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

Percentage Accurate: 41.8% → 89.5%

Time: 8.4s

Alternatives: 7

Speedup: 2.0×

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: 41.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.5%

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

   3. Taylor expanded in re around inf 93.7%

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

      1. *-commutative 93.7%

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

      2. associate-*l* 94.2%

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

   5. Simplified 94.2%

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

      1. add-cube-cbrt 93.5%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)} \]

      2. pow3 93.5%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)}^{3}} \]

      3. sqrt-div 93.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}}\right)}^{3} \]

      4. metadata-eval 93.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}}}\right)}^{3} \]

      5. un-div-inv 93.7%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)}{\sqrt{re}}}}\right)}^{3} \]

      6. associate-*r* 93.7%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot im}}{\sqrt{re}}}\right)}^{3} \]

      7. sqrt-unprod 93.5%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\sqrt{0.5 \cdot 2}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      8. metadata-eval 93.5%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{1}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      9. metadata-eval 93.5%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{1} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      10. *-un-lft-identity 93.5%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{im}}{\sqrt{re}}}\right)}^{3} \]

   7. Applied egg-rr 93.5%

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

      1. rem-cube-cbrt 95.0%

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

      2. clear-num 93.2%

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

      3. associate-/r/ 95.0%

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

      4. pow1/2 95.0%

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

      5. pow-flip 95.2%

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

      6. metadata-eval 95.2%

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

   9. Applied egg-rr 95.2%

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

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

   1. Initial program 44.2%

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

      1. sub-neg 44.2%

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

      2. sqr-neg 44.2%

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

      3. sub-neg 44.2%

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

      4. sqr-neg 44.2%

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

      5. hypot-define 89.9%

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

   3. Simplified 89.9%

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

4. Final simplification 90.7%

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

Alternative 2: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.18:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.18)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.4e-149)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 2.3e-97)
       (* 0.5 (/ im (sqrt re)))
       (if (<= re 2.35e+67)
         (* 0.5 (sqrt (* im 2.0)))
         (* 0.5 (* im (pow re -0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.18) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.4e-149) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 2.3e-97) {
		tmp = 0.5 * (im / sqrt(re));
	} else if (re <= 2.35e+67) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.18d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.4d-149) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 2.3d-97) then
        tmp = 0.5d0 * (im / sqrt(re))
    else if (re <= 2.35d+67) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.18) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.4e-149) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 2.3e-97) {
		tmp = 0.5 * (im / Math.sqrt(re));
	} else if (re <= 2.35e+67) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.18:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.4e-149:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 2.3e-97:
		tmp = 0.5 * (im / math.sqrt(re))
	elif re <= 2.35e+67:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.18)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.4e-149)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 2.3e-97)
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	elseif (re <= 2.35e+67)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.18)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.4e-149)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 2.3e-97)
		tmp = 0.5 * (im / sqrt(re));
	elseif (re <= 2.35e+67)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.18], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.4e-149], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.3e-97], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.35e+67], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -0.17999999999999999

   1. Initial program 37.2%

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

   3. Taylor expanded in re around -inf 81.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 81.9%

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

   5. Simplified 81.9%

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

   if -0.17999999999999999 < re < 2.4000000000000001e-149

   1. Initial program 60.1%

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

   3. Taylor expanded in re around 0 82.8%

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

   if 2.4000000000000001e-149 < re < 2.29999999999999994e-97

   1. Initial program 33.5%

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

   3. Taylor expanded in re around inf 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-*l* 71.4%

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

   5. Simplified 71.4%

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

      1. sqrt-div 71.6%

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

      2. metadata-eval 71.6%

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

      3. un-div-inv 71.4%

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

      4. associate-*r* 71.2%

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

      5. sqrt-unprod 72.1%

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

      6. metadata-eval 72.1%

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

      7. metadata-eval 72.1%

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

      8. *-un-lft-identity 72.1%

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

   7. Applied egg-rr 72.1%

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

   if 2.29999999999999994e-97 < re < 2.35000000000000009e67

   1. Initial program 27.8%

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

   3. Taylor expanded in re around 0 70.4%

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

      1. *-commutative 70.4%

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

   5. Simplified 70.4%

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

   if 2.35000000000000009e67 < re

   1. Initial program 7.8%

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

   3. Taylor expanded in re around inf 90.4%

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

      1. *-commutative 90.4%

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

      2. associate-*l* 91.0%

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

   5. Simplified 91.0%

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

      1. add-cube-cbrt 90.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)} \]

      2. pow3 90.4%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)}^{3}} \]

      3. sqrt-div 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}}\right)}^{3} \]

      4. metadata-eval 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}}}\right)}^{3} \]

      5. un-div-inv 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)}{\sqrt{re}}}}\right)}^{3} \]

      6. associate-*r* 90.5%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot im}}{\sqrt{re}}}\right)}^{3} \]

      7. sqrt-unprod 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\sqrt{0.5 \cdot 2}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      8. metadata-eval 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{1}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      9. metadata-eval 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{1} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      10. *-un-lft-identity 90.0%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{im}}{\sqrt{re}}}\right)}^{3} \]

   7. Applied egg-rr 90.0%

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

      1. rem-cube-cbrt 91.5%

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

      2. clear-num 88.9%

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

      3. associate-/r/ 91.3%

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

      4. pow1/2 91.3%

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

      5. pow-flip 91.6%

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

      6. metadata-eval 91.6%

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

   9. Applied egg-rr 91.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.18:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.4 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 2.3 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \mathbf{elif}\;re \leq 2.35 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1050:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.55 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \frac{1}{\sqrt{re}}\right)\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1050.0)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.55e-149)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 1.65e-99)
       (* 0.5 (* im (/ 1.0 (sqrt re))))
       (if (<= re 7e+67)
         (* 0.5 (sqrt (* im 2.0)))
         (* 0.5 (* im (pow re -0.5))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1050.0) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.55e-149) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 1.65e-99) {
		tmp = 0.5 * (im * (1.0 / sqrt(re)));
	} else if (re <= 7e+67) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1050.0d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.55d-149) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 1.65d-99) then
        tmp = 0.5d0 * (im * (1.0d0 / sqrt(re)))
    else if (re <= 7d+67) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1050.0) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.55e-149) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 1.65e-99) {
		tmp = 0.5 * (im * (1.0 / Math.sqrt(re)));
	} else if (re <= 7e+67) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1050.0:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.55e-149:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 1.65e-99:
		tmp = 0.5 * (im * (1.0 / math.sqrt(re)))
	elif re <= 7e+67:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1050.0)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.55e-149)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 1.65e-99)
		tmp = Float64(0.5 * Float64(im * Float64(1.0 / sqrt(re))));
	elseif (re <= 7e+67)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1050.0)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.55e-149)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 1.65e-99)
		tmp = 0.5 * (im * (1.0 / sqrt(re)));
	elseif (re <= 7e+67)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1050.0], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.55e-149], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.65e-99], N[(0.5 * N[(im * N[(1.0 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7e+67], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if re < -1050

   1. Initial program 37.2%

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

   3. Taylor expanded in re around -inf 81.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 81.9%

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

   5. Simplified 81.9%

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

   if -1050 < re < 2.54999999999999991e-149

   1. Initial program 60.1%

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

   3. Taylor expanded in re around 0 82.8%

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

   if 2.54999999999999991e-149 < re < 1.64999999999999993e-99

   1. Initial program 33.5%

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

   3. Taylor expanded in re around inf 71.0%

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

      1. *-commutative 71.0%

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

      2. associate-*l* 71.4%

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

   5. Simplified 71.4%

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

      1. add-cube-cbrt 70.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)} \]

      2. pow3 71.1%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)}^{3}} \]

      3. sqrt-div 71.4%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}}\right)}^{3} \]

      4. metadata-eval 71.4%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}}}\right)}^{3} \]

      5. un-div-inv 71.4%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)}{\sqrt{re}}}}\right)}^{3} \]

      6. associate-*r* 71.2%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot im}}{\sqrt{re}}}\right)}^{3} \]

      7. sqrt-unprod 71.1%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\sqrt{0.5 \cdot 2}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      8. metadata-eval 71.1%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{1}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      9. metadata-eval 71.1%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{1} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      10. *-un-lft-identity 71.1%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{im}}{\sqrt{re}}}\right)}^{3} \]

   7. Applied egg-rr 71.1%

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

      1. rem-cube-cbrt 72.1%

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

      2. frac-2neg 72.1%

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

      3. div-inv 72.2%

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

   9. Applied egg-rr 72.2%

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

   if 1.64999999999999993e-99 < re < 7e67

   1. Initial program 27.8%

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

   3. Taylor expanded in re around 0 70.4%

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

      1. *-commutative 70.4%

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

   5. Simplified 70.4%

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

   if 7e67 < re

   1. Initial program 7.8%

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

   3. Taylor expanded in re around inf 90.4%

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

      1. *-commutative 90.4%

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

      2. associate-*l* 91.0%

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

   5. Simplified 91.0%

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

      1. add-cube-cbrt 90.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)} \]

      2. pow3 90.4%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)}^{3}} \]

      3. sqrt-div 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}}\right)}^{3} \]

      4. metadata-eval 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}}}\right)}^{3} \]

      5. un-div-inv 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)}{\sqrt{re}}}}\right)}^{3} \]

      6. associate-*r* 90.5%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot im}}{\sqrt{re}}}\right)}^{3} \]

      7. sqrt-unprod 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\sqrt{0.5 \cdot 2}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      8. metadata-eval 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{1}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      9. metadata-eval 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{1} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      10. *-un-lft-identity 90.0%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{im}}{\sqrt{re}}}\right)}^{3} \]

   7. Applied egg-rr 90.0%

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

      1. rem-cube-cbrt 91.5%

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

      2. clear-num 88.9%

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

      3. associate-/r/ 91.3%

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

      4. pow1/2 91.3%

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

      5. pow-flip 91.6%

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

      6. metadata-eval 91.6%

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

   9. Applied egg-rr 91.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1050:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.55 \cdot 10^{-149}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 1.65 \cdot 10^{-99}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \frac{1}{\sqrt{re}}\right)\\ \mathbf{elif}\;re \leq 7 \cdot 10^{+67}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -27.5:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -27.5)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 2.5e+71)
     (* 0.5 (sqrt (* im 2.0)))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -27.5) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 2.5e+71) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-27.5d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 2.5d+71) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -27.5) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 2.5e+71) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -27.5:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 2.5e+71:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -27.5)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 2.5e+71)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -27.5)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 2.5e+71)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -27.5], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.5e+71], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -27.5

   1. Initial program 37.2%

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

   3. Taylor expanded in re around -inf 81.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 81.9%

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

   5. Simplified 81.9%

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

   if -27.5 < re < 2.49999999999999986e71

   1. Initial program 49.2%

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

   3. Taylor expanded in re around 0 74.6%

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

      1. *-commutative 74.6%

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

   5. Simplified 74.6%

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

   if 2.49999999999999986e71 < re

   1. Initial program 7.8%

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

   3. Taylor expanded in re around inf 90.4%

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

      1. *-commutative 90.4%

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

      2. associate-*l* 91.0%

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

   5. Simplified 91.0%

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

      1. add-cube-cbrt 90.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}} \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right) \cdot \sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)} \]

      2. pow3 90.4%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \sqrt{\frac{1}{re}}}\right)}^{3}} \]

      3. sqrt-div 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{re}}}}\right)}^{3} \]

      4. metadata-eval 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\left(\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{re}}}\right)}^{3} \]

      5. un-div-inv 90.3%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\color{blue}{\frac{\sqrt{0.5} \cdot \left(\sqrt{2} \cdot im\right)}{\sqrt{re}}}}\right)}^{3} \]

      6. associate-*r* 90.5%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot im}}{\sqrt{re}}}\right)}^{3} \]

      7. sqrt-unprod 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{\sqrt{0.5 \cdot 2}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      8. metadata-eval 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\sqrt{\color{blue}{1}} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      9. metadata-eval 90.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{1} \cdot im}{\sqrt{re}}}\right)}^{3} \]

      10. *-un-lft-identity 90.0%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\frac{\color{blue}{im}}{\sqrt{re}}}\right)}^{3} \]

   7. Applied egg-rr 90.0%

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

      1. rem-cube-cbrt 91.5%

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

      2. clear-num 88.9%

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

      3. associate-/r/ 91.3%

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

      4. pow1/2 91.3%

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

      5. pow-flip 91.6%

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

      6. metadata-eval 91.6%

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

   9. Applied egg-rr 91.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -27.5:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 2.5 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -640:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{+76}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -640.0)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 4.6e+76) (* 0.5 (sqrt (* im 2.0))) (* 0.5 (/ im (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -640.0) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 4.6e+76) {
		tmp = 0.5 * sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-640.0d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 4.6d+76) then
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    else
        tmp = 0.5d0 * (im / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -640.0) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 4.6e+76) {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -640.0:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 4.6e+76:
		tmp = 0.5 * math.sqrt((im * 2.0))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -640.0)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 4.6e+76)
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	else
		tmp = Float64(0.5 * Float64(im / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -640.0)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 4.6e+76)
		tmp = 0.5 * sqrt((im * 2.0));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -640.0], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 4.6e+76], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -640

   1. Initial program 37.2%

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

   3. Taylor expanded in re around -inf 81.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 81.9%

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

   5. Simplified 81.9%

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

   if -640 < re < 4.60000000000000002e76

   1. Initial program 49.2%

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

   3. Taylor expanded in re around 0 74.6%

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

      1. *-commutative 74.6%

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

   5. Simplified 74.6%

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

   if 4.60000000000000002e76 < re

   1. Initial program 7.8%

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

   3. Taylor expanded in re around inf 90.4%

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

      1. *-commutative 90.4%

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

      2. associate-*l* 91.0%

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

   5. Simplified 91.0%

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

      1. sqrt-div 91.0%

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

      2. metadata-eval 91.0%

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

      3. un-div-inv 91.0%

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

      4. associate-*r* 90.5%

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

      5. sqrt-unprod 91.5%

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

      6. metadata-eval 91.5%

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

      7. metadata-eval 91.5%

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

      8. *-un-lft-identity 91.5%

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

   7. Applied egg-rr 91.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -640:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 4.6 \cdot 10^{+76}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0185:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0185) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* im 2.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0185) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0185d0)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0185) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0185:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0185)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0185)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0185], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -0.0184999999999999991

   1. Initial program 37.2%

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

   3. Taylor expanded in re around -inf 81.9%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-4 \cdot re}} \]
   4. Step-by-step derivation

      1. *-commutative 81.9%

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

   5. Simplified 81.9%

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

   if -0.0184999999999999991 < re

   1. Initial program 39.0%

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

   3. Taylor expanded in re around 0 59.7%

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

      1. *-commutative 59.7%

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

   5. Simplified 59.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0185:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.6%

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

3. Taylor expanded in re around 0 51.2%

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

   1. *-commutative 51.2%

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

5. Simplified 51.2%

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

6. Final simplification 51.2%

   \[\leadsto 0.5 \cdot \sqrt{im \cdot 2} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024075 
(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)))))