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

Percentage Accurate: 40.6% → 90.0%

Time: 11.7s

Alternatives: 6

Speedup: 1.9×

Specification

\[im > 0\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.6% 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: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (* im (sqrt (/ 1.0 re))))
   (* 0.5 (sqrt (* 2.0 (- (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 = 0.5 * (im * sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * sqrt((2.0 * (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 = 0.5 * (im * Math.sqrt((1.0 / re)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (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 = 0.5 * (im * math.sqrt((1.0 / re)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (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(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 = 0.5 * (im * sqrt((1.0 / re)));
	else
		tmp = 0.5 * sqrt((2.0 * (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[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $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 6.3%

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

      1. add-cbrt-cube 6.3%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. pow1/3 6.3%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 6.3%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 6.3%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 6.3%

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

      3. pow1/2 6.3%

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

      4. sqrt-unprod 6.3%

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

      5. *-commutative 6.3%

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

      6. add-sqr-sqrt 6.3%

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

      7. associate-*l* 6.3%

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

      8. pow1/2 6.3%

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

      9. metadata-eval 6.3%

         \[\leadsto 0.5 \cdot \left(\sqrt{{2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      10. sqrt-pow1 6.3%

          \[\leadsto 0.5 \cdot \left(\color{blue}{{2}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      11. metadata-eval 6.3%

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

      12. metadata-eval 6.3%

          \[\leadsto 0.5 \cdot \left({2}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      13. pow1/2 6.3%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      14. metadata-eval 6.3%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\sqrt{{2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      15. sqrt-pow1 6.3%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      16. metadata-eval 6.3%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left({2}^{\left(\frac{\color{blue}{0.5}}{2}\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      17. metadata-eval 6.3%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left({2}^{\color{blue}{0.25}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

   6. Applied egg-rr 6.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right)} \]

   7. Taylor expanded in re around inf 99.6%

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

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

   1. Initial program 49.7%

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

      1. sub-neg 49.7%

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

      2. sqr-neg 49.7%

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

      3. sub-neg 49.7%

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

      4. sqr-neg 49.7%

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

      5. hypot-define 89.5%

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

   3. Simplified 89.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(2 + \frac{re}{im} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.65e-40)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 1.1e-79)
     (* 0.5 (sqrt (* im (+ 2.0 (* (/ re im) -2.0)))))
     (* 0.5 (* im (sqrt (/ 1.0 re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.65e-40) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 1.1e-79) {
		tmp = 0.5 * sqrt((im * (2.0 + ((re / im) * -2.0))));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / 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 <= (-1.65d-40)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 1.1d-79) then
        tmp = 0.5d0 * sqrt((im * (2.0d0 + ((re / im) * (-2.0d0)))))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.65e-40) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 1.1e-79) {
		tmp = 0.5 * Math.sqrt((im * (2.0 + ((re / im) * -2.0))));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.65e-40:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 1.1e-79:
		tmp = 0.5 * math.sqrt((im * (2.0 + ((re / im) * -2.0))))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.65e-40)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 1.1e-79)
		tmp = Float64(0.5 * sqrt(Float64(im * Float64(2.0 + Float64(Float64(re / im) * -2.0)))));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.65e-40)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 1.1e-79)
		tmp = 0.5 * sqrt((im * (2.0 + ((re / im) * -2.0))));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.65e-40], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.1e-79], N[(0.5 * N[Sqrt[N[(im * N[(2.0 + N[(N[(re / im), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -1.64999999999999996e-40

   1. Initial program 48.1%

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

      1. sub-neg 48.1%

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

      2. sqr-neg 48.1%

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

      3. sub-neg 48.1%

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

      4. sqr-neg 48.1%

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

      5. hypot-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in re around -inf 78.1%

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

      1. *-commutative 78.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 78.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -1.64999999999999996e-40 < re < 1.0999999999999999e-79

   1. Initial program 65.0%

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

      1. sub-neg 65.0%

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

      2. sqr-neg 65.0%

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

      3. sub-neg 65.0%

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

      4. sqr-neg 65.0%

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

      5. hypot-define 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in im around inf 84.4%

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

      1. *-commutative 84.4%

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

   7. Simplified 84.4%

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

   if 1.0999999999999999e-79 < re

   1. Initial program 12.3%

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

      1. add-cbrt-cube 12.2%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. pow1/3 11.8%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 20.6%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 36.5%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 36.5%

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

      3. pow1/2 36.5%

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

      4. sqrt-unprod 36.3%

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

      5. *-commutative 36.3%

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

      6. add-sqr-sqrt 36.3%

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

      7. associate-*l* 36.4%

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

      8. pow1/2 36.4%

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

      9. metadata-eval 36.4%

         \[\leadsto 0.5 \cdot \left(\sqrt{{2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      10. sqrt-pow1 36.4%

          \[\leadsto 0.5 \cdot \left(\color{blue}{{2}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      11. metadata-eval 36.4%

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

      12. metadata-eval 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      13. pow1/2 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      14. metadata-eval 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\sqrt{{2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      15. sqrt-pow1 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      16. metadata-eval 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left({2}^{\left(\frac{\color{blue}{0.5}}{2}\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      17. metadata-eval 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left({2}^{\color{blue}{0.25}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

   6. Applied egg-rr 36.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right)} \]

   7. Taylor expanded in re around inf 76.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.65 \cdot 10^{-40}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 1.1 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot \left(2 + \frac{re}{im} \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4.2e-31)
   (* 0.5 (sqrt (* re -4.0)))
   (if (<= re 8.2e-80)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (sqrt (/ 1.0 re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4.2e-31) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else if (re <= 8.2e-80) {
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= (-4.2d-31)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else if (re <= 8.2d-80) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im - re)))
    else
        tmp = 0.5d0 * (im * sqrt((1.0d0 / re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -4.2e-31) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else if (re <= 8.2e-80) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.sqrt((1.0 / re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4.2e-31:
		tmp = 0.5 * math.sqrt((re * -4.0))
	elif re <= 8.2e-80:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.sqrt((1.0 / re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4.2e-31)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	elseif (re <= 8.2e-80)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im - re))));
	else
		tmp = Float64(0.5 * Float64(im * sqrt(Float64(1.0 / re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -4.2e-31)
		tmp = 0.5 * sqrt((re * -4.0));
	elseif (re <= 8.2e-80)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * sqrt((1.0 / re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4.2e-31], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.2e-80], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Sqrt[N[(1.0 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.19999999999999982e-31

   1. Initial program 48.1%

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

      1. sub-neg 48.1%

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

      2. sqr-neg 48.1%

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

      3. sub-neg 48.1%

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

      4. sqr-neg 48.1%

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

      5. hypot-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in re around -inf 78.1%

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

      1. *-commutative 78.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 78.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -4.19999999999999982e-31 < re < 8.1999999999999999e-80

   1. Initial program 65.0%

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

   3. Taylor expanded in re around 0 84.4%

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

   if 8.1999999999999999e-80 < re

   1. Initial program 12.3%

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

      1. add-cbrt-cube 12.2%

         \[\leadsto 0.5 \cdot \color{blue}{\sqrt[3]{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}}} \]

      2. pow1/3 11.8%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 20.6%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 36.5%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 36.5%

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

      3. pow1/2 36.5%

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

      4. sqrt-unprod 36.3%

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

      5. *-commutative 36.3%

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

      6. add-sqr-sqrt 36.3%

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

      7. associate-*l* 36.4%

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

      8. pow1/2 36.4%

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

      9. metadata-eval 36.4%

         \[\leadsto 0.5 \cdot \left(\sqrt{{2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      10. sqrt-pow1 36.4%

          \[\leadsto 0.5 \cdot \left(\color{blue}{{2}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      11. metadata-eval 36.4%

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

      12. metadata-eval 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      13. pow1/2 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      14. metadata-eval 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\sqrt{{2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      15. sqrt-pow1 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      16. metadata-eval 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left({2}^{\left(\frac{\color{blue}{0.5}}{2}\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

      17. metadata-eval 36.4%

          \[\leadsto 0.5 \cdot \left({2}^{0.25} \cdot \left({2}^{\color{blue}{0.25}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right) \]

   6. Applied egg-rr 36.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) - re}\right)\right)} \]

   7. Taylor expanded in re around inf 76.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot \sqrt{\frac{1}{re}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.2 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{elif}\;re \leq 8.2 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \sqrt{\frac{1}{re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-232}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im 1.75e-232) (* 0.5 (sqrt 0.0)) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-232) {
		tmp = 0.5 * sqrt(0.0);
	} else {
		tmp = 0.5 * sqrt((2.0 * im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= 1.75d-232) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= 1.75e-232) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= 1.75e-232:
		tmp = 0.5 * math.sqrt(0.0)
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= 1.75e-232)
		tmp = Float64(0.5 * sqrt(0.0));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= 1.75e-232)
		tmp = 0.5 * sqrt(0.0);
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, 1.75e-232], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.7499999999999999e-232

   1. Initial program 40.2%

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

      1. add-sqr-sqrt 38.9%

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

      2. add-sqr-sqrt 2.7%

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

      3. difference-of-squares 2.7%

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

      4. hypot-define 2.7%

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

      5. hypot-define 13.7%

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

   4. Applied egg-rr 13.7%

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

   5. Taylor expanded in re around inf 15.7%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]

   if 1.7499999999999999e-232 < im

   1. Initial program 44.9%

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

      1. sub-neg 44.9%

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

      2. sqr-neg 44.9%

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

      3. sub-neg 44.9%

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

      4. sqr-neg 44.9%

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

      5. hypot-define 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in re around 0 57.8%

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

      1. *-commutative 57.8%

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

   7. Simplified 57.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.75 \cdot 10^{-232}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -9.2e-29) (* 0.5 (sqrt (* re -4.0))) (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -9.2e-29) {
		tmp = 0.5 * sqrt((re * -4.0));
	} else {
		tmp = 0.5 * sqrt((2.0 * 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 <= (-9.2d-29)) then
        tmp = 0.5d0 * sqrt((re * (-4.0d0)))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -9.2e-29) {
		tmp = 0.5 * Math.sqrt((re * -4.0));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -9.2e-29:
		tmp = 0.5 * math.sqrt((re * -4.0))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -9.2e-29)
		tmp = Float64(0.5 * sqrt(Float64(re * -4.0)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -9.2e-29)
		tmp = 0.5 * sqrt((re * -4.0));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -9.2e-29], N[(0.5 * N[Sqrt[N[(re * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -9.19999999999999965e-29

   1. Initial program 48.1%

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

      1. sub-neg 48.1%

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

      2. sqr-neg 48.1%

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

      3. sub-neg 48.1%

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

      4. sqr-neg 48.1%

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

      5. hypot-define 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in re around -inf 78.1%

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

      1. *-commutative 78.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   7. Simplified 78.1%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{re \cdot -4}} \]

   if -9.19999999999999965e-29 < re

   1. Initial program 42.2%

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

      1. sub-neg 42.2%

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

      2. sqr-neg 42.2%

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

      3. sub-neg 42.2%

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

      4. sqr-neg 42.2%

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

      5. hypot-define 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in re around 0 59.2%

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

      1. *-commutative 59.2%

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

   7. Simplified 59.2%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -9.2 \cdot 10^{-29}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 5.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{0} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (sqrt 0.0)))

C

double code(double re, double im) {
	return 0.5 * sqrt(0.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * sqrt(0.0d0)
end function

Java

public static double code(double re, double im) {
	return 0.5 * Math.sqrt(0.0);
}

Python

def code(re, im):
	return 0.5 * math.sqrt(0.0)

Julia

function code(re, im)
	return Float64(0.5 * sqrt(0.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * sqrt(0.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 44.1%

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

   1. add-sqr-sqrt 43.7%

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

   2. add-sqr-sqrt 14.0%

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

   3. difference-of-squares 14.0%

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

   4. hypot-define 14.0%

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

   5. hypot-define 26.0%

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

4. Applied egg-rr 26.0%

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

5. Taylor expanded in re around inf 5.5%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{0}} \]

6. Final simplification 5.5%

   \[\leadsto 0.5 \cdot \sqrt{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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)))))