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

Percentage Accurate: 41.9% → 90.4%

Time: 11.1s

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: 41.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* im (sqrt (/ 0.25 re)))
   (sqrt (* 0.5 (- (hypot re im) re)))))

C

double code(double re, double im) {
	double tmp;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = im * sqrt((0.25 / re));
	} else {
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.sqrt((2.0 * (Math.sqrt(((re * re) + (im * im))) - re))) <= 0.0) {
		tmp = im * Math.sqrt((0.25 / re));
	} else {
		tmp = Math.sqrt((0.5 * (Math.hypot(re, im) - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.sqrt((2.0 * (math.sqrt(((re * re) + (im * im))) - re))) <= 0.0:
		tmp = im * math.sqrt((0.25 / re))
	else:
		tmp = math.sqrt((0.5 * (math.hypot(re, im) - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (sqrt(Float64(2.0 * Float64(sqrt(Float64(Float64(re * re) + Float64(im * im))) - re))) <= 0.0)
		tmp = Float64(im * sqrt(Float64(0.25 / re)));
	else
		tmp = sqrt(Float64(0.5 * Float64(hypot(re, im) - re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (sqrt((2.0 * (sqrt(((re * re) + (im * im))) - re))) <= 0.0)
		tmp = im * sqrt((0.25 / re));
	else
		tmp = sqrt((0.5 * (hypot(re, im) - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[N[Sqrt[N[(2.0 * N[(N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(im * N[Sqrt[N[(0.25 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.7%

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

   3. Taylor expanded in re around inf 48.4%

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

      1. div-inv 48.6%

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

      2. sqrt-prod 57.8%

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

      3. sqrt-pow1 99.7%

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

      4. metadata-eval 99.7%

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

      5. pow1 99.7%

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

      6. *-rgt-identity 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. sqrt-unprod 98.1%

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

      10. associate-*l* 98.1%

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

      11. sqrt-div 98.1%

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

      12. metadata-eval 98.1%

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

      13. un-div-inv 98.2%

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

      14. sqrt-unprod 99.6%

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

      15. metadata-eval 99.6%

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

      16. metadata-eval 99.6%

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

      17. *-rgt-identity 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.4%

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

   7. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-/r/ 98.0%

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

      3. div-inv 98.0%

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

      4. associate-/r* 99.4%

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

      5. add-sqr-sqrt 99.1%

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

      6. sqrt-unprod 99.4%

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

      7. frac-times 99.5%

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

      8. metadata-eval 99.5%

         \[\leadsto \frac{\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}}}{\frac{1}{im}} \]

      9. add-sqr-sqrt 99.5%

         \[\leadsto \frac{\sqrt{\frac{0.25}{\color{blue}{re}}}}{\frac{1}{im}} \]

   9. Applied egg-rr 99.5%

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

       1. associate-/r/ 99.7%

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{0.25}{re}}}{1} \cdot im} \]

       2. /-rgt-identity 99.7%

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

       3. *-commutative 99.7%

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

   11. Simplified 99.7%

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

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

   1. Initial program 44.2%

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

      1. pow1 44.2%

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

   4. Applied egg-rr 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* 92.2%

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

      4. metadata-eval 92.2%

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

   6. Simplified 92.2%

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

Alternative 2: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{im \cdot \left(0.5 + -0.5 \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -2.5e+16)
   (sqrt (- re))
   (if (<= re 6.8e-67)
     (sqrt (* im (+ 0.5 (* -0.5 (/ re im)))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -2.5e+16) {
		tmp = sqrt(-re);
	} else if (re <= 6.8e-67) {
		tmp = sqrt((im * (0.5 + (-0.5 * (re / im)))));
	} else {
		tmp = 0.5 * (im * 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 <= (-2.5d+16)) then
        tmp = sqrt(-re)
    else if (re <= 6.8d-67) then
        tmp = sqrt((im * (0.5d0 + ((-0.5d0) * (re / im)))))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -2.5e+16) {
		tmp = Math.sqrt(-re);
	} else if (re <= 6.8e-67) {
		tmp = Math.sqrt((im * (0.5 + (-0.5 * (re / im)))));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -2.5e+16:
		tmp = math.sqrt(-re)
	elif re <= 6.8e-67:
		tmp = math.sqrt((im * (0.5 + (-0.5 * (re / im)))))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -2.5e+16)
		tmp = sqrt(Float64(-re));
	elseif (re <= 6.8e-67)
		tmp = sqrt(Float64(im * Float64(0.5 + Float64(-0.5 * Float64(re / im)))));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -2.5e+16)
		tmp = sqrt(-re);
	elseif (re <= 6.8e-67)
		tmp = sqrt((im * (0.5 + (-0.5 * (re / im)))));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -2.5e+16], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 6.8e-67], N[Sqrt[N[(im * N[(0.5 + N[(-0.5 * N[(re / im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.5e16

   1. Initial program 30.5%

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

      1. pow1 30.5%

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

   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 80.8%

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

      1. neg-mul-1 80.8%

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

   9. Simplified 80.8%

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

   if -2.5e16 < re < 6.8000000000000002e-67

   1. Initial program 57.2%

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

      1. pow1 57.2%

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

   4. Applied egg-rr 97.6%

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

      1. unpow1 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r* 97.6%

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

      4. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in im around inf 84.9%

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

   if 6.8000000000000002e-67 < re

   1. Initial program 20.2%

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

   3. Taylor expanded in re around inf 37.1%

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

      1. div-inv 37.2%

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

      2. sqrt-prod 47.2%

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

      3. sqrt-pow1 67.5%

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

      4. metadata-eval 67.5%

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

      5. pow1 67.5%

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

      6. *-commutative 67.5%

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

      7. inv-pow 67.5%

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

      8. sqrt-pow1 67.5%

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

      9. metadata-eval 67.5%

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

   5. Applied egg-rr 67.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.5 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{im \cdot \left(0.5 + -0.5 \cdot \frac{re}{im}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.3e+17)
   (sqrt (- re))
   (if (<= re 7.5e-67) (sqrt (* 0.5 (- im re))) (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.3e+17) {
		tmp = sqrt(-re);
	} else if (re <= 7.5e-67) {
		tmp = sqrt((0.5 * (im - re)));
	} else {
		tmp = 0.5 * (im * 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 <= (-1.3d+17)) then
        tmp = sqrt(-re)
    else if (re <= 7.5d-67) then
        tmp = sqrt((0.5d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * (re ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.3e+17) {
		tmp = Math.sqrt(-re);
	} else if (re <= 7.5e-67) {
		tmp = Math.sqrt((0.5 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.3e+17:
		tmp = math.sqrt(-re)
	elif re <= 7.5e-67:
		tmp = math.sqrt((0.5 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.3e+17)
		tmp = sqrt(Float64(-re));
	elseif (re <= 7.5e-67)
		tmp = sqrt(Float64(0.5 * Float64(im - re)));
	else
		tmp = Float64(0.5 * Float64(im * (re ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.3e+17)
		tmp = sqrt(-re);
	elseif (re <= 7.5e-67)
		tmp = sqrt((0.5 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.3e+17], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 7.5e-67], N[Sqrt[N[(0.5 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.3e17

   1. Initial program 30.5%

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

      1. pow1 30.5%

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

   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 80.8%

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

      1. neg-mul-1 80.8%

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

   9. Simplified 80.8%

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

   if -1.3e17 < re < 7.5000000000000005e-67

   1. Initial program 57.2%

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

      1. pow1 57.2%

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

   4. Applied egg-rr 97.6%

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

      1. unpow1 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r* 97.6%

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

      4. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in re around 0 84.9%

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

      1. neg-mul-1 84.9%

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

      2. unsub-neg 84.9%

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

   9. Simplified 84.9%

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

   if 7.5000000000000005e-67 < re

   1. Initial program 20.2%

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

   3. Taylor expanded in re around inf 37.1%

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

      1. div-inv 37.2%

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

      2. sqrt-prod 47.2%

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

      3. sqrt-pow1 67.5%

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

      4. metadata-eval 67.5%

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

      5. pow1 67.5%

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

      6. *-commutative 67.5%

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

      7. inv-pow 67.5%

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

      8. sqrt-pow1 67.5%

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

      9. metadata-eval 67.5%

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

   5. Applied egg-rr 67.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -6.3 \cdot 10^{+16}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{elif}\;re \leq 6.4 \cdot 10^{-67}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;im \cdot \sqrt{\frac{0.25}{re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -6.3e+16)
   (sqrt (- re))
   (if (<= re 6.4e-67) (sqrt (* 0.5 (- im re))) (* im (sqrt (/ 0.25 re))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -6.3e+16) {
		tmp = sqrt(-re);
	} else if (re <= 6.4e-67) {
		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 <= (-6.3d+16)) then
        tmp = sqrt(-re)
    else if (re <= 6.4d-67) 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 <= -6.3e+16) {
		tmp = Math.sqrt(-re);
	} else if (re <= 6.4e-67) {
		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 <= -6.3e+16:
		tmp = math.sqrt(-re)
	elif re <= 6.4e-67:
		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 <= -6.3e+16)
		tmp = sqrt(Float64(-re));
	elseif (re <= 6.4e-67)
		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 <= -6.3e+16)
		tmp = sqrt(-re);
	elseif (re <= 6.4e-67)
		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, -6.3e+16], N[Sqrt[(-re)], $MachinePrecision], If[LessEqual[re, 6.4e-67], 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 < -6.3e16

   1. Initial program 30.5%

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

      1. pow1 30.5%

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

   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 80.8%

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

      1. neg-mul-1 80.8%

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

   9. Simplified 80.8%

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

   if -6.3e16 < re < 6.40000000000000043e-67

   1. Initial program 57.2%

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

      1. pow1 57.2%

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

   4. Applied egg-rr 97.6%

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

      1. unpow1 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r* 97.6%

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

      4. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in re around 0 84.9%

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

      1. neg-mul-1 84.9%

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

      2. unsub-neg 84.9%

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

   9. Simplified 84.9%

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

   if 6.40000000000000043e-67 < re

   1. Initial program 20.2%

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

   3. Taylor expanded in re around inf 37.1%

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

      1. div-inv 37.2%

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

      2. sqrt-prod 47.2%

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

      3. sqrt-pow1 67.5%

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

      4. metadata-eval 67.5%

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

      5. pow1 67.5%

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

      6. *-rgt-identity 67.5%

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

      7. metadata-eval 67.5%

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

      8. metadata-eval 67.5%

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

      9. sqrt-unprod 66.7%

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

      10. associate-*l* 66.7%

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

      11. sqrt-div 66.7%

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

      12. metadata-eval 66.7%

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

      13. un-div-inv 66.6%

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

      14. sqrt-unprod 67.4%

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

      15. metadata-eval 67.4%

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

      16. metadata-eval 67.4%

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

      17. *-rgt-identity 67.4%

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

   5. Applied egg-rr 67.4%

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

      1. *-commutative 67.4%

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

      2. associate-/l* 67.3%

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

   7. Simplified 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-/r/ 66.9%

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

      3. div-inv 66.8%

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

      4. associate-/r* 67.3%

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

      5. add-sqr-sqrt 67.1%

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

      6. sqrt-unprod 67.3%

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

      7. frac-times 67.3%

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

      8. metadata-eval 67.3%

         \[\leadsto \frac{\sqrt{\frac{\color{blue}{0.25}}{\sqrt{re} \cdot \sqrt{re}}}}{\frac{1}{im}} \]

      9. add-sqr-sqrt 67.4%

         \[\leadsto \frac{\sqrt{\frac{0.25}{\color{blue}{re}}}}{\frac{1}{im}} \]

   9. Applied egg-rr 67.4%

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

       1. associate-/r/ 67.5%

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{0.25}{re}}}{1} \cdot im} \]

       2. /-rgt-identity 67.5%

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

       3. *-commutative 67.5%

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

   11. Simplified 67.5%

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

Alternative 5: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.26 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.26e+15) (sqrt (- re)) (sqrt (* im 0.5))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.26e+15) {
		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 <= (-1.26d+15)) 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 <= -1.26e+15) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = Math.sqrt((im * 0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.26e+15:
		tmp = math.sqrt(-re)
	else:
		tmp = math.sqrt((im * 0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.26e+15)
		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 <= -1.26e+15)
		tmp = sqrt(-re);
	else
		tmp = sqrt((im * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.26e+15], N[Sqrt[(-re)], $MachinePrecision], N[Sqrt[N[(im * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.26e15

   1. Initial program 30.5%

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

      1. pow1 30.5%

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

   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 80.8%

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

      1. neg-mul-1 80.8%

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

   9. Simplified 80.8%

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

   if -1.26e15 < re

   1. Initial program 43.5%

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

      1. pow1 43.5%

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

   4. Applied egg-rr 77.9%

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

      1. unpow1 77.9%

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

      2. *-commutative 77.9%

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

      3. associate-*r* 77.9%

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

      4. metadata-eval 77.9%

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

   6. Simplified 77.9%

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

   7. Taylor expanded in re around 0 66.3%

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

      1. *-commutative 66.3%

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

   9. Simplified 66.3%

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

Alternative 6: 31.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{-re}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (if (<= re -5e-310) (sqrt (- re)) 0.0))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = Math.sqrt(-re);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e-310:
		tmp = math.sqrt(-re)
	else:
		tmp = 0.0
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e-310)
		tmp = sqrt(Float64(-re));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[re_, im_] := If[LessEqual[re, -5e-310], N[Sqrt[(-re)], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if re < -4.999999999999985e-310

   1. Initial program 48.3%

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

      1. pow1 48.3%

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

   4. Applied egg-rr 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 51.6%

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

      1. neg-mul-1 51.6%

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

   9. Simplified 51.6%

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

   if -4.999999999999985e-310 < re

   1. Initial program 31.1%

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

   3. Taylor expanded in re around inf 7.4%

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

   4. Taylor expanded in re around 0 7.4%

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

Alternative 7: 6.3% 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 40.3%

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

3. Taylor expanded in re around inf 5.0%

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

4. Taylor expanded in re around 0 5.0%

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

Reproduce

herbie shell --seed 2024165 
(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)))))