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

Percentage Accurate: 41.7% → 88.4%

Time: 30.3s

Alternatives: 6

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: 41.7% 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: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 7.5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 7.5e-23)
   (sqrt (* 0.5 (- (hypot re im) re)))
   (/ (* 0.5 im) (sqrt re))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 7.5e-23) {
		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 <= 7.5e-23) {
		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 <= 7.5e-23:
		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 <= 7.5e-23)
		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 <= 7.5e-23)
		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, 7.5e-23], 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 < 7.4999999999999998e-23

   1. Initial program 52.6%

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

      1. pow1 52.6%

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

   4. Applied egg-rr 92.2%

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

      1. unpow1 92.2%

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

      2. *-commutative 92.2%

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

      3. associate-*r* 92.7%

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

      4. metadata-eval 92.7%

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

   6. Simplified 92.7%

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

   if 7.4999999999999998e-23 < re

   1. Initial program 8.6%

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

      1. pow1 8.6%

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

   4. Applied egg-rr 33.7%

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

      1. unpow1 33.7%

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

      2. *-commutative 33.7%

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

      3. associate-*r* 33.7%

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

      4. metadata-eval 33.7%

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

   6. Simplified 33.7%

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

   7. Taylor expanded in re around inf 78.6%

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

      1. unpow2 78.6%

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

      2. rem-square-sqrt 79.5%

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

   9. Simplified 79.5%

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

       1. sqrt-div 79.4%

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

       2. metadata-eval 79.4%

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

       3. un-div-inv 79.6%

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

   11. Applied egg-rr 79.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 7.5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.8e+35)
   (sqrt (- re))
   (if (<= re 1.4e-22) (sqrt (* 0.5 (- im re))) (/ (* 0.5 im) (sqrt re)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.8e+35) {
		tmp = sqrt(-re);
	} else if (re <= 1.4e-22) {
		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 <= (-9.8d+35)) then
        tmp = sqrt(-re)
    else if (re <= 1.4d-22) 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 <= -9.8e+35) {
		tmp = Math.sqrt(-re);
	} else if (re <= 1.4e-22) {
		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 <= -9.8e+35:
		tmp = math.sqrt(-re)
	elif re <= 1.4e-22:
		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 <= -9.8e+35)
		tmp = sqrt(Float64(-re));
	elseif (re <= 1.4e-22)
		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 <= -9.8e+35)
		tmp = sqrt(-re);
	elseif (re <= 1.4e-22)
		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, -9.8e+35], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 1.4e-22], 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 < -9.8000000000000005e35

   1. Initial program 42.6%

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

      1. pow1 42.6%

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

   4. Applied egg-rr 98.4%

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

      1. unpow1 98.4%

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

      2. *-commutative 98.4%

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

      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 87.8%

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

      1. neg-mul-1 87.8%

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

   9. Simplified 87.8%

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

   if -9.8000000000000005e35 < re < 1.39999999999999997e-22

   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 89.2%

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

      1. unpow1 89.2%

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

      2. *-commutative 89.2%

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

      3. associate-*r* 89.2%

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

      4. metadata-eval 89.2%

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

   6. Simplified 89.2%

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

   7. Taylor expanded in re around 0 74.9%

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

      1. neg-mul-1 74.9%

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

      2. unsub-neg 74.9%

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

   9. Simplified 74.9%

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

   if 1.39999999999999997e-22 < re

   1. Initial program 8.6%

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

      1. pow1 8.6%

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

   4. Applied egg-rr 33.7%

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

      1. unpow1 33.7%

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

      2. *-commutative 33.7%

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

      3. associate-*r* 33.7%

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

      4. metadata-eval 33.7%

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

   6. Simplified 33.7%

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

   7. Taylor expanded in re around inf 78.6%

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

      1. unpow2 78.6%

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

      2. rem-square-sqrt 79.5%

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

   9. Simplified 79.5%

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

       1. sqrt-div 79.4%

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

       2. metadata-eval 79.4%

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

       3. un-div-inv 79.6%

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

   11. Applied egg-rr 79.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 1.4 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot im}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.8e+35)
   (sqrt (- re))
   (if (<= re 5e-23) (sqrt (* 0.5 (- im re))) (* im (/ 0.5 (sqrt re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.8e+35) {
		tmp = sqrt(-re);
	} else if (re <= 5e-23) {
		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 <= (-9.8d+35)) then
        tmp = sqrt(-re)
    else if (re <= 5d-23) 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 <= -9.8e+35) {
		tmp = Math.sqrt(-re);
	} else if (re <= 5e-23) {
		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 <= -9.8e+35:
		tmp = math.sqrt(-re)
	elif re <= 5e-23:
		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 <= -9.8e+35)
		tmp = sqrt(Float64(-re));
	elseif (re <= 5e-23)
		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 <= -9.8e+35)
		tmp = sqrt(-re);
	elseif (re <= 5e-23)
		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, -9.8e+35], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 5e-23], 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 < -9.8000000000000005e35

   1. Initial program 42.6%

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

      1. pow1 42.6%

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

   4. Applied egg-rr 98.4%

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

      1. unpow1 98.4%

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

      2. *-commutative 98.4%

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

      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 87.8%

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

      1. neg-mul-1 87.8%

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

   9. Simplified 87.8%

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

   if -9.8000000000000005e35 < re < 5.0000000000000002e-23

   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 89.2%

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

      1. unpow1 89.2%

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

      2. *-commutative 89.2%

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

      3. associate-*r* 89.2%

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

      4. metadata-eval 89.2%

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

   6. Simplified 89.2%

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

   7. Taylor expanded in re around 0 74.9%

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

      1. neg-mul-1 74.9%

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

      2. unsub-neg 74.9%

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

   9. Simplified 74.9%

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

   if 5.0000000000000002e-23 < re

   1. Initial program 8.6%

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

      1. pow1 8.6%

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

   4. Applied egg-rr 33.7%

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

      1. unpow1 33.7%

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

      2. *-commutative 33.7%

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

      3. associate-*r* 33.7%

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

      4. metadata-eval 33.7%

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

   6. Simplified 33.7%

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

   7. Taylor expanded in re around inf 78.6%

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

      1. unpow2 78.6%

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

      2. rem-square-sqrt 79.5%

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

   9. Simplified 79.5%

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

       1. add-cube-cbrt 78.5%

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

       2. pow3 78.5%

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

       3. associate-*l* 78.5%

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

       4. sqrt-div 78.5%

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

       5. metadata-eval 78.5%

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

       6. un-div-inv 78.5%

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

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{im \cdot \frac{0.5}{\sqrt{re}}}\right)}^{3}} \]
   12. Step-by-step derivation

       1. rem-cube-cbrt 79.4%

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

       2. *-commutative 79.4%

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

   13. Applied egg-rr 79.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.8 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \frac{0.5}{\sqrt{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -3e+22) (sqrt (- re)) (sqrt (* 0.5 im))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -3e+22) {
		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 <= (-3d+22)) 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 <= -3e+22) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((0.5 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -3e+22:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((0.5 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -3e+22)
		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 <= -3e+22)
		tmp = sqrt(-re);
	else
		tmp = sqrt((0.5 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -3e+22], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(0.5 * im), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -3e22

   1. Initial program 44.7%

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

      1. pow1 44.7%

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

   4. Applied egg-rr 98.5%

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

      1. unpow1 98.5%

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

      2. *-commutative 98.5%

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

      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 85.7%

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

      1. neg-mul-1 85.7%

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

   9. Simplified 85.7%

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

   if -3e22 < re

   1. Initial program 39.9%

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

   3. Taylor expanded in re around 0 57.0%

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

      1. *-commutative 57.0%

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

   5. Simplified 57.0%

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

      1. pow1 57.0%

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

      2. add-sqr-sqrt 56.6%

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

      3. sqrt-unprod 57.0%

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

      4. *-commutative 57.0%

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

      5. *-commutative 57.0%

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

      6. swap-sqr 57.0%

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

      7. add-sqr-sqrt 57.0%

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

      8. metadata-eval 57.0%

         \[\leadsto {\left(\sqrt{\left(im \cdot 2\right) \cdot \color{blue}{0.25}}\right)}^{1} \]

   7. Applied egg-rr 57.0%

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

      1. unpow1 57.0%

         \[\leadsto \color{blue}{\sqrt{\left(im \cdot 2\right) \cdot 0.25}} \]

      2. associate-*l* 57.0%

         \[\leadsto \sqrt{\color{blue}{im \cdot \left(2 \cdot 0.25\right)}} \]

      3. metadata-eval 57.0%

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

   9. Simplified 57.0%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3 \cdot 10^{+22}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 28.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e-310) (sqrt (- re)) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-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 <= (-5d-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 <= -5e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-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 <= -5e-310)
		tmp = sqrt(-re);
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 56.2%

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

      1. pow1 56.2%

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

   4. Applied egg-rr 99.3%

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

      1. unpow1 99.3%

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

      2. *-commutative 99.3%

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

      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 56.8%

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

      1. neg-mul-1 56.8%

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

   9. Simplified 56.8%

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

   if -4.999999999999985e-310 < re

   1. Initial program 24.4%

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

      1. pow1 24.4%

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

   4. Applied egg-rr 52.2%

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

      1. unpow1 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-*r* 52.2%

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

      4. metadata-eval 52.2%

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

   6. Simplified 52.2%

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

   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. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 4.7%

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

       3. sqr-neg 4.7%

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

       4. sqrt-unprod 5.7%

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

       5. add-sqr-sqrt 5.7%

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

       6. *-un-lft-identity 5.7%

          \[\leadsto \color{blue}{1 \cdot \sqrt{re}} \]

   11. Applied egg-rr 5.7%

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

       1. *-lft-identity 5.7%

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

   13. Simplified 5.7%

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

Alternative 6: 2.9% 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 41.1%

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

   1. pow1 41.1%

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

4. Applied egg-rr 76.8%

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

   1. unpow1 76.8%

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

   2. *-commutative 76.8%

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

   3. associate-*r* 77.2%

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

   4. metadata-eval 77.2%

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

6. Simplified 77.2%

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

7. Taylor expanded in re around -inf 29.7%

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

   1. neg-mul-1 29.7%

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

9. Simplified 29.7%

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

    1. add-sqr-sqrt 29.7%

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

    2. sqrt-unprod 18.9%

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

    3. sqr-neg 18.9%

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

    4. sqrt-unprod 2.7%

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

    5. add-sqr-sqrt 2.7%

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

    6. *-un-lft-identity 2.7%

       \[\leadsto \color{blue}{1 \cdot \sqrt{re}} \]

11. Applied egg-rr 2.7%

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

    1. *-lft-identity 2.7%

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

13. Simplified 2.7%

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

Reproduce

herbie shell --seed 2024143 
(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)))))