
		tmp = sqrt(re);
	elseif (im <= -1.35e-277)
		tmp = Float64(0.5 * Float64(im * Float64(1.0 / Float64(-sqrt(Float64(0.5 * Float64(re * -2.0)))))));
	elseif (im <= 1e-305)
		tmp = sqrt(re);
	elseif (im <= 2.55e-197)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(-1.0 / re))));
	elseif (im <= 1.6e-112)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -5.8e-118)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= -2.8e-237)
		tmp = sqrt(re);
	elseif (im <= -1.35e-277)
		tmp = 0.5 * (im * (1.0 / -sqrt((0.5 * (re * -2.0)))));
	elseif (im <= 1e-305)
		tmp = sqrt(re);
	elseif (im <= 2.55e-197)
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	elseif (im <= 1.6e-112)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -5.8e-118], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -2.8e-237], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, -1.35e-277], N[(0.5 * N[(im * N[(1.0 / (-N[Sqrt[N[(0.5 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1e-305], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 2.55e-197], N[(0.5 * N[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.6e-112], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if im < -5.79999999999999961e-118

   1. Initial program 43.2%

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

   2. Taylor expanded in im around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. sub-neg 77.5%

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

   4. Simplified 77.5%

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

   if -5.79999999999999961e-118 < im < -2.79999999999999997e-237 or -1.34999999999999988e-277 < im < 9.99999999999999996e-306 or 2.5500000000000001e-197 < im < 1.59999999999999997e-112

   1. Initial program 47.6%

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

   2. Taylor expanded in im around 0 60.2%

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

      1. associate-*r* 60.2%

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

      2. unpow2 60.2%

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

      3. rem-square-sqrt 61.4%

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

      4. metadata-eval 61.4%

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

      5. *-lft-identity 61.4%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   4. Simplified 61.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if -2.79999999999999997e-237 < im < -1.34999999999999988e-277

   1. Initial program 14.2%

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

   2. Taylor expanded in re around -inf 26.1%

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

      1. *-commutative 26.1%

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

      2. associate-*l/ 26.1%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot -0.5}{re}}} \]

      3. unpow2 26.1%

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

   4. Simplified 26.1%

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

      1. sqrt-prod 26.1%

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

      2. associate-/l* 26.1%

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

      3. sqrt-div 25.3%

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

      4. sqrt-prod 0.0%

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

      5. add-sqr-sqrt 24.4%

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

      6. div-inv 24.4%

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

      7. metadata-eval 24.4%

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

   6. Applied egg-rr 24.4%

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

      1. *-commutative 24.4%

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

      2. associate-/r/ 24.4%

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

   8. Simplified 24.4%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}} \]
   9. Step-by-step derivation

      1. frac-2neg 24.4%

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

      2. div-inv 24.4%

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

      3. add-sqr-sqrt 24.4%

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

      4. sqrt-unprod 25.3%

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

      5. sqr-neg 25.3%

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

      6. sqrt-prod 0.0%

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

      7. add-sqr-sqrt 77.6%

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

      8. sqrt-undiv 77.6%

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

      9. div-inv 77.6%

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

      10. metadata-eval 77.6%

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

   10. Applied egg-rr 77.6%

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

   if 9.99999999999999996e-306 < im < 2.5500000000000001e-197

   1. Initial program 12.2%

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

   2. Taylor expanded in re around -inf 26.6%

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

      1. *-commutative 26.6%

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

      2. associate-*l/ 26.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot -0.5}{re}}} \]

      3. unpow2 26.6%

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

   4. Simplified 26.6%

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

      1. sqrt-prod 26.6%

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

      2. associate-/l* 26.6%

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

      3. sqrt-div 25.8%

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

      4. sqrt-prod 68.6%

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

      5. add-sqr-sqrt 68.7%

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

      6. div-inv 68.7%

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

      7. metadata-eval 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-/r/ 68.7%

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

   8. Simplified 68.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}} \]
   9. Step-by-step derivation

      1. div-inv 68.8%

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

      2. clear-num 68.8%

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

      3. sqrt-undiv 69.0%

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

   10. Applied egg-rr 69.0%

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

       1. *-commutative 69.0%

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

       2. associate-/r* 69.0%

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

       3. metadata-eval 69.0%

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

   12. Simplified 69.0%

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

   if 1.59999999999999997e-112 < im

   1. Initial program 47.7%

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

   2. Taylor expanded in re around 0 73.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(re + im\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.8 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -2.8 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -1.35 \cdot 10^{-277}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \frac{1}{-\sqrt{0.5 \cdot \left(re \cdot -2\right)}}\right)\\ \mathbf{elif}\;im \leq 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.55 \cdot 10^{-197}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 1.6 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 4: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 2.65 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.3e-118)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im 1.95e-305)
     (sqrt re)
     (if (<= im 1.45e-194)
       (* 0.5 (/ im (sqrt (- re))))
       (if (<= im 2.65e-110) (sqrt re) (* 0.5 (sqrt (* 2.0 (+ re im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.3e-118) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= 1.95e-305) {
		tmp = sqrt(re);
	} else if (im <= 1.45e-194) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (im <= 2.65e-110) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (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 (im <= (-1.3d-118)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= 1.95d-305) then
        tmp = sqrt(re)
    else if (im <= 1.45d-194) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (im <= 2.65d-110) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.3e-118) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= 1.95e-305) {
		tmp = Math.sqrt(re);
	} else if (im <= 1.45e-194) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (im <= 2.65e-110) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.3e-118:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= 1.95e-305:
		tmp = math.sqrt(re)
	elif im <= 1.45e-194:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif im <= 2.65e-110:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.3e-118)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= 1.95e-305)
		tmp = sqrt(re);
	elseif (im <= 1.45e-194)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (im <= 2.65e-110)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.3e-118)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= 1.95e-305)
		tmp = sqrt(re);
	elseif (im <= 1.45e-194)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (im <= 2.65e-110)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.3e-118], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 1.95e-305], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 1.45e-194], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.65e-110], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.3e-118

   1. Initial program 43.2%

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

   2. Taylor expanded in im around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. sub-neg 77.5%

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

   4. Simplified 77.5%

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

   if -1.3e-118 < im < 1.95000000000000013e-305 or 1.44999999999999985e-194 < im < 2.65e-110

   1. Initial program 42.8%

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

   2. Taylor expanded in im around 0 53.2%

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

      1. associate-*r* 53.2%

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

      2. unpow2 53.2%

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

      3. rem-square-sqrt 54.3%

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

      4. metadata-eval 54.3%

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

      5. *-lft-identity 54.3%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   4. Simplified 54.3%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if 1.95000000000000013e-305 < im < 1.44999999999999985e-194

   1. Initial program 12.2%

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

   2. Taylor expanded in re around -inf 26.6%

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

      1. *-commutative 26.6%

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

      2. associate-*l/ 26.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot -0.5}{re}}} \]

      3. unpow2 26.6%

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

   4. Simplified 26.6%

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

      1. expm1-log1p-u 26.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{\left(im \cdot im\right) \cdot -0.5}{re}}\right)\right)} \]

      2. expm1-udef 26.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{\left(im \cdot im\right) \cdot -0.5}{re}}\right)} - 1\right)} \]

      3. associate-*r/ 26.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{2 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)}{re}}}\right)} - 1\right) \]

      4. *-commutative 26.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{2 \cdot \color{blue}{\left(-0.5 \cdot \left(im \cdot im\right)\right)}}{re}}\right)} - 1\right) \]

      5. associate-*r* 26.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{\left(2 \cdot -0.5\right) \cdot \left(im \cdot im\right)}}{re}}\right)} - 1\right) \]

      6. metadata-eval 26.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{-1} \cdot \left(im \cdot im\right)}{re}}\right)} - 1\right) \]

   6. Applied egg-rr 26.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 \cdot \left(im \cdot im\right)}{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 26.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{-1 \cdot \left(im \cdot im\right)}{re}}\right)\right)} \]

      2. expm1-log1p 26.6%

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

      3. rem-exp-log 26.6%

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

      4. rem-exp-log 26.6%

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

      5. mul-1-neg 26.6%

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

      6. distribute-rgt-neg-in 26.6%

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

   8. Simplified 26.6%

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

      1. frac-2neg 26.6%

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

      2. sqrt-div 25.8%

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

      3. distribute-rgt-neg-out 25.8%

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

      4. remove-double-neg 25.8%

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

      5. sqrt-prod 68.6%

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

      6. add-sqr-sqrt 68.8%

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

   10. Applied egg-rr 68.8%

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

   if 2.65e-110 < im

   1. Initial program 47.7%

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

   2. Taylor expanded in re around 0 73.7%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.3 \cdot 10^{-118}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 1.95 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 1.45 \cdot 10^{-194}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 2.65 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 5: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.55 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.02 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.55e-117)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im 2.1e-305)
     (sqrt re)
     (if (<= im 2.02e-190)
       (* 0.5 (* im (sqrt (/ -1.0 re))))
       (if (<= im 8.2e-111) (sqrt re) (* 0.5 (sqrt (* 2.0 (+ re im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.55e-117) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= 2.1e-305) {
		tmp = sqrt(re);
	} else if (im <= 2.02e-190) {
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	} else if (im <= 8.2e-111) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (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 (im <= (-1.55d-117)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= 2.1d-305) then
        tmp = sqrt(re)
    else if (im <= 2.02d-190) then
        tmp = 0.5d0 * (im * sqrt(((-1.0d0) / re)))
    else if (im <= 8.2d-111) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.55e-117) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= 2.1e-305) {
		tmp = Math.sqrt(re);
	} else if (im <= 2.02e-190) {
		tmp = 0.5 * (im * Math.sqrt((-1.0 / re)));
	} else if (im <= 8.2e-111) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.55e-117:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= 2.1e-305:
		tmp = math.sqrt(re)
	elif im <= 2.02e-190:
		tmp = 0.5 * (im * math.sqrt((-1.0 / re)))
	elif im <= 8.2e-111:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.55e-117)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= 2.1e-305)
		tmp = sqrt(re);
	elseif (im <= 2.02e-190)
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(-1.0 / re))));
	elseif (im <= 8.2e-111)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.55e-117)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= 2.1e-305)
		tmp = sqrt(re);
	elseif (im <= 2.02e-190)
		tmp = 0.5 * (im * sqrt((-1.0 / re)));
	elseif (im <= 8.2e-111)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.55e-117], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 2.1e-305], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 2.02e-190], N[(0.5 * N[(im * N[Sqrt[N[(-1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 8.2e-111], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.55000000000000005e-117

   1. Initial program 43.2%

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

   2. Taylor expanded in im around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. sub-neg 77.5%

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

   4. Simplified 77.5%

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

   if -1.55000000000000005e-117 < im < 2.1e-305 or 2.0200000000000001e-190 < im < 8.19999999999999936e-111

   1. Initial program 42.8%

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

   2. Taylor expanded in im around 0 53.2%

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

      1. associate-*r* 53.2%

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

      2. unpow2 53.2%

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

      3. rem-square-sqrt 54.3%

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

      4. metadata-eval 54.3%

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

      5. *-lft-identity 54.3%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   4. Simplified 54.3%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if 2.1e-305 < im < 2.0200000000000001e-190

   1. Initial program 12.2%

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

   2. Taylor expanded in re around -inf 26.6%

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

      1. *-commutative 26.6%

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

      2. associate-*l/ 26.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot -0.5}{re}}} \]

      3. unpow2 26.6%

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

   4. Simplified 26.6%

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

      1. sqrt-prod 26.6%

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

      2. associate-/l* 26.6%

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

      3. sqrt-div 25.8%

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

      4. sqrt-prod 68.6%

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

      5. add-sqr-sqrt 68.7%

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

      6. div-inv 68.7%

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

      7. metadata-eval 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. *-commutative 68.7%

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

      2. associate-/r/ 68.7%

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

   8. Simplified 68.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{im}{\frac{\sqrt{re \cdot -2}}{\sqrt{2}}}} \]
   9. Step-by-step derivation

      1. div-inv 68.8%

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

      2. clear-num 68.8%

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

      3. sqrt-undiv 69.0%

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

   10. Applied egg-rr 69.0%

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

       1. *-commutative 69.0%

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

       2. associate-/r* 69.0%

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

       3. metadata-eval 69.0%

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

   12. Simplified 69.0%

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

   if 8.19999999999999936e-111 < im

   1. Initial program 47.7%

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

   2. Taylor expanded in re around 0 73.7%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.55 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq 2.1 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.02 \cdot 10^{-190}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{-1}{re}}\right)\\ \mathbf{elif}\;im \leq 8.2 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6: 51.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5.1e-60)
   (* 0.5 (/ im (sqrt (- re))))
   (if (<= re 6.5e-98) (* 0.5 (sqrt (* 2.0 (+ re im)))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5.1e-60) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (re <= 6.5e-98) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 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 <= (-5.1d-60)) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (re <= 6.5d-98) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5.1e-60) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (re <= 6.5e-98) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5.1e-60:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif re <= 6.5e-98:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5.1e-60)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (re <= 6.5e-98)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5.1e-60)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (re <= 6.5e-98)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5.1e-60], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e-98], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -5.09999999999999982e-60

   1. Initial program 8.3%

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

   2. Taylor expanded in re around -inf 40.4%

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

      1. *-commutative 40.4%

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

      2. associate-*l/ 40.4%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot -0.5}{re}}} \]

      3. unpow2 40.4%

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

   4. Simplified 40.4%

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

      1. expm1-log1p-u 40.4%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{\left(im \cdot im\right) \cdot -0.5}{re}}\right)\right)} \]

      2. expm1-udef 21.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{\left(im \cdot im\right) \cdot -0.5}{re}}\right)} - 1\right)} \]

      3. associate-*r/ 21.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{2 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)}{re}}}\right)} - 1\right) \]

      4. *-commutative 21.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{2 \cdot \color{blue}{\left(-0.5 \cdot \left(im \cdot im\right)\right)}}{re}}\right)} - 1\right) \]

      5. associate-*r* 21.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{\left(2 \cdot -0.5\right) \cdot \left(im \cdot im\right)}}{re}}\right)} - 1\right) \]

      6. metadata-eval 21.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{-1} \cdot \left(im \cdot im\right)}{re}}\right)} - 1\right) \]

   6. Applied egg-rr 21.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 \cdot \left(im \cdot im\right)}{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 40.5%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{-1 \cdot \left(im \cdot im\right)}{re}}\right)\right)} \]

      2. expm1-log1p 40.5%

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

      3. rem-exp-log 39.2%

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

      4. rem-exp-log 40.5%

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

      5. mul-1-neg 40.5%

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

      6. distribute-rgt-neg-in 40.5%

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

   8. Simplified 40.5%

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

      1. frac-2neg 40.5%

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

      2. sqrt-div 53.2%

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

      3. distribute-rgt-neg-out 53.2%

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

      4. remove-double-neg 53.2%

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

      5. sqrt-prod 39.7%

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

      6. add-sqr-sqrt 46.2%

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

   10. Applied egg-rr 46.2%

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

   if -5.09999999999999982e-60 < re < 6.50000000000000017e-98

   1. Initial program 53.7%

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

   2. Taylor expanded in re around 0 38.8%

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

   if 6.50000000000000017e-98 < re

   1. Initial program 50.9%

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

   2. Taylor expanded in im around 0 65.0%

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

      1. associate-*r* 65.0%

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

      2. unpow2 65.0%

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

      3. rem-square-sqrt 66.2%

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

      4. metadata-eval 66.2%

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

      5. *-lft-identity 66.2%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   4. Simplified 66.2%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.1 \cdot 10^{-60}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-98}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 7: 50.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.4e-54)
   (* 0.5 (/ im (sqrt (- re))))
   (if (<= re 2.4e-155) (* 0.5 (sqrt (* im 2.0))) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.4e-54) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (re <= 2.4e-155) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.4d-54)) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (re <= 2.4d-155) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.4e-54) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (re <= 2.4e-155) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.4e-54:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif re <= 2.4e-155:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.4e-54)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (re <= 2.4e-155)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.4e-54)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (re <= 2.4e-155)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.4e-54], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.4e-155], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.3999999999999999e-54

   1. Initial program 8.5%

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

   2. Taylor expanded in re around -inf 41.6%

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

      1. *-commutative 41.6%

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

      2. associate-*l/ 41.6%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot -0.5}{re}}} \]

      3. unpow2 41.6%

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

   4. Simplified 41.6%

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

      1. expm1-log1p-u 41.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2 \cdot \frac{\left(im \cdot im\right) \cdot -0.5}{re}}\right)\right)} \]

      2. expm1-udef 22.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2 \cdot \frac{\left(im \cdot im\right) \cdot -0.5}{re}}\right)} - 1\right)} \]

      3. associate-*r/ 22.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{2 \cdot \left(\left(im \cdot im\right) \cdot -0.5\right)}{re}}}\right)} - 1\right) \]

      4. *-commutative 22.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{2 \cdot \color{blue}{\left(-0.5 \cdot \left(im \cdot im\right)\right)}}{re}}\right)} - 1\right) \]

      5. associate-*r* 22.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{\left(2 \cdot -0.5\right) \cdot \left(im \cdot im\right)}}{re}}\right)} - 1\right) \]

      6. metadata-eval 22.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{-1} \cdot \left(im \cdot im\right)}{re}}\right)} - 1\right) \]

   6. Applied egg-rr 22.2%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 \cdot \left(im \cdot im\right)}{re}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 41.6%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{-1 \cdot \left(im \cdot im\right)}{re}}\right)\right)} \]

      2. expm1-log1p 41.7%

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

      3. rem-exp-log 40.3%

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

      4. rem-exp-log 41.7%

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

      5. mul-1-neg 41.7%

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

      6. distribute-rgt-neg-in 41.7%

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

   8. Simplified 41.7%

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

      1. frac-2neg 41.7%

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

      2. sqrt-div 54.8%

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

      3. distribute-rgt-neg-out 54.8%

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

      4. remove-double-neg 54.8%

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

      5. sqrt-prod 39.3%

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

      6. add-sqr-sqrt 46.0%

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

   10. Applied egg-rr 46.0%

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

   if -4.3999999999999999e-54 < re < 2.4e-155

   1. Initial program 49.1%

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

   2. Taylor expanded in re around 0 39.3%

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

      1. *-commutative 39.3%

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

      2. sqrt-unprod 39.5%

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

   4. Applied egg-rr 39.5%

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

   if 2.4e-155 < re

   1. Initial program 55.0%

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

   2. Taylor expanded in im around 0 60.3%

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

      1. associate-*r* 60.3%

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

      2. unpow2 60.3%

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

      3. rem-square-sqrt 61.5%

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

      4. metadata-eval 61.5%

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

      5. *-lft-identity 61.5%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   4. Simplified 61.5%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{-54}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 8: 43.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.6e-110) (sqrt re) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.6e-110) {
		tmp = sqrt(re);
	} 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 (im <= 2.6d-110) then
        tmp = sqrt(re)
    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 (im <= 2.6e-110) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.6e-110:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.6e-110)
		tmp = sqrt(re);
	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 (im <= 2.6e-110)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.6e-110], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.5999999999999999e-110

   1. Initial program 39.2%

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

   2. Taylor expanded in im around 0 29.8%

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

      1. associate-*r* 29.8%

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

      2. unpow2 29.8%

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

      3. rem-square-sqrt 30.4%

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

      4. metadata-eval 30.4%

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

      5. *-lft-identity 30.4%

         \[\leadsto \color{blue}{\sqrt{re}} \]

   4. Simplified 30.4%

      \[\leadsto \color{blue}{\sqrt{re}} \]

   if 2.5999999999999999e-110 < im

   1. Initial program 47.7%

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

   2. Taylor expanded in re around 0 72.3%

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

      1. *-commutative 72.3%

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

      2. sqrt-unprod 72.7%

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

   4. Applied egg-rr 72.7%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 9: 26.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 41.8%

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

2. Taylor expanded in im around 0 25.5%

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

   1. associate-*r* 25.5%

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

   2. unpow2 25.5%

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

   3. rem-square-sqrt 26.0%

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

   4. metadata-eval 26.0%

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

   5. *-lft-identity 26.0%

      \[\leadsto \color{blue}{\sqrt{re}} \]

4. Simplified 26.0%

   \[\leadsto \color{blue}{\sqrt{re}} \]

5. Final simplification 26.0%

   \[\leadsto \sqrt{re} \]

Developer target: 47.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + 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(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

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

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2023174 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))