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

Percentage Accurate: 40.8% → 90.1%

Time: 9.4s

Alternatives: 7

Speedup: 2.0×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.7%

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

      1. pow1 6.7%

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

   4. Applied egg-rr 21.7%

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

      1. unpow1 21.7%

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

      2. *-commutative 21.7%

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

      3. associate-*r* 21.7%

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

      4. metadata-eval 21.7%

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

      5. hypot-undefine 6.7%

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

      6. unpow2 6.7%

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

      7. unpow2 6.7%

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

      8. +-commutative 6.7%

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

      9. unpow2 6.7%

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

      10. unpow2 6.7%

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

      11. hypot-undefine 21.7%

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

   6. Simplified 21.7%

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

   7. Taylor expanded in im around 0 85.8%

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

      1. associate-*l* 85.8%

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

      2. unpow2 85.8%

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

      3. rem-square-sqrt 87.1%

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

      4. unpow-1 87.1%

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

      5. metadata-eval 87.1%

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

      6. pow-sqr 87.2%

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

      7. rem-sqrt-square 87.2%

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

      8. rem-square-sqrt 86.7%

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

      9. fabs-sqr 86.7%

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

      10. rem-square-sqrt 87.2%

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

   9. Simplified 87.2%

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

   if 0.0 < (-.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. Add Preprocessing
   3. Step-by-step derivation

      1. pow1 49.8%

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

   4. Applied egg-rr 91.4%

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

      1. unpow1 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-*r* 91.4%

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

      4. metadata-eval 91.4%

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

      5. hypot-undefine 49.8%

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

      6. unpow2 49.8%

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

      7. unpow2 49.8%

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

      8. +-commutative 49.8%

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

      9. unpow2 49.8%

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

      10. unpow2 49.8%

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

      11. hypot-undefine 91.4%

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

   6. Simplified 91.4%

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

Alternative 2: 75.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 3.5 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(0.5 \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9e-8)
   (sqrt (- re))
   (if (<= re 3.5e-103)
     (sqrt (* 0.5 (- im re)))
     (* im (* 0.5 (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9e-8) {
		tmp = sqrt(-re);
	} else if (re <= 3.5e-103) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 * pow(re, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-9d-8)) then
        tmp = sqrt(-re)
    else if (re <= 3.5d-103) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * (0.5d0 * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9e-8) {
		tmp = Math.sqrt(-re);
	} else if (re <= 3.5e-103) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9e-8:
		tmp = math.sqrt(-re)
	elif re <= 3.5e-103:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9e-8)
		tmp = sqrt(Float64(-re));
	elseif (re <= 3.5e-103)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64(0.5 * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9e-8)
		tmp = sqrt(-re);
	elseif (re <= 3.5e-103)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9e-8], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 3.5e-103], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -8.99999999999999986e-8

   1. Initial program 43.8%

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

      1. pow1 43.8%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.8%

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

      6. unpow2 43.8%

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

      7. unpow2 43.8%

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

      8. +-commutative 43.8%

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

      9. unpow2 43.8%

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

      10. unpow2 43.8%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 81.9%

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

      1. neg-mul-1 81.9%

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

   9. Simplified 81.9%

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

   if -8.99999999999999986e-8 < re < 3.50000000000000016e-103

   1. Initial program 54.2%

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

      1. pow1 54.2%

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

   4. Applied egg-rr 91.4%

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

      1. unpow1 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-*r* 91.4%

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

      4. metadata-eval 91.4%

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

      5. hypot-undefine 54.2%

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

      6. unpow2 54.2%

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

      7. unpow2 54.2%

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

      8. +-commutative 54.2%

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

      9. unpow2 54.2%

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

      10. unpow2 54.2%

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

      11. hypot-undefine 91.4%

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

   6. Simplified 91.4%

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

   7. Taylor expanded in re around 0 82.1%

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

      1. neg-mul-1 82.1%

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

      2. unsub-neg 82.1%

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

   9. Simplified 82.1%

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

   if 3.50000000000000016e-103 < re

   1. Initial program 23.0%

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

      1. pow1 23.0%

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

   4. Applied egg-rr 43.8%

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

      1. unpow1 43.8%

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

      2. *-commutative 43.8%

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

      3. associate-*r* 43.8%

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

      4. metadata-eval 43.8%

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

      5. hypot-undefine 23.0%

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

      6. unpow2 23.0%

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

      7. unpow2 23.0%

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

      8. +-commutative 23.0%

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

      9. unpow2 23.0%

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

      10. unpow2 23.0%

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

      11. hypot-undefine 43.8%

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

   6. Simplified 43.8%

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

   7. Taylor expanded in im around 0 69.9%

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

      1. associate-*l* 69.9%

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

      2. unpow2 69.9%

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

      3. rem-square-sqrt 70.8%

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

      4. unpow-1 70.8%

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

      5. metadata-eval 70.8%

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

      6. pow-sqr 70.9%

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

      7. rem-sqrt-square 70.9%

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

      8. rem-square-sqrt 70.6%

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

      9. fabs-sqr 70.6%

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

      10. rem-square-sqrt 70.9%

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

   9. Simplified 70.9%

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

Alternative 3: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.2e-8)
   (sqrt (- re))
   (if (<= re 1.8e-103) (sqrt (* 0.5 (- im re))) (/ (* im 0.5) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.2e-8) {
		tmp = sqrt(-re);
	} else if (re <= 1.8e-103) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.2d-8)) then
        tmp = sqrt(-re)
    else if (re <= 1.8d-103) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (im * 0.5d0) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.2e-8) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.8e-103) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (im * 0.5) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.2e-8:
		tmp = math.sqrt(-re)
	elif re <= 1.8e-103:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (im * 0.5) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.2e-8)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.8e-103)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(im * 0.5) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.2e-8)
		tmp = sqrt(-re);
	elseif (re <= 1.8e-103)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (im * 0.5) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.2e-8], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.8e-103], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(im * 0.5), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.19999999999999989e-8

   1. Initial program 43.8%

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

      1. pow1 43.8%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.8%

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

      6. unpow2 43.8%

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

      7. unpow2 43.8%

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

      8. +-commutative 43.8%

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

      9. unpow2 43.8%

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

      10. unpow2 43.8%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 81.9%

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

      1. neg-mul-1 81.9%

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

   9. Simplified 81.9%

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

   if -4.19999999999999989e-8 < re < 1.7999999999999999e-103

   1. Initial program 54.2%

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

      1. pow1 54.2%

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

   4. Applied egg-rr 91.4%

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

      1. unpow1 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-*r* 91.4%

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

      4. metadata-eval 91.4%

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

      5. hypot-undefine 54.2%

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

      6. unpow2 54.2%

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

      7. unpow2 54.2%

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

      8. +-commutative 54.2%

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

      9. unpow2 54.2%

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

      10. unpow2 54.2%

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

      11. hypot-undefine 91.4%

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

   6. Simplified 91.4%

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

   7. Taylor expanded in re around 0 82.1%

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

      1. neg-mul-1 82.1%

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

      2. unsub-neg 82.1%

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

   9. Simplified 82.1%

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

   if 1.7999999999999999e-103 < re

   1. Initial program 23.0%

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

   3. Taylor expanded in re around inf 38.6%

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

      1. sqrt-div 44.0%

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

      2. sqrt-pow1 70.8%

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

      3. metadata-eval 70.8%

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

      4. pow1 70.8%

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

      5. associate-*r/ 70.8%

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

   5. Applied egg-rr 70.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.8 \cdot 10^{-103}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{im \cdot 0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5.1 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.4e-8)
   (sqrt (- re))
   (if (<= re 5.1e-104) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-8) {
		tmp = sqrt(-re);
	} else if (re <= 5.1e-104) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-3.4d-8)) then
        tmp = sqrt(-re)
    else if (re <= 5.1d-104) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.4e-8) {
		tmp = Math.sqrt(-re);
	} else if (re <= 5.1e-104) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.4e-8:
		tmp = math.sqrt(-re)
	elif re <= 5.1e-104:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.4e-8)
		tmp = sqrt(Float64(-re));
	elseif (re <= 5.1e-104)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * Float64(0.5 / sqrt(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.4e-8)
		tmp = sqrt(-re);
	elseif (re <= 5.1e-104)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.4e-8], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 5.1e-104], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[(0.5 / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.4e-8

   1. Initial program 43.8%

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

      1. pow1 43.8%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.8%

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

      6. unpow2 43.8%

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

      7. unpow2 43.8%

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

      8. +-commutative 43.8%

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

      9. unpow2 43.8%

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

      10. unpow2 43.8%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 81.9%

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

      1. neg-mul-1 81.9%

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

   9. Simplified 81.9%

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

   if -3.4e-8 < re < 5.09999999999999992e-104

   1. Initial program 54.2%

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

      1. pow1 54.2%

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

   4. Applied egg-rr 91.4%

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

      1. unpow1 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-*r* 91.4%

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

      4. metadata-eval 91.4%

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

      5. hypot-undefine 54.2%

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

      6. unpow2 54.2%

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

      7. unpow2 54.2%

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

      8. +-commutative 54.2%

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

      9. unpow2 54.2%

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

      10. unpow2 54.2%

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

      11. hypot-undefine 91.4%

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

   6. Simplified 91.4%

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

   7. Taylor expanded in re around 0 82.1%

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

      1. neg-mul-1 82.1%

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

      2. unsub-neg 82.1%

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

   9. Simplified 82.1%

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

   if 5.09999999999999992e-104 < re

   1. Initial program 23.0%

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

   3. Taylor expanded in re around inf 38.6%

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

      1. sqrt-div 44.0%

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

      2. sqrt-pow1 70.8%

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

      3. metadata-eval 70.8%

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

      4. pow1 70.8%

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

      5. associate-*r/ 70.8%

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

   5. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-/l* 70.8%

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

   7. Simplified 70.8%

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

Alternative 5: 64.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.2e-8) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.2e-8) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt((im * 0.5));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.2e-8) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.2e-8:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.2e-8)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(im * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.2e-8)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.2e-8], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -2.1999999999999998e-8

   1. Initial program 43.8%

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

      1. pow1 43.8%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 43.8%

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

      6. unpow2 43.8%

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

      7. unpow2 43.8%

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

      8. +-commutative 43.8%

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

      9. unpow2 43.8%

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

      10. unpow2 43.8%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 81.9%

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

      1. neg-mul-1 81.9%

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

   9. Simplified 81.9%

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

   if -2.1999999999999998e-8 < re

   1. Initial program 41.8%

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

      1. pow1 41.8%

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

   4. Applied egg-rr 72.4%

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

      1. unpow1 72.4%

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

      2. *-commutative 72.4%

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

      3. associate-*r* 72.4%

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

      4. metadata-eval 72.4%

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

      5. hypot-undefine 41.8%

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

      6. unpow2 41.8%

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

      7. unpow2 41.8%

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

      8. +-commutative 41.8%

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

      9. unpow2 41.8%

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

      10. unpow2 41.8%

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

      11. hypot-undefine 72.4%

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

   6. Simplified 72.4%

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

   7. Taylor expanded in im around inf 61.1%

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

      1. *-commutative 61.1%

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

   9. Simplified 61.1%

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

Alternative 6: 30.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -1e-309) (sqrt (- re)) 0.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1e-309) {
		tmp = sqrt(-re);
	} else {
		tmp = 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 <= (-1d-309)) then
        tmp = sqrt(-re)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1e-309) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1e-309:
		tmp = math.sqrt(-re)
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1e-309)
		tmp = sqrt(Float64(-re));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1e-309)
		tmp = sqrt(-re);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1e-309], N[Sqrt[(-re)], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if re < -1.000000000000002e-309

   1. Initial program 54.2%

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

      1. pow1 54.2%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

      5. hypot-undefine 54.2%

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

      6. unpow2 54.2%

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

      7. unpow2 54.2%

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

      8. +-commutative 54.2%

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

      9. unpow2 54.2%

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

      10. unpow2 54.2%

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

      11. hypot-undefine 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 53.3%

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

      1. neg-mul-1 53.3%

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

   9. Simplified 53.3%

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

   if -1.000000000000002e-309 < re

   1. Initial program 31.2%

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

   3. Taylor expanded in re around inf 8.6%

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

   4. Taylor expanded in re around 0 8.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 6.0% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

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

Fortran

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

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

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

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in re around inf 5.9%

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

4. Taylor expanded in re around 0 5.9%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(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)))))