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

Percentage Accurate: 41.5% → 90.6%

Time: 9.5s

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: 41.5% 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: 90.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.3%

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

   3. Taylor expanded in re around inf 98.4%

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

      1. associate-*l* 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-*r* 98.8%

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

   5. Simplified 98.8%

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

      1. pow1 98.8%

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

      2. add-log-exp 13.6%

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

      3. associate-*l* 13.6%

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

      4. exp-prod 13.6%

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

      5. sqrt-unprod 13.6%

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

      6. metadata-eval 13.6%

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

      7. metadata-eval 13.6%

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

      8. pow1 13.6%

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

      9. add-log-exp 99.5%

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

      10. inv-pow 99.5%

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

      11. sqrt-pow1 99.7%

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

      12. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. unpow1 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   9. Simplified 99.7%

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

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

   1. Initial program 49.9%

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

      1. pow1 49.9%

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

   4. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 90.8%

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

      2. *-commutative 90.8%

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

      3. associate-*r* 90.8%

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

      4. metadata-eval 90.8%

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

   6. Simplified 90.8%

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

Alternative 2: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5.8 \cdot 10^{-36}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(im \cdot 0.5\right) \cdot {re}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e+17)
   (sqrt (- re))
   (if (<= re 5.8e-36) (sqrt (* 0.5 (- im re))) (* (* im 0.5) (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+17) {
		tmp = sqrt(-re);
	} else if (re <= 5.8e-36) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) * 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 <= (-4.5d+17)) then
        tmp = sqrt(-re)
    else if (re <= 5.8d-36) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (im * 0.5d0) * (re ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+17) {
		tmp = Math.sqrt(-re);
	} else if (re <= 5.8e-36) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) * Math.pow(re, -0.5);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e+17:
		tmp = math.sqrt(-re)
	elif re <= 5.8e-36:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (im * 0.5) * math.pow(re, -0.5)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e+17)
		tmp = sqrt(Float64(-re));
	elseif (re <= 5.8e-36)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) * (re ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e+17)
		tmp = sqrt(-re);
	elseif (re <= 5.8e-36)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (im * 0.5) * (re ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e+17], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 5.8e-36], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.5e17

   1. Initial program 41.5%

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

      1. pow1 41.5%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 76.2%

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

      1. neg-mul-1 76.2%

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

   9. Simplified 76.2%

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

   if -4.5e17 < re < 5.80000000000000026e-36

   1. Initial program 60.9%

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

      1. pow1 60.9%

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

   4. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*r* 91.8%

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

      4. metadata-eval 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in re around 0 82.4%

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

      1. neg-mul-1 82.4%

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

      2. sub-neg 82.4%

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

   9. Simplified 82.4%

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

   if 5.80000000000000026e-36 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf 71.5%

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

      1. associate-*l* 71.5%

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

      2. *-commutative 71.5%

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

      3. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

      1. pow1 71.7%

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

      2. add-log-exp 17.9%

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

      3. associate-*l* 17.9%

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

      4. exp-prod 17.9%

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

      5. sqrt-unprod 17.9%

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

      6. metadata-eval 17.9%

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

      7. metadata-eval 17.9%

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

      8. pow1 17.9%

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

      9. add-log-exp 72.2%

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

      10. inv-pow 72.2%

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

      11. sqrt-pow1 72.2%

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

      12. metadata-eval 72.2%

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

   7. Applied egg-rr 72.2%

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

      1. unpow1 72.2%

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

      2. associate-*r* 72.2%

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

      3. *-commutative 72.2%

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

   9. Simplified 72.2%

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

Alternative 3: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.46 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.46e+18)
   (sqrt (- re))
   (if (<= re 1.12e-34) (sqrt (* 0.5 (- im re))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.46e+18) {
		tmp = sqrt(-re);
	} else if (re <= 1.12e-34) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.46d+18)) then
        tmp = sqrt(-re)
    else if (re <= 1.12d-34) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.46e+18) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.12e-34) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.46e+18:
		tmp = math.sqrt(-re)
	elif re <= 1.12e-34:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.46e+18)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.12e-34)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.46e+18)
		tmp = sqrt(-re);
	elseif (re <= 1.12e-34)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.46e+18], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.12e-34], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.46e18

   1. Initial program 41.5%

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

      1. pow1 41.5%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 76.2%

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

      1. neg-mul-1 76.2%

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

   9. Simplified 76.2%

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

   if -1.46e18 < re < 1.12e-34

   1. Initial program 60.9%

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

      1. pow1 60.9%

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

   4. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*r* 91.8%

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

      4. metadata-eval 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in re around 0 82.4%

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

      1. neg-mul-1 82.4%

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

      2. sub-neg 82.4%

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

   9. Simplified 82.4%

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

   if 1.12e-34 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf 71.5%

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

      1. associate-*l* 71.5%

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

      2. *-commutative 71.5%

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

      3. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

      1. add-log-exp 17.9%

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

      2. associate-*l* 17.9%

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

      3. exp-prod 17.9%

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

      4. sqrt-unprod 17.9%

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

      5. metadata-eval 17.9%

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

      6. metadata-eval 17.9%

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

      7. pow1 17.9%

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

      8. add-log-exp 72.2%

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

      9. sqrt-div 72.0%

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

      10. metadata-eval 72.0%

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

   7. Applied egg-rr 72.0%

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

      1. associate-*r* 72.0%

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

      2. un-div-inv 72.2%

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

   9. Applied egg-rr 72.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.46 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.6 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 3.05 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.6e+18)
   (sqrt (- re))
   (if (<= re 3.05e-34) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.6e+18) {
		tmp = sqrt(-re);
	} else if (re <= 3.05e-34) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2.6d+18)) then
        tmp = sqrt(-re)
    else if (re <= 3.05d-34) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.6e+18) {
		tmp = Math.sqrt(-re);
	} else if (re <= 3.05e-34) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.6e+18:
		tmp = math.sqrt(-re)
	elif re <= 3.05e-34:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.6e+18)
		tmp = sqrt(Float64(-re));
	elseif (re <= 3.05e-34)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.6e+18)
		tmp = sqrt(-re);
	elseif (re <= 3.05e-34)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.6e+18], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 3.05e-34], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.6e18

   1. Initial program 41.5%

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

      1. pow1 41.5%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 76.2%

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

      1. neg-mul-1 76.2%

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

   9. Simplified 76.2%

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

   if -2.6e18 < re < 3.0499999999999999e-34

   1. Initial program 60.9%

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

      1. pow1 60.9%

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

   4. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 91.8%

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

      2. *-commutative 91.8%

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

      3. associate-*r* 91.8%

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

      4. metadata-eval 91.8%

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

   6. Simplified 91.8%

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

   7. Taylor expanded in re around 0 82.4%

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

      1. neg-mul-1 82.4%

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

      2. sub-neg 82.4%

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

   9. Simplified 82.4%

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

   if 3.0499999999999999e-34 < re

   1. Initial program 11.2%

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

   3. Taylor expanded in re around inf 71.5%

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

      1. associate-*l* 71.5%

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

      2. *-commutative 71.5%

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

      3. associate-*r* 71.7%

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

   5. Simplified 71.7%

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

      1. add-log-exp 17.9%

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

      2. associate-*l* 17.9%

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

      3. exp-prod 17.9%

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

      4. sqrt-unprod 17.9%

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

      5. metadata-eval 17.9%

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

      6. metadata-eval 17.9%

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

      7. pow1 17.9%

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

      8. add-log-exp 72.2%

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

      9. sqrt-div 72.0%

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

      10. metadata-eval 72.0%

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

   7. Applied egg-rr 72.0%

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

   8. Taylor expanded in im around 0 72.2%

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

      1. associate-*r* 72.2%

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

      2. unpow-1 72.2%

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

      3. metadata-eval 72.2%

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

      4. pow-sqr 72.2%

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

      5. rem-sqrt-square 72.2%

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

      6. metadata-eval 72.2%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left|{re}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \]

      7. pow-sqr 72.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \left|\color{blue}{{re}^{-0.25} \cdot {re}^{-0.25}}\right| \]

      8. fabs-sqr 72.1%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{\left({re}^{-0.25} \cdot {re}^{-0.25}\right)} \]

      9. pow-sqr 72.2%

         \[\leadsto \left(0.5 \cdot im\right) \cdot \color{blue}{{re}^{\left(2 \cdot -0.25\right)}} \]

      10. metadata-eval 72.2%

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

      11. exp-to-pow 68.2%

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

      12. metadata-eval 68.2%

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

      13. distribute-rgt-neg-in 68.2%

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

      14. rec-exp 68.2%

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

      15. exp-to-pow 72.0%

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

      16. unpow1/2 72.0%

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

      17. associate-/l* 72.2%

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

      18. *-rgt-identity 72.2%

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

      19. *-commutative 72.2%

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

      20. associate-/l* 72.0%

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

   10. Simplified 72.0%

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

Alternative 5: 65.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6e+17) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+17) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt((im * 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 <= (-4.6d+17)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt((im * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+17) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6e+17:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6e+17)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.6e+17)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6e+17], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.6e17

   1. Initial program 41.5%

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

      1. pow1 41.5%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 76.2%

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

      1. neg-mul-1 76.2%

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

   9. Simplified 76.2%

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

   if -4.6e17 < re

   1. Initial program 46.9%

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

      1. pow1 46.9%

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

   4. Applied egg-rr 78.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 78.2%

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

      2. *-commutative 78.2%

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

      3. associate-*r* 78.2%

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

      4. metadata-eval 78.2%

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

   6. Simplified 78.2%

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

   7. Taylor expanded in re around 0 67.8%

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

      1. *-commutative 67.8%

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

   9. Simplified 67.8%

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

Alternative 6: 29.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310) (sqrt (- re)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = sqrt(-re);
	} 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 <= (-5d-310)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e-310)
		tmp = sqrt(-re);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 57.6%

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

      1. pow1 57.6%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 44.0%

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

      1. neg-mul-1 44.0%

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

   9. Simplified 44.0%

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

   if -4.999999999999985e-310 < re

   1. Initial program 32.3%

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

      1. pow1 32.3%

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

   4. Applied egg-rr 63.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
   5. Step-by-step derivation

      1. unpow1 63.3%

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

      2. *-commutative 63.3%

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

      3. associate-*r* 63.3%

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

      4. metadata-eval 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in re around -inf 0.0%

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

      1. neg-mul-1 0.0%

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

   9. Simplified 0.0%

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

   10. Taylor expanded in re around 0 0.0%

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

       1. mul-1-neg 0.0%

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

       2. rem-square-sqrt 0.0%

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

       3. distribute-rgt-neg-out 0.0%

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

       4. neg-mul-1 0.0%

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

       5. associate-*r* 0.0%

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

       6. *-commutative 0.0%

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

       7. rem-square-sqrt 0.0%

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

       8. unpow1/2 0.0%

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

       9. metadata-eval 0.0%

          \[\leadsto \sqrt{\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {re}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \cdot \sqrt{re}} \]

       10. pow-sqr 0.0%

           \[\leadsto \sqrt{\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({re}^{0.25} \cdot {re}^{0.25}\right)}\right) \cdot \sqrt{re}} \]

       11. unswap-sqr 0.0%

           \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{-1} \cdot {re}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {re}^{0.25}\right)\right)} \cdot \sqrt{re}} \]

       12. fabs-sqr 0.0%

           \[\leadsto \sqrt{\color{blue}{\left|\left(\sqrt{-1} \cdot {re}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {re}^{0.25}\right)\right|} \cdot \sqrt{re}} \]

       13. unswap-sqr 0.0%

           \[\leadsto \sqrt{\left|\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({re}^{0.25} \cdot {re}^{0.25}\right)}\right| \cdot \sqrt{re}} \]

       14. rem-square-sqrt 6.0%

           \[\leadsto \sqrt{\left|\color{blue}{-1} \cdot \left({re}^{0.25} \cdot {re}^{0.25}\right)\right| \cdot \sqrt{re}} \]

       15. pow-sqr 6.0%

           \[\leadsto \sqrt{\left|-1 \cdot \color{blue}{{re}^{\left(2 \cdot 0.25\right)}}\right| \cdot \sqrt{re}} \]

       16. metadata-eval 6.0%

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

       17. unpow1/2 6.0%

           \[\leadsto \sqrt{\left|-1 \cdot \color{blue}{\sqrt{re}}\right| \cdot \sqrt{re}} \]

       18. neg-mul-1 6.0%

           \[\leadsto \sqrt{\left|\color{blue}{-\sqrt{re}}\right| \cdot \sqrt{re}} \]

       19. fabs-neg 6.0%

           \[\leadsto \sqrt{\color{blue}{\left|\sqrt{re}\right|} \cdot \sqrt{re}} \]

       20. unpow1/2 6.0%

           \[\leadsto \sqrt{\left|\color{blue}{{re}^{0.5}}\right| \cdot \sqrt{re}} \]

       21. metadata-eval 6.0%

           \[\leadsto \sqrt{\left|{re}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right| \cdot \sqrt{re}} \]

       22. pow-sqr 6.0%

           \[\leadsto \sqrt{\left|\color{blue}{{re}^{0.25} \cdot {re}^{0.25}}\right| \cdot \sqrt{re}} \]

       23. fabs-sqr 6.0%

           \[\leadsto \sqrt{\color{blue}{\left({re}^{0.25} \cdot {re}^{0.25}\right)} \cdot \sqrt{re}} \]

   12. Simplified 6.0%

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

Alternative 7: 2.8% 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 45.7%

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

   1. pow1 45.7%

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

4. Applied egg-rr 82.8%

   \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)\right) \cdot 0.25}\right)}^{1}} \]
5. Step-by-step derivation

   1. unpow1 82.8%

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

   2. *-commutative 82.8%

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

   3. associate-*r* 82.8%

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

   4. metadata-eval 82.8%

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

6. Simplified 82.8%

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

7. Taylor expanded in re around -inf 23.4%

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

   1. neg-mul-1 23.4%

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

9. Simplified 23.4%

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

10. Taylor expanded in re around 0 23.4%

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

    1. mul-1-neg 23.4%

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

    2. rem-square-sqrt 0.0%

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

    3. distribute-rgt-neg-out 0.0%

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

    4. neg-mul-1 0.0%

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

    5. associate-*r* 0.0%

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

    6. *-commutative 0.0%

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

    7. rem-square-sqrt 0.0%

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

    8. unpow1/2 0.0%

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

    9. metadata-eval 0.0%

       \[\leadsto \sqrt{\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {re}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \cdot \sqrt{re}} \]

    10. pow-sqr 0.0%

        \[\leadsto \sqrt{\left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({re}^{0.25} \cdot {re}^{0.25}\right)}\right) \cdot \sqrt{re}} \]

    11. unswap-sqr 0.0%

        \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{-1} \cdot {re}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {re}^{0.25}\right)\right)} \cdot \sqrt{re}} \]

    12. fabs-sqr 0.0%

        \[\leadsto \sqrt{\color{blue}{\left|\left(\sqrt{-1} \cdot {re}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {re}^{0.25}\right)\right|} \cdot \sqrt{re}} \]

    13. unswap-sqr 0.0%

        \[\leadsto \sqrt{\left|\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({re}^{0.25} \cdot {re}^{0.25}\right)}\right| \cdot \sqrt{re}} \]

    14. rem-square-sqrt 2.8%

        \[\leadsto \sqrt{\left|\color{blue}{-1} \cdot \left({re}^{0.25} \cdot {re}^{0.25}\right)\right| \cdot \sqrt{re}} \]

    15. pow-sqr 2.8%

        \[\leadsto \sqrt{\left|-1 \cdot \color{blue}{{re}^{\left(2 \cdot 0.25\right)}}\right| \cdot \sqrt{re}} \]

    16. metadata-eval 2.8%

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

    17. unpow1/2 2.8%

        \[\leadsto \sqrt{\left|-1 \cdot \color{blue}{\sqrt{re}}\right| \cdot \sqrt{re}} \]

    18. neg-mul-1 2.8%

        \[\leadsto \sqrt{\left|\color{blue}{-\sqrt{re}}\right| \cdot \sqrt{re}} \]

    19. fabs-neg 2.8%

        \[\leadsto \sqrt{\color{blue}{\left|\sqrt{re}\right|} \cdot \sqrt{re}} \]

    20. unpow1/2 2.8%

        \[\leadsto \sqrt{\left|\color{blue}{{re}^{0.5}}\right| \cdot \sqrt{re}} \]

    21. metadata-eval 2.8%

        \[\leadsto \sqrt{\left|{re}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right| \cdot \sqrt{re}} \]

    22. pow-sqr 2.8%

        \[\leadsto \sqrt{\left|\color{blue}{{re}^{0.25} \cdot {re}^{0.25}}\right| \cdot \sqrt{re}} \]

    23. fabs-sqr 2.8%

        \[\leadsto \sqrt{\color{blue}{\left({re}^{0.25} \cdot {re}^{0.25}\right)} \cdot \sqrt{re}} \]

12. Simplified 2.8%

    \[\leadsto \sqrt{\color{blue}{re}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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)))))