math.sqrt on complex, real part

Percentage Accurate: 41.1% → 84.1%

Time: 4.6s

Alternatives: 7

Speedup: 2.0×

Specification

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: 41.1% 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: 84.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.0%

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

      1. +-commutative 6.0%

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

      2. hypot-def 6.0%

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

   3. Simplified 6.0%

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

   4. Taylor expanded in re around -inf 57.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. unpow2 57.1%

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

      2. associate-*r/ 57.1%

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

      3. neg-mul-1 57.1%

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

      4. distribute-rgt-neg-in 57.1%

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

   6. Simplified 57.1%

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

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

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. hypot-def 87.7%

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

   3. Simplified 87.7%

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

4. Final simplification 84.4%

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

Alternative 2: 52.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right) + re \cdot \frac{re}{im}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0085)
   (* 0.5 (sqrt (* 2.0 (* (/ im (/ re im)) -0.5))))
   (if (<= re 2.25e-102)
     (* 0.5 (sqrt (+ (* 2.0 (+ re im)) (* re (/ re im)))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0085) {
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= 2.25e-102) {
		tmp = 0.5 * sqrt(((2.0 * (re + im)) + (re * (re / im))));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0085d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((im / (re / im)) * (-0.5d0))))
    else if (re <= 2.25d-102) then
        tmp = 0.5d0 * sqrt(((2.0d0 * (re + im)) + (re * (re / im))))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0085) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= 2.25e-102) {
		tmp = 0.5 * Math.sqrt(((2.0 * (re + im)) + (re * (re / im))));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0085:
		tmp = 0.5 * math.sqrt((2.0 * ((im / (re / im)) * -0.5)))
	elif re <= 2.25e-102:
		tmp = 0.5 * math.sqrt(((2.0 * (re + im)) + (re * (re / im))))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0085)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im / Float64(re / im)) * -0.5))));
	elseif (re <= 2.25e-102)
		tmp = Float64(0.5 * sqrt(Float64(Float64(2.0 * Float64(re + im)) + Float64(re * Float64(re / im)))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0085)
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	elseif (re <= 2.25e-102)
		tmp = 0.5 * sqrt(((2.0 * (re + im)) + (re * (re / im))));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0085], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.25e-102], N[(0.5 * N[Sqrt[N[(N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision] + N[(re * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0085000000000000006

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

      2. hypot-def 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in re around -inf 46.6%

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

      1. *-commutative 46.6%

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

      2. unpow2 46.6%

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

      3. associate-/l* 52.6%

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

   6. Simplified 52.6%

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

   if -0.0085000000000000006 < re < 2.25e-102

   1. Initial program 50.6%

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

      1. +-commutative 50.6%

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

      2. hypot-def 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in re around 0 48.6%

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

      1. unpow2 48.6%

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

      2. distribute-lft-out 48.6%

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

   6. Simplified 48.6%

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

      1. associate-/l* 48.2%

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

      2. associate-/r/ 48.2%

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

   8. Applied egg-rr 48.2%

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

   if 2.25e-102 < re

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 75.6%

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

      1. unpow2 75.6%

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

      2. rem-square-sqrt 76.9%

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

   6. Simplified 76.9%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right) + re \cdot \frac{re}{im}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 3: 52.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0028:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re \cdot re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0028)
   (* 0.5 (sqrt (* 2.0 (* (/ im (/ re im)) -0.5))))
   (if (<= re 1.8e-102)
     (* 0.5 (sqrt (+ (/ (* re re) im) (* 2.0 (+ re im)))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0028) {
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= 1.8e-102) {
		tmp = 0.5 * sqrt((((re * re) / im) + (2.0 * (re + im))));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0028d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((im / (re / im)) * (-0.5d0))))
    else if (re <= 1.8d-102) then
        tmp = 0.5d0 * sqrt((((re * re) / im) + (2.0d0 * (re + im))))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0028) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= 1.8e-102) {
		tmp = 0.5 * Math.sqrt((((re * re) / im) + (2.0 * (re + im))));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0028:
		tmp = 0.5 * math.sqrt((2.0 * ((im / (re / im)) * -0.5)))
	elif re <= 1.8e-102:
		tmp = 0.5 * math.sqrt((((re * re) / im) + (2.0 * (re + im))))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0028)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im / Float64(re / im)) * -0.5))));
	elseif (re <= 1.8e-102)
		tmp = Float64(0.5 * sqrt(Float64(Float64(Float64(re * re) / im) + Float64(2.0 * Float64(re + im)))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0028)
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	elseif (re <= 1.8e-102)
		tmp = 0.5 * sqrt((((re * re) / im) + (2.0 * (re + im))));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0028], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.8e-102], N[(0.5 * N[Sqrt[N[(N[(N[(re * re), $MachinePrecision] / im), $MachinePrecision] + N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.00279999999999999997

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

      2. hypot-def 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in re around -inf 46.6%

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

      1. *-commutative 46.6%

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

      2. unpow2 46.6%

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

      3. associate-/l* 52.6%

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

   6. Simplified 52.6%

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

   if -0.00279999999999999997 < re < 1.8e-102

   1. Initial program 50.6%

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

      1. +-commutative 50.6%

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

      2. hypot-def 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in re around 0 48.6%

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

      1. unpow2 48.6%

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

      2. distribute-lft-out 48.6%

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

   6. Simplified 48.6%

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

   if 1.8e-102 < re

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 75.6%

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

      1. unpow2 75.6%

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

      2. rem-square-sqrt 76.9%

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

   6. Simplified 76.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0028:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{re \cdot re}{im} + 2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 4: 52.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0046:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0046)
   (* 0.5 (sqrt (* 2.0 (* (/ im (/ re im)) -0.5))))
   (if (<= re 1.55e-102)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0046) {
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= 1.55e-102) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0046d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((im / (re / im)) * (-0.5d0))))
    else if (re <= 1.55d-102) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0046) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im / (re / im)) * -0.5)));
	} else if (re <= 1.55e-102) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0046:
		tmp = 0.5 * math.sqrt((2.0 * ((im / (re / im)) * -0.5)))
	elif re <= 1.55e-102:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0046)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im / Float64(re / im)) * -0.5))));
	elseif (re <= 1.55e-102)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0046)
		tmp = 0.5 * sqrt((2.0 * ((im / (re / im)) * -0.5)));
	elseif (re <= 1.55e-102)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0046], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im / N[(re / im), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.55e-102], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0045999999999999999

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

      2. hypot-def 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in re around -inf 46.6%

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

      1. *-commutative 46.6%

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

      2. unpow2 46.6%

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

      3. associate-/l* 52.6%

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

   6. Simplified 52.6%

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

   if -0.0045999999999999999 < re < 1.55000000000000006e-102

   1. Initial program 50.6%

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

      1. +-commutative 50.6%

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

      2. hypot-def 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in re around 0 48.4%

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

      1. distribute-lft-out 48.4%

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

      2. +-commutative 48.4%

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

      3. *-commutative 48.4%

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

      4. +-commutative 48.4%

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

   6. Simplified 48.4%

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

   if 1.55000000000000006e-102 < re

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 75.6%

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

      1. unpow2 75.6%

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

      2. rem-square-sqrt 76.9%

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

   6. Simplified 76.9%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0046:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\frac{im}{\frac{re}{im}} \cdot -0.5\right)}\\ \mathbf{elif}\;re \leq 1.55 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 5: 51.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.0085)
   (* 0.5 (sqrt (/ (* im (- im)) re)))
   (if (<= re 2.25e-102)
     (* 0.5 (sqrt (* 2.0 (+ re im))))
     (* 0.5 (* 2.0 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.0085) {
		tmp = 0.5 * sqrt(((im * -im) / re));
	} else if (re <= 2.25e-102) {
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-0.0085d0)) then
        tmp = 0.5d0 * sqrt(((im * -im) / re))
    else if (re <= 2.25d-102) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -0.0085) {
		tmp = 0.5 * Math.sqrt(((im * -im) / re));
	} else if (re <= 2.25e-102) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.0085:
		tmp = 0.5 * math.sqrt(((im * -im) / re))
	elif re <= 2.25e-102:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.0085)
		tmp = Float64(0.5 * sqrt(Float64(Float64(im * Float64(-im)) / re)));
	elseif (re <= 2.25e-102)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -0.0085)
		tmp = 0.5 * sqrt(((im * -im) / re));
	elseif (re <= 2.25e-102)
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.0085], N[(0.5 * N[Sqrt[N[(N[(im * (-im)), $MachinePrecision] / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 2.25e-102], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -0.0085000000000000006

   1. Initial program 8.3%

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

      1. +-commutative 8.3%

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

      2. hypot-def 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in re around -inf 46.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. unpow2 46.7%

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

      2. associate-*r/ 46.7%

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

      3. neg-mul-1 46.7%

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

      4. distribute-rgt-neg-in 46.7%

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

   6. Simplified 46.7%

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

   if -0.0085000000000000006 < re < 2.25e-102

   1. Initial program 50.6%

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

      1. +-commutative 50.6%

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

      2. hypot-def 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in re around 0 48.4%

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

      1. distribute-lft-out 48.4%

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

      2. +-commutative 48.4%

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

      3. *-commutative 48.4%

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

      4. +-commutative 48.4%

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

   6. Simplified 48.4%

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

   if 2.25e-102 < re

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 75.6%

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

      1. unpow2 75.6%

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

      2. rem-square-sqrt 76.9%

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

   6. Simplified 76.9%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.0085:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{im \cdot \left(-im\right)}{re}}\\ \mathbf{elif}\;re \leq 2.25 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 6: 42.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 2.25 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 2.25e-102) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* 2.0 (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 2.25e-102) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * 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 <= 2.25d-102) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * (2.0d0 * sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 2.25e-102) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (2.0 * Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 2.25e-102:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (2.0 * math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 2.25e-102)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * Float64(2.0 * sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 2.25e-102)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (2.0 * sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 2.25e-102], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 2.25e-102

   1. Initial program 33.6%

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

      1. +-commutative 33.6%

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

      2. hypot-def 68.3%

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

   3. Simplified 68.3%

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

   4. Taylor expanded in re around 0 35.3%

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

      1. *-commutative 35.3%

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

   6. Simplified 35.3%

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

   if 2.25e-102 < re

   1. Initial program 60.3%

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

      1. +-commutative 60.3%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 75.6%

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

      1. unpow2 75.6%

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

      2. rem-square-sqrt 76.9%

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

   6. Simplified 76.9%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 2.25 \cdot 10^{-102}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \sqrt{re}\right)\\ \end{array} \]

Alternative 7: 26.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.7%

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

   1. +-commutative 42.7%

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

   2. hypot-def 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in re around 0 27.2%

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

   1. *-commutative 27.2%

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

6. Simplified 27.2%

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

7. Final simplification 27.2%

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

Developer target: 48.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023238 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))