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

Percentage Accurate: 41.2% → 89.8%

Time: 8.9s

Alternatives: 9

Speedup: 1.9×

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: 41.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(\left(\sqrt{0.5} \cdot \left(im \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (* (* (sqrt 0.5) (* im (sqrt 2.0))) (sqrt (/ 1.0 re))))
   (* 0.5 (sqrt (* 2.0 (- (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 = 0.5 * ((sqrt(0.5) * (im * sqrt(2.0))) * sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

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

Python

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = 0.5 * ((math.sqrt(0.5) * (im * math.sqrt(2.0))) * math.sqrt((1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (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(0.5 * Float64(Float64(sqrt(0.5) * Float64(im * sqrt(2.0))) * sqrt(Float64(1.0 / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = 0.5 * ((sqrt(0.5) * (im * sqrt(2.0))) * sqrt((1.0 / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (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[(0.5 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(im * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 10.4%

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

   3. Taylor expanded in re around inf 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*l* 98.6%

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

   5. Simplified 98.6%

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

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

   1. Initial program 49.8%

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

      1. sub-neg 49.8%

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

      2. sqr-neg 49.8%

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

      3. sub-neg 49.8%

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

      4. sqr-neg 49.8%

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

      5. hypot-define 89.1%

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

   3. Simplified 89.1%

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

4. Final simplification 90.3%

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

Alternative 2: 89.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (* im (* (sqrt 0.5) (* (sqrt 2.0) (pow re -0.5)))))
   (* 0.5 (sqrt (* 2.0 (- (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 = 0.5 * (im * (sqrt(0.5) * (sqrt(2.0) * pow(re, -0.5))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 10.4%

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

   3. Taylor expanded in im around 0 56.6%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left({im}^{2} \cdot \left(-0.125 \cdot \frac{{im}^{2}}{{re}^{3}} + 0.5 \cdot \frac{1}{re}\right)\right)}} \]

   4. Taylor expanded in im around 0 98.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(im \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)\right) \cdot \sqrt{\frac{1}{re}}\right)} \]
   5. 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. associate-*l* 98.7%

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

      3. rem-exp-log 94.6%

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

      4. exp-neg 94.6%

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

      5. unpow1/2 94.6%

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

      6. exp-prod 94.6%

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

      7. distribute-lft-neg-out 94.6%

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

      8. distribute-rgt-neg-in 94.6%

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

      9. metadata-eval 94.6%

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

      10. exp-to-pow 98.6%

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

   6. Simplified 98.6%

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

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

   1. Initial program 49.8%

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

      1. sub-neg 49.8%

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

      2. sqr-neg 49.8%

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

      3. sub-neg 49.8%

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

      4. sqr-neg 49.8%

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

      5. hypot-define 89.1%

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

   3. Simplified 89.1%

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

4. Final simplification 90.3%

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

Alternative 3: 84.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (sqrt (* 2.0 (* 0.5 (/ (pow im 2.0) re)))))
   (* 0.5 (sqrt (* 2.0 (- (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 = 0.5 * sqrt((2.0 * (0.5 * (pow(im, 2.0) / re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

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

Python

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = 0.5 * math.sqrt((2.0 * (0.5 * (math.pow(im, 2.0) / re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (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(0.5 * sqrt(Float64(2.0 * Float64(0.5 * Float64((im ^ 2.0) / re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = 0.5 * sqrt((2.0 * (0.5 * ((im ^ 2.0) / re))));
	else
		tmp = 0.5 * sqrt((2.0 * (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[(0.5 * N[Sqrt[N[(2.0 * N[(0.5 * N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 10.4%

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

   3. Taylor expanded in re around inf 56.8%

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

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

   1. Initial program 49.8%

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

      1. sub-neg 49.8%

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

      2. sqr-neg 49.8%

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

      3. sub-neg 49.8%

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

      4. sqr-neg 49.8%

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

      5. hypot-define 89.1%

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

   3. Simplified 89.1%

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

4. Final simplification 85.2%

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

Alternative 4: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+198}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 10^{+285}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.6e+65)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 3.2e+198)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 1e+285)
       (* 0.5 (sqrt (* 2.0 (- re re))))
       (* 0.5 (* (sqrt 2.0) (sqrt im)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+65) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.2e+198) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 1e+285) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else {
		tmp = 0.5 * (sqrt(2.0) * sqrt(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.6d+65)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 3.2d+198) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 1d+285) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.6e+65) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 3.2e+198) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 1e+285) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.6e+65:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 3.2e+198:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 1e+285:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	else:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.6e+65)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 3.2e+198)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 1e+285)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	else
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.6e+65)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 3.2e+198)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 1e+285)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	else
		tmp = 0.5 * (sqrt(2.0) * sqrt(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.6e+65], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 3.2e+198], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1e+285], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.6e65

   1. Initial program 38.1%

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

   3. Taylor expanded in re around -inf 92.0%

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

      1. *-commutative 92.0%

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

   5. Simplified 92.0%

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

   if -4.6e65 < re < 3.1999999999999998e198

   1. Initial program 53.2%

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

   3. Taylor expanded in re around 0 68.2%

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

   if 3.1999999999999998e198 < re < 9.9999999999999998e284

   1. Initial program 2.2%

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

   3. Taylor expanded in re around inf 25.0%

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

   if 9.9999999999999998e284 < re

   1. Initial program 2.8%

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

   3. Taylor expanded in re around 0 17.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.6 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 3.2 \cdot 10^{+198}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 10^{+285}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

   1. sub-neg 45.1%

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

   2. sqr-neg 45.1%

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

   3. sub-neg 45.1%

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

   4. sqr-neg 45.1%

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

   5. hypot-define 79.6%

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

3. Simplified 79.6%

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

5. Final simplification 79.6%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \]
6. Add Preprocessing

Alternative 6: 64.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7.4 \cdot 10^{+195}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+288}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(0.5 \cdot \frac{re}{im} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.5e+65)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 7.4e+195)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (if (<= re 9.5e+288)
       (* 0.5 (sqrt (* 2.0 (- re re))))
       (* 0.5 (sqrt (* 2.0 (+ im (* re (+ (* 0.5 (/ re im)) -1.0))))))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+65) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 7.4e+195) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else if (re <= 9.5e+288) {
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.5d+65)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 7.4d+195) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else if (re <= 9.5d+288) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im + (re * ((0.5d0 * (re / im)) + (-1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.5e+65) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 7.4e+195) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else if (re <= 9.5e+288) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.5e+65:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 7.4e+195:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	elif re <= 9.5e+288:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.5e+65)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 7.4e+195)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	elseif (re <= 9.5e+288)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im + Float64(re * Float64(Float64(0.5 * Float64(re / im)) + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.5e+65)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 7.4e+195)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	elseif (re <= 9.5e+288)
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	else
		tmp = 0.5 * sqrt((2.0 * (im + (re * ((0.5 * (re / im)) + -1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.5e+65], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 7.4e+195], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e+288], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im + N[(re * N[(N[(0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if re < -4.5e65

   1. Initial program 38.1%

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

   3. Taylor expanded in re around -inf 92.0%

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

      1. *-commutative 92.0%

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

   5. Simplified 92.0%

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

   if -4.5e65 < re < 7.40000000000000001e195

   1. Initial program 53.2%

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

   3. Taylor expanded in re around 0 68.2%

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

   if 7.40000000000000001e195 < re < 9.49999999999999994e288

   1. Initial program 2.2%

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

   3. Taylor expanded in re around inf 23.9%

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

   if 9.49999999999999994e288 < re

   1. Initial program 2.8%

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

   3. Taylor expanded in re around 0 16.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(im + re \cdot \left(0.5 \cdot \frac{re}{im} - 1\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 7.4 \cdot 10^{+195}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+288}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im + re \cdot \left(0.5 \cdot \frac{re}{im} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.9 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+196} \lor \neg \left(re \leq 3.1 \cdot 10^{+288}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.9e+65)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (or (<= re 4.8e+196) (not (<= re 3.1e+288)))
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (sqrt (* 2.0 (- re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.9e+65) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 4.8e+196) || !(re <= 3.1e+288)) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re - 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 <= (-6.9d+65)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if ((re <= 4.8d+196) .or. (.not. (re <= 3.1d+288))) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -6.9e+65) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if ((re <= 4.8e+196) || !(re <= 3.1e+288)) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -6.9e+65:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif (re <= 4.8e+196) or not (re <= 3.1e+288):
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -6.9e+65)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif ((re <= 4.8e+196) || !(re <= 3.1e+288))
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -6.9e+65)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif ((re <= 4.8e+196) || ~((re <= 3.1e+288)))
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -6.9e+65], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, 4.8e+196], N[Not[LessEqual[re, 3.1e+288]], $MachinePrecision]], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -6.9e65

   1. Initial program 38.1%

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

   3. Taylor expanded in re around -inf 92.0%

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

      1. *-commutative 92.0%

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

   5. Simplified 92.0%

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

   if -6.9e65 < re < 4.8000000000000001e196 or 3.1e288 < re

   1. Initial program 51.3%

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

   3. Taylor expanded in re around 0 66.2%

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

   if 4.8000000000000001e196 < re < 3.1e288

   1. Initial program 2.2%

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

   3. Taylor expanded in re around inf 23.9%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -6.9 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 4.8 \cdot 10^{+196} \lor \neg \left(re \leq 3.1 \cdot 10^{+288}\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e+65)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (sqrt (* 2.0 (- im re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e+65) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-5d+65)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e+65) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e+65:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e+65)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e+65)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e+65], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.99999999999999973e65

   1. Initial program 38.1%

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

   3. Taylor expanded in re around -inf 92.0%

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

      1. *-commutative 92.0%

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

   5. Simplified 92.0%

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

   if -4.99999999999999973e65 < re

   1. Initial program 46.8%

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

   3. Taylor expanded in re around 0 60.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+65}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.1%

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

3. Taylor expanded in re around -inf 29.1%

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

   1. *-commutative 29.1%

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

5. Simplified 29.1%

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

6. Final simplification 29.1%

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

Reproduce

herbie shell --seed 2024115 
(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)))))