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

Percentage Accurate: 42.2% → 88.5%

Time: 10.1s

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: 42.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 1.12 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 1.12e-54)
   (sqrt (* 0.5 (- (hypot re im) re)))
   (/ (* 0.5 im) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 1.12e-54) {
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / sqrt(re);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 1.12e-54) {
		tmp = Math.sqrt((0.5 * (Math.hypot(re, im) - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 1.12e-54:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 1.12e-54)
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, im) - re)));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 1.12e-54)
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 1.12e-54], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 1.11999999999999994e-54

   1. Initial program 57.3%

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

      1. pow1 57.3%

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

   4. Applied egg-rr 96.8%

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

      1. unpow1 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*r* 96.8%

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

      4. metadata-eval 96.8%

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

   6. Simplified 96.8%

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

   if 1.11999999999999994e-54 < re

   1. Initial program 18.3%

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

   3. Taylor expanded in re around inf 48.1%

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

      1. *-commutative 48.1%

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

      2. sqrt-div 58.8%

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

      3. sqrt-pow1 75.9%

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

      4. metadata-eval 75.9%

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

      5. pow1 75.9%

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

      6. associate-*l/ 75.9%

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

   5. Applied egg-rr 75.9%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 1.12 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -7.2e+61)
   (sqrt (- re))
   (if (<= re 5e-50) (sqrt (* 0.5 (- im re))) (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -7.2e+61) {
		tmp = sqrt(-re);
	} else if (re <= 5e-50) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = (0.5 * im) / 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 <= (-7.2d+61)) then
        tmp = sqrt(-re)
    else if (re <= 5d-50) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = (0.5d0 * im) / sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -7.2e+61) {
		tmp = Math.sqrt(-re);
	} else if (re <= 5e-50) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = (0.5 * im) / Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -7.2e+61:
		tmp = math.sqrt(-re)
	elif re <= 5e-50:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = (0.5 * im) / math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -7.2e+61)
		tmp = sqrt(Float64(-re));
	elseif (re <= 5e-50)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(Float64(0.5 * im) / sqrt(re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -7.2e+61)
		tmp = sqrt(-re);
	elseif (re <= 5e-50)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = (0.5 * im) / sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -7.2e+61], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 5e-50], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 * im), $MachinePrecision] / N[Sqrt[re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -7.20000000000000021e61

   1. Initial program 36.8%

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

      1. pow1 36.8%

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

   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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 82.5%

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

      1. neg-mul-1 82.5%

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

   9. Simplified 82.5%

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

   if -7.20000000000000021e61 < re < 4.99999999999999968e-50

   1. Initial program 64.6%

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

      1. pow1 64.6%

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

   4. Applied egg-rr 95.6%

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

      1. unpow1 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r* 95.6%

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

      4. metadata-eval 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in re around 0 80.3%

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

      1. neg-mul-1 80.3%

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

      2. sub-neg 80.3%

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

   9. Simplified 80.3%

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

   if 4.99999999999999968e-50 < re

   1. Initial program 18.3%

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

   3. Taylor expanded in re around inf 48.1%

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

      1. *-commutative 48.1%

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

      2. sqrt-div 58.8%

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

      3. sqrt-pow1 75.9%

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

      4. metadata-eval 75.9%

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

      5. pow1 75.9%

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

      6. associate-*l/ 75.9%

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

   5. Applied egg-rr 75.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -7.2 \cdot 10^{+61}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-50}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.26 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3.1e+64)
   (sqrt (- re))
   (if (<= re 1.26e-51) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+64) {
		tmp = sqrt(-re);
	} else if (re <= 1.26e-51) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * sqrt((0.25 / 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 <= (-3.1d+64)) then
        tmp = sqrt(-re)
    else if (re <= 1.26d-51) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * sqrt((0.25d0 / re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -3.1e+64) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.26e-51) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * Math.sqrt((0.25 / re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3.1e+64:
		tmp = math.sqrt(-re)
	elif re <= 1.26e-51:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * math.sqrt((0.25 / re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3.1e+64)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.26e-51)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -3.1e+64)
		tmp = sqrt(-re);
	elseif (re <= 1.26e-51)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * sqrt((0.25 / re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3.1e+64], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.26e-51], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.0999999999999999e64

   1. Initial program 36.8%

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

      1. pow1 36.8%

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

   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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 82.5%

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

      1. neg-mul-1 82.5%

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

   9. Simplified 82.5%

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

   if -3.0999999999999999e64 < re < 1.2600000000000001e-51

   1. Initial program 64.6%

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

      1. pow1 64.6%

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

   4. Applied egg-rr 95.6%

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

      1. unpow1 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r* 95.6%

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

      4. metadata-eval 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in re around 0 80.3%

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

      1. neg-mul-1 80.3%

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

      2. sub-neg 80.3%

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

   9. Simplified 80.3%

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

   if 1.2600000000000001e-51 < re

   1. Initial program 18.3%

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

   3. Taylor expanded in re around inf 48.1%

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

      1. *-commutative 48.1%

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

      2. sqrt-div 58.8%

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

      3. sqrt-pow1 75.9%

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

      4. metadata-eval 75.9%

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

      5. pow1 75.9%

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

      6. associate-*l/ 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. associate-/l* 75.8%

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

      2. *-commutative 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. *-un-lft-identity 75.8%

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

      2. *-commutative 75.8%

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

      3. add-sqr-sqrt 75.5%

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

      4. sqrt-unprod 75.8%

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

      5. frac-times 75.7%

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

      6. metadata-eval 75.7%

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

      7. add-sqr-sqrt 75.9%

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

   9. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{0.25}{re}} \cdot 1\right)} \cdot im \]
   10. Step-by-step derivation

       1. *-rgt-identity 75.9%

          \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re}}} \cdot im \]

   11. Simplified 75.9%

       \[\leadsto \color{blue}{\sqrt{\frac{0.25}{re}}} \cdot im \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.1 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.26 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.8 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.42 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.8e+64)
   (sqrt (- re))
   (if (<= re 1.42e-52) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+64) {
		tmp = sqrt(-re);
	} else if (re <= 1.42e-52) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / sqrt(re));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-1.8d+64)) then
        tmp = sqrt(-re)
    else if (re <= 1.42d-52) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = im * (0.5d0 / sqrt(re))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.8e+64) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.42e-52) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = im * (0.5 / Math.sqrt(re));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.8e+64:
		tmp = math.sqrt(-re)
	elif re <= 1.42e-52:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = im * (0.5 / math.sqrt(re))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.8e+64)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.42e-52)
		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 <= -1.8e+64)
		tmp = sqrt(-re);
	elseif (re <= 1.42e-52)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = im * (0.5 / sqrt(re));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.8e+64], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.42e-52], 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 < -1.80000000000000007e64

   1. Initial program 36.8%

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

      1. pow1 36.8%

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

   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%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 82.5%

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

      1. neg-mul-1 82.5%

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

   9. Simplified 82.5%

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

   if -1.80000000000000007e64 < re < 1.4200000000000001e-52

   1. Initial program 64.6%

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

      1. pow1 64.6%

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

   4. Applied egg-rr 95.6%

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

      1. unpow1 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r* 95.6%

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

      4. metadata-eval 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in re around 0 80.3%

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

      1. neg-mul-1 80.3%

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

      2. sub-neg 80.3%

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

   9. Simplified 80.3%

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

   if 1.4200000000000001e-52 < re

   1. Initial program 18.3%

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

   3. Taylor expanded in re around inf 48.1%

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

      1. sqrt-div 58.8%

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

      2. sqrt-pow1 75.9%

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

      3. metadata-eval 75.9%

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

      4. pow1 75.9%

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

      5. associate-*r/ 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-/l* 75.8%

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

   7. Simplified 75.8%

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

Alternative 5: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -8.5e-118) (sqrt (- re)) (sqrt (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -8.5e-118) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt((0.5 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-8.5d-118)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt((0.5d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -8.5e-118) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -8.5e-118:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -8.5e-118)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(Float64(0.5 * im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -8.5e-118)
		tmp = sqrt(-re);
	else
		tmp = sqrt((0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -8.5e-118], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(0.5 * im), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -8.50000000000000087e-118

   1. Initial program 54.5%

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

      1. pow1 54.5%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 70.7%

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

      1. neg-mul-1 70.7%

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

   9. Simplified 70.7%

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

   if -8.50000000000000087e-118 < re

   1. Initial program 41.6%

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

      1. pow1 41.6%

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

   4. Applied egg-rr 71.1%

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

      1. unpow1 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*r* 71.1%

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

      4. metadata-eval 71.1%

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

   6. Simplified 71.1%

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

   7. Taylor expanded in re around 0 62.0%

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

      1. *-commutative 62.0%

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

   9. Simplified 62.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -8.5 \cdot 10^{-118}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2e-310) (sqrt (- re)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2e-310) {
		tmp = sqrt(-re);
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-2d-310)) then
        tmp = sqrt(-re)
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2e-310)
		tmp = sqrt(-re);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2e-310], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.999999999999994e-310

   1. Initial program 57.4%

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

      1. pow1 57.4%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in re around -inf 49.7%

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

      1. neg-mul-1 49.7%

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

   9. Simplified 49.7%

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

   if -1.999999999999994e-310 < re

   1. Initial program 32.6%

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

      1. pow1 32.6%

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

   4. Applied egg-rr 58.3%

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

      1. unpow1 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-*r* 58.3%

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

      4. metadata-eval 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in re around -inf 0.0%

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

      1. neg-mul-1 0.0%

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

   9. Simplified 0.0%

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

       1. neg-sub0 0.0%

          \[\leadsto \sqrt{\color{blue}{0 - re}} \]

       2. sub-neg 0.0%

          \[\leadsto \sqrt{\color{blue}{0 + \left(-re\right)}} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto \sqrt{0 + \color{blue}{\sqrt{-re} \cdot \sqrt{-re}}} \]

       4. sqrt-unprod 4.4%

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

       5. sqr-neg 4.4%

          \[\leadsto \sqrt{0 + \sqrt{\color{blue}{re \cdot re}}} \]

       6. sqrt-unprod 5.5%

          \[\leadsto \sqrt{0 + \color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

       7. add-sqr-sqrt 5.5%

          \[\leadsto \sqrt{0 + \color{blue}{re}} \]

   11. Applied egg-rr 5.5%

       \[\leadsto \sqrt{\color{blue}{0 + re}} \]
   12. Step-by-step derivation

       1. +-lft-identity 5.5%

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

   13. Simplified 5.5%

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

Alternative 7: 2.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 45.6%

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

   1. pow1 45.6%

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

4. Applied egg-rr 80.1%

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

   1. unpow1 80.1%

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

   2. *-commutative 80.1%

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

   3. associate-*r* 80.1%

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

   4. metadata-eval 80.1%

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

6. Simplified 80.1%

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

7. Taylor expanded in re around -inf 26.0%

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

   1. neg-mul-1 26.0%

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

9. Simplified 26.0%

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

    1. neg-sub0 26.0%

       \[\leadsto \sqrt{\color{blue}{0 - re}} \]

    2. sub-neg 26.0%

       \[\leadsto \sqrt{\color{blue}{0 + \left(-re\right)}} \]

    3. add-sqr-sqrt 26.0%

       \[\leadsto \sqrt{0 + \color{blue}{\sqrt{-re} \cdot \sqrt{-re}}} \]

    4. sqrt-unprod 16.7%

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

    5. sqr-neg 16.7%

       \[\leadsto \sqrt{0 + \sqrt{\color{blue}{re \cdot re}}} \]

    6. sqrt-unprod 2.6%

       \[\leadsto \sqrt{0 + \color{blue}{\sqrt{re} \cdot \sqrt{re}}} \]

    7. add-sqr-sqrt 2.6%

       \[\leadsto \sqrt{0 + \color{blue}{re}} \]

11. Applied egg-rr 2.6%

    \[\leadsto \sqrt{\color{blue}{0 + re}} \]
12. Step-by-step derivation

    1. +-lft-identity 2.6%

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

13. Simplified 2.6%

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

Reproduce

herbie shell --seed 2024123 
(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)))))