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

Percentage Accurate: 41.9% → 88.0%

Time: 8.5s

Alternatives: 8

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: 41.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 10^{-38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1e-38)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (* (* (* im (sqrt 0.5)) (sqrt 2.0)) (sqrt (/ 1.0 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1e-38) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (((im * sqrt(0.5)) * sqrt(2.0)) * sqrt((1.0 / re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1e-38) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (((im * Math.sqrt(0.5)) * Math.sqrt(2.0)) * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1e-38:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (((im * math.sqrt(0.5)) * math.sqrt(2.0)) * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1e-38)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(im * sqrt(0.5)) * sqrt(2.0)) * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1e-38)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (((im * sqrt(0.5)) * sqrt(2.0)) * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1e-38], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(im * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 9.9999999999999996e-39

   1. Initial program 48.6%

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

      1. sub-neg 48.6%

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

      2. sqr-neg 48.6%

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

      3. sub-neg 48.6%

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

      4. sqr-neg 48.6%

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

      5. hypot-define 94.0%

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

   3. Simplified 94.0%

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

   if 9.9999999999999996e-39 < re

   1. Initial program 17.0%

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

   3. Taylor expanded in re around inf 77.3%

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

      1. associate-*r* 77.6%

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

   5. Simplified 77.6%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 10^{-38}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(im \cdot \sqrt{0.5}\right) \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% 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 11.8%

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

   3. Taylor expanded in re around inf 46.1%

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

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

   1. Initial program 43.3%

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

      1. sub-neg 43.3%

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

      2. sqr-neg 43.3%

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

      3. sub-neg 43.3%

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

      4. sqr-neg 43.3%

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

      5. hypot-define 87.3%

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

   3. Simplified 87.3%

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

4. Final simplification 82.7%

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

Alternative 3: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 5.5 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot {re}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 5.5e-37)
   (* 0.5 (sqrt (* 2.0 (- (hypot re im) re))))
   (* 0.5 (* im (* (* (sqrt 0.5) (sqrt 2.0)) (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 5.5e-37) {
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * ((sqrt(0.5) * sqrt(2.0)) * pow(re, -0.5)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 5.5e-37) {
		tmp = 0.5 * Math.sqrt((2.0 * (Math.hypot(re, im) - re)));
	} else {
		tmp = 0.5 * (im * ((Math.sqrt(0.5) * Math.sqrt(2.0)) * Math.pow(re, -0.5)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 5.5e-37:
		tmp = 0.5 * math.sqrt((2.0 * (math.hypot(re, im) - re)))
	else:
		tmp = 0.5 * (im * ((math.sqrt(0.5) * math.sqrt(2.0)) * math.pow(re, -0.5)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 5.5e-37)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(hypot(re, im) - re))));
	else
		tmp = Float64(0.5 * Float64(im * Float64(Float64(sqrt(0.5) * sqrt(2.0)) * (re ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 5.5e-37)
		tmp = 0.5 * sqrt((2.0 * (hypot(re, im) - re)));
	else
		tmp = 0.5 * (im * ((sqrt(0.5) * sqrt(2.0)) * (re ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 5.5e-37], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.4999999999999998e-37

   1. Initial program 48.6%

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

      1. sub-neg 48.6%

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

      2. sqr-neg 48.6%

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

      3. sub-neg 48.6%

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

      4. sqr-neg 48.6%

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

      5. hypot-define 94.0%

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

   3. Simplified 94.0%

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

   if 5.4999999999999998e-37 < re

   1. Initial program 17.0%

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

      1. sub-neg 17.0%

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

      2. +-commutative 17.0%

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

      3. add-sqr-sqrt 15.2%

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

      4. distribute-rgt-neg-in 15.2%

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

      5. fma-define 13.5%

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

      6. hypot-define 25.7%

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

   4. Applied egg-rr 25.7%

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

   5. Taylor expanded in re around inf 77.3%

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

      1. associate-*l* 77.3%

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

      2. *-commutative 77.3%

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

      3. rem-exp-log 73.8%

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

      4. exp-neg 73.8%

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

      5. unpow1/2 73.8%

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

      6. exp-prod 73.8%

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

      7. distribute-lft-neg-out 73.8%

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

      8. distribute-rgt-neg-in 73.8%

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

      9. metadata-eval 73.8%

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

      10. exp-to-pow 77.3%

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

   7. Simplified 77.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.5 \cdot 10^{-37}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(\left(\sqrt{0.5} \cdot \sqrt{2}\right) \cdot {re}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% 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 39.9%

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

   1. sub-neg 39.9%

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

   2. sqr-neg 39.9%

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

   3. sub-neg 39.9%

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

   4. sqr-neg 39.9%

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

   5. hypot-define 79.0%

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

3. Simplified 79.0%

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

5. Final simplification 79.0%

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

Alternative 5: 64.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.2e-14)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 9.5e+116) (* 0.5 (sqrt (* 2.0 (- im re)))) (* 0.5 (sqrt 0.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.2e-14) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 9.5e+116) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * sqrt(0.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-8.2d-14)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 9.5d+116) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * sqrt(0.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.2e-14) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 9.5e+116) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.2e-14:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 9.5e+116:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * math.sqrt(0.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.2e-14)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 9.5e+116)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * sqrt(0.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.2e-14)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 9.5e+116)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.2e-14], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e+116], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.2000000000000004e-14

   1. Initial program 44.0%

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

   3. Taylor expanded in re around -inf 80.2%

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

      1. *-commutative 80.2%

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

   5. Simplified 80.2%

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

   if -8.2000000000000004e-14 < re < 9.5000000000000004e116

   1. Initial program 48.1%

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

   3. Taylor expanded in re around 0 72.0%

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

   if 9.5000000000000004e116 < re

   1. Initial program 7.3%

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

      1. sub-neg 7.3%

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

      2. +-commutative 7.3%

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

      3. add-sqr-sqrt 4.9%

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

      4. distribute-rgt-neg-in 4.9%

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

      5. fma-define 2.5%

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

      6. hypot-define 13.3%

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

   4. Applied egg-rr 13.3%

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

   5. Taylor expanded in im around 0 24.2%

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

      1. distribute-rgt1-in 24.2%

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

      2. metadata-eval 24.2%

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

      3. mul0-lft 24.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]

   7. Simplified 24.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.2 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{+116}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.3e-14)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.15e+138) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (sqrt 0.0)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-14) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.15e+138) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * sqrt(0.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.3d-14)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.15d+138) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * sqrt(0.0d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.3e-14) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.15e+138) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * Math.sqrt(0.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.3e-14:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.15e+138:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * math.sqrt(0.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.3e-14)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.15e+138)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * sqrt(0.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.3e-14)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.15e+138)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * sqrt(0.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.3e-14], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.15e+138], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.2999999999999998e-14

   1. Initial program 44.0%

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

   3. Taylor expanded in re around -inf 80.2%

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

      1. *-commutative 80.2%

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

   5. Simplified 80.2%

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

   if -3.2999999999999998e-14 < re < 1.15000000000000004e138

   1. Initial program 47.9%

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

      1. sub-neg 47.9%

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

      2. +-commutative 47.9%

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

      3. add-sqr-sqrt 25.0%

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

      4. distribute-rgt-neg-in 25.0%

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

      5. fma-define 24.2%

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

      6. hypot-define 36.4%

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

   4. Applied egg-rr 36.4%

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

   5. Taylor expanded in re around 0 70.8%

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

   if 1.15000000000000004e138 < re

   1. Initial program 5.2%

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

      1. sub-neg 5.2%

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

      2. +-commutative 5.2%

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

      3. add-sqr-sqrt 2.7%

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

      4. distribute-rgt-neg-in 2.7%

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

      5. fma-define 2.5%

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

      6. hypot-define 11.7%

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

   4. Applied egg-rr 11.7%

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

   5. Taylor expanded in im around 0 23.3%

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

      1. distribute-rgt1-in 23.3%

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

      2. metadata-eval 23.3%

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

      3. mul0-lft 23.3%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]

   7. Simplified 23.3%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.3 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.15 \cdot 10^{+138}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 2.2e-246) (* 0.5 (sqrt 0.0)) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 2.2e-246) {
		tmp = 0.5 * sqrt(0.0);
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 2.2d-246) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 2.2e-246) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 2.2e-246:
		tmp = 0.5 * math.sqrt(0.0)
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 2.2e-246)
		tmp = Float64(0.5 * sqrt(0.0));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 2.2e-246)
		tmp = 0.5 * sqrt(0.0);
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 2.2e-246], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 2.19999999999999998e-246

   1. Initial program 39.0%

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

      1. sub-neg 39.0%

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

      2. +-commutative 39.0%

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

      3. add-sqr-sqrt 4.4%

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

      4. distribute-rgt-neg-in 4.4%

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

      5. fma-define 0.7%

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

      6. hypot-define 3.9%

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

   4. Applied egg-rr 3.9%

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

   5. Taylor expanded in im around 0 37.2%

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

      1. distribute-rgt1-in 37.2%

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

      2. metadata-eval 37.2%

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

      3. mul0-lft 37.2%

         \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]

   7. Simplified 37.2%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]

   if 2.19999999999999998e-246 < im

   1. Initial program 40.0%

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

      1. sub-neg 40.0%

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

      2. +-commutative 40.0%

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

      3. add-sqr-sqrt 16.2%

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

      4. distribute-rgt-neg-in 16.2%

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

      5. fma-define 16.1%

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

      6. hypot-define 25.3%

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

   4. Applied egg-rr 25.3%

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

   5. Taylor expanded in re around 0 54.7%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 2.2 \cdot 10^{-246}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 5.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{0} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (sqrt 0.0)))

C

double code(double re, double im) {
	return 0.5 * sqrt(0.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt(0.0d0)
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt(0.0);
}

Python

def code(re, im):
	return 0.5 * math.sqrt(0.0)

Julia

function code(re, im)
	return Float64(0.5 * sqrt(0.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt(0.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

   1. sub-neg 39.9%

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

   2. +-commutative 39.9%

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

   3. add-sqr-sqrt 15.0%

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

   4. distribute-rgt-neg-in 15.0%

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

   5. fma-define 14.5%

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

   6. hypot-define 23.1%

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

4. Applied egg-rr 23.1%

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

5. Taylor expanded in im around 0 6.8%

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

   1. distribute-rgt1-in 6.8%

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

   2. metadata-eval 6.8%

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

   3. mul0-lft 6.8%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]

7. Simplified 6.8%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]

8. Final simplification 6.8%

   \[\leadsto 0.5 \cdot \sqrt{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(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)))))