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

Percentage Accurate: 41.7% → 88.7%

Time: 6.7s

Alternatives: 6

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: 41.7% 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: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 170000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 170000.0)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 170000.0) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * pow(re, -0.5));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 170000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 170000.0:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 170000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	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 <= 170000.0)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 170000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.7e5

   1. Initial program 53.0%

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

      1. add-log-exp 6.1%

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

      2. *-un-lft-identity 6.1%

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

      3. log-prod 6.1%

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

      4. metadata-eval 6.1%

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

      5. add-log-exp 53.0%

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

      6. hypot-def 93.1%

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

   3. Applied egg-rr 93.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\left(0 + \mathsf{hypot}\left(re, im\right)\right)} - re\right)} \]
   4. Step-by-step derivation

      1. +-lft-identity 93.1%

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

   5. Simplified 93.1%

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

   if 1.7e5 < re

   1. Initial program 10.4%

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

   2. Taylor expanded in re around inf 58.6%

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

      1. unpow2 58.6%

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

      2. associate-/l* 60.4%

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

   4. Simplified 60.4%

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

   5. Taylor expanded in im around 0 83.9%

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

      1. *-commutative 83.9%

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

      2. unpow1/2 83.9%

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

      3. exp-to-pow 79.5%

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

      4. *-commutative 79.5%

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

      5. log-rec 79.5%

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

      6. neg-mul-1 79.5%

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

      7. associate-*r* 79.5%

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

      8. metadata-eval 79.5%

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

      9. log-pow 79.5%

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

      10. rem-exp-log 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 170000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 2: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.4e+47)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.9e-9)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.4e+47) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.9e-9) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} 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 <= (-6.4d+47)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.9d-9) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    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 <= -6.4e+47) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.9e-9) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.4e+47:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.9e-9:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.4e+47)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.9e-9)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	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 <= -6.4e+47)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.9e-9)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.4e+47], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.9e-9], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $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 < -6.4e47

   1. Initial program 23.4%

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

   2. Taylor expanded in re around -inf 81.7%

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

      1. *-commutative 81.7%

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

   4. Simplified 81.7%

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

   if -6.4e47 < re < 1.90000000000000006e-9

   1. Initial program 65.1%

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

   2. Taylor expanded in re around 0 77.1%

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

   if 1.90000000000000006e-9 < re

   1. Initial program 10.1%

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

   2. Taylor expanded in re around inf 56.9%

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

      1. unpow2 56.9%

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

      2. associate-/l* 58.6%

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

   4. Simplified 58.6%

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

   5. Taylor expanded in im around 0 82.1%

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

      1. *-commutative 82.1%

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

      2. unpow1/2 82.1%

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

      3. exp-to-pow 77.9%

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

      4. *-commutative 77.9%

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

      5. log-rec 77.9%

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

      6. neg-mul-1 77.9%

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

      7. associate-*r* 77.9%

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

      8. metadata-eval 77.9%

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

      9. log-pow 77.9%

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

      10. rem-exp-log 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 3: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 50000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.4e+43)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 50000.0)
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.4e+43) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 50000.0) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= (-6.4d+43)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 50000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= -6.4e+43) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 50000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.4e+43:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 50000.0:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.4e+43)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 50000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= -6.4e+43)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 50000.0)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.4e+43], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 50000.0], N[(0.5 * N[Sqrt[N[(2.0 * im), $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 < -6.40000000000000029e43

   1. Initial program 23.4%

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

   2. Taylor expanded in re around -inf 81.7%

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

      1. *-commutative 81.7%

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

   4. Simplified 81.7%

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

   if -6.40000000000000029e43 < re < 5e4

   1. Initial program 63.4%

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

   2. Taylor expanded in re around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. sqrt-unprod 74.1%

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

   4. Applied egg-rr 74.1%

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

   if 5e4 < re

   1. Initial program 10.4%

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

   2. Taylor expanded in re around inf 58.6%

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

      1. unpow2 58.6%

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

      2. associate-/l* 60.4%

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

   4. Simplified 60.4%

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

   5. Taylor expanded in im around 0 83.9%

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

      1. *-commutative 83.9%

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

      2. unpow1/2 83.9%

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

      3. exp-to-pow 79.5%

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

      4. *-commutative 79.5%

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

      5. log-rec 79.5%

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

      6. neg-mul-1 79.5%

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

      7. associate-*r* 79.5%

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

      8. metadata-eval 79.5%

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

      9. log-pow 79.5%

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

      10. rem-exp-log 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.4 \cdot 10^{+43}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 50000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 4: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 15000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 15000.0) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 15000.0) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= 15000.0d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= 15000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 15000.0:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 15000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= 15000.0)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 15000.0], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 15000

   1. Initial program 53.0%

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

   2. Taylor expanded in re around 0 60.8%

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

      1. *-commutative 60.8%

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

      2. sqrt-unprod 61.2%

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

   4. Applied egg-rr 61.2%

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

   if 15000 < re

   1. Initial program 10.4%

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

   2. Taylor expanded in re around inf 58.6%

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

      1. unpow2 58.6%

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

      2. associate-/l* 60.4%

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

   4. Simplified 60.4%

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

   5. Taylor expanded in im around 0 83.9%

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

      1. *-commutative 83.9%

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

      2. unpow1/2 83.9%

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

      3. exp-to-pow 79.5%

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

      4. *-commutative 79.5%

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

      5. log-rec 79.5%

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

      6. neg-mul-1 79.5%

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

      7. associate-*r* 79.5%

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

      8. metadata-eval 79.5%

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

      9. log-pow 79.5%

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

      10. rem-exp-log 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 15000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]

Alternative 5: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 0.0062:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 0.0062) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (/ im (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 0.0062) {
		tmp = 0.5 * sqrt((2.0 * im));
	} 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 <= 0.0062d0) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    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 <= 0.0062) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 0.0062:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 0.0062)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	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 <= 0.0062)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 0.0062], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 0.00619999999999999978

   1. Initial program 53.5%

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

   2. Taylor expanded in re around 0 60.9%

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

      1. *-commutative 60.9%

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

      2. sqrt-unprod 61.3%

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

   4. Applied egg-rr 61.3%

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

   if 0.00619999999999999978 < re

   1. Initial program 10.3%

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

   2. Taylor expanded in re around inf 57.1%

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

      1. unpow2 57.1%

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

      2. associate-/l* 58.8%

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

   4. Simplified 58.8%

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

   5. Taylor expanded in im around 0 83.0%

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

      1. *-commutative 83.0%

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

      2. unpow1/2 83.0%

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

      3. exp-to-pow 78.7%

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

      4. *-commutative 78.7%

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

      5. log-rec 78.7%

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

      6. neg-mul-1 78.7%

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

      7. associate-*r* 78.7%

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

      8. metadata-eval 78.7%

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

      9. log-pow 78.7%

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

      10. rem-exp-log 83.0%

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

   7. Simplified 83.0%

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

      1. add-sqr-sqrt 82.7%

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

      2. sqrt-unprod 58.9%

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

      3. swap-sqr 57.1%

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

      4. pow-prod-up 57.1%

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

      5. metadata-eval 57.1%

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

      6. inv-pow 57.1%

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

      7. associate-*l* 58.9%

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

      8. div-inv 58.9%

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

      9. clear-num 58.8%

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

      10. div-inv 58.9%

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

      11. *-un-lft-identity 58.9%

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

      12. div-inv 59.0%

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

      13. frac-times 57.2%

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

      14. *-un-lft-identity 57.2%

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

      15. metadata-eval 57.2%

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

      16. associate-*r* 57.1%

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

      17. add-log-exp 22.5%

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

      18. *-un-lft-identity 22.5%

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

      19. log-prod 22.5%

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

      20. metadata-eval 22.5%

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

   9. Applied egg-rr 83.0%

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

       1. +-lft-identity 83.0%

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

   11. Simplified 83.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 0.0062:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{re}}\\ \end{array} \]

Alternative 6: 51.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

2. Taylor expanded in re around 0 50.9%

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

   1. *-commutative 50.9%

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

   2. sqrt-unprod 51.3%

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

4. Applied egg-rr 51.3%

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

5. Final simplification 51.3%

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

Reproduce

herbie shell --seed 2023174 
(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)))))