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

Percentage Accurate: 41.6% → 90.3%

Time: 12.7s

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: 41.6% 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: 90.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (* im (pow re -0.5)))
   (* 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 * pow(re, -0.5));
	} 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.pow(re, -0.5));
	} 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.pow(re, -0.5))
	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 * (re ^ -0.5)));
	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 * (re ^ -0.5));
	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[Power[re, -0.5], $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 9.8%

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

      1. add-sqr-sqrt 9.8%

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

      2. pow2 9.8%

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

      3. pow1/2 9.8%

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

      4. sqrt-pow1 9.8%

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

      5. *-commutative 9.8%

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

      6. hypot-define 9.8%

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

      7. metadata-eval 9.8%

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

   4. Applied egg-rr 9.8%

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

   5. Taylor expanded in re around inf 99.5%

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

      1. unpow1/2 99.5%

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

      2. exp-to-pow 94.2%

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

      3. log-rec 94.2%

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

      4. distribute-lft-neg-out 94.2%

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

      5. distribute-rgt-neg-in 94.2%

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

      6. metadata-eval 94.2%

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

      7. exp-to-pow 99.7%

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

   7. Simplified 99.7%

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

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

   1. Initial program 49.0%

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

      1. sub-neg 49.0%

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

      2. sqr-neg 49.0%

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

      3. sub-neg 49.0%

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

      4. sqr-neg 49.0%

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

      5. hypot-define 90.2%

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

   3. Simplified 90.2%

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

4. Final simplification 91.6%

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

Alternative 2: 77.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -4100000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \left(1 - \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -4100000000.0)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 8.5e-53)
     (* 0.5 (sqrt (* 2.0 (* im (- 1.0 (/ re im))))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -4100000000.0) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 8.5e-53) {
		tmp = 0.5 * sqrt((2.0 * (im * (1.0 - (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 <= (-4100000000.0d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 8.5d-53) then
        tmp = 0.5d0 * sqrt((2.0d0 * (im * (1.0d0 - (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 <= -4100000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 8.5e-53) {
		tmp = 0.5 * Math.sqrt((2.0 * (im * (1.0 - (re / im)))));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -4100000000.0:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 8.5e-53:
		tmp = 0.5 * math.sqrt((2.0 * (im * (1.0 - (re / im)))))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -4100000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 8.5e-53)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(im * Float64(1.0 - 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 <= -4100000000.0)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 8.5e-53)
		tmp = 0.5 * sqrt((2.0 * (im * (1.0 - (re / im)))));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -4100000000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 8.5e-53], N[(0.5 * N[Sqrt[N[(2.0 * N[(im * N[(1.0 - N[(re / im), $MachinePrecision]), $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 < -4.1e9

   1. Initial program 50.1%

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

   3. Taylor expanded in re around -inf 81.9%

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

      1. *-commutative 81.9%

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

   5. Simplified 81.9%

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

   if -4.1e9 < re < 8.50000000000000044e-53

   1. Initial program 59.5%

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

   3. Taylor expanded in im around inf 81.0%

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

      1. mul-1-neg 81.0%

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

      2. unsub-neg 81.0%

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

   5. Simplified 81.0%

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

   if 8.50000000000000044e-53 < re

   1. Initial program 12.7%

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

      1. add-sqr-sqrt 12.6%

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

      2. pow2 12.6%

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

      3. pow1/2 12.6%

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

      4. sqrt-pow1 12.6%

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

      5. *-commutative 12.6%

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

      6. hypot-define 35.7%

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

      7. metadata-eval 35.7%

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

   4. Applied egg-rr 35.7%

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

   5. Taylor expanded in re around inf 78.7%

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

      1. unpow1/2 78.7%

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

      2. exp-to-pow 74.3%

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

      3. log-rec 74.3%

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

      4. distribute-lft-neg-out 74.3%

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

      5. distribute-rgt-neg-in 74.3%

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

      6. metadata-eval 74.3%

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

      7. exp-to-pow 78.8%

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

   7. Simplified 78.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4100000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 8.5 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im \cdot \left(1 - \frac{re}{im}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -700000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -700000000.0)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 6.5e-50)
     (* 0.5 (sqrt (* 2.0 (- im re))))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -700000000.0) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 6.5e-50) {
		tmp = 0.5 * sqrt((2.0 * (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 <= (-700000000.0d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 6.5d-50) then
        tmp = 0.5d0 * sqrt((2.0d0 * (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 <= -700000000.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 6.5e-50) {
		tmp = 0.5 * Math.sqrt((2.0 * (im - re)));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -700000000.0:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 6.5e-50:
		tmp = 0.5 * math.sqrt((2.0 * (im - re)))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -700000000.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 6.5e-50)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= -700000000.0)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 6.5e-50)
		tmp = 0.5 * sqrt((2.0 * (im - re)));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -700000000.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 6.5e-50], N[(0.5 * N[Sqrt[N[(2.0 * N[(im - re), $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 < -7e8

   1. Initial program 50.1%

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

   3. Taylor expanded in re around -inf 81.9%

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

      1. *-commutative 81.9%

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

   5. Simplified 81.9%

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

   if -7e8 < re < 6.49999999999999987e-50

   1. Initial program 59.5%

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

   3. Taylor expanded in re around 0 81.0%

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

   if 6.49999999999999987e-50 < re

   1. Initial program 12.7%

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

      1. add-sqr-sqrt 12.6%

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

      2. pow2 12.6%

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

      3. pow1/2 12.6%

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

      4. sqrt-pow1 12.6%

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

      5. *-commutative 12.6%

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

      6. hypot-define 35.7%

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

      7. metadata-eval 35.7%

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

   4. Applied egg-rr 35.7%

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

   5. Taylor expanded in re around inf 78.7%

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

      1. unpow1/2 78.7%

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

      2. exp-to-pow 74.3%

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

      3. log-rec 74.3%

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

      4. distribute-lft-neg-out 74.3%

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

      5. distribute-rgt-neg-in 74.3%

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

      6. metadata-eval 74.3%

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

      7. exp-to-pow 78.8%

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

   7. Simplified 78.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -700000000:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 6.5 \cdot 10^{-50}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.32e-52)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 9.5e-45)
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (* im (pow re -0.5))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.32e-52) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 9.5e-45) {
		tmp = 0.5 * sqrt((2.0 * 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 <= (-1.32d-52)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 9.5d-45) then
        tmp = 0.5d0 * sqrt((2.0d0 * 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 <= -1.32e-52) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 9.5e-45) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.32e-52:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 9.5e-45:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.32e-52)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 9.5e-45)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= -1.32e-52)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 9.5e-45)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.32e-52], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e-45], N[(0.5 * N[Sqrt[N[(2.0 * im), $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 < -1.32000000000000002e-52

   1. Initial program 51.5%

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

   3. Taylor expanded in re around -inf 79.1%

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

      1. *-commutative 79.1%

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

   5. Simplified 79.1%

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

   if -1.32000000000000002e-52 < re < 9.5000000000000002e-45

   1. Initial program 58.1%

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

   3. Taylor expanded in re around 0 80.1%

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

      1. sqrt-unprod 80.6%

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

      2. pow1 80.6%

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

   5. Applied egg-rr 80.6%

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

      1. unpow1 80.6%

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

   7. Simplified 80.6%

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

   if 9.5000000000000002e-45 < re

   1. Initial program 12.9%

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

      1. add-sqr-sqrt 12.9%

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

      2. pow2 12.9%

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

      3. pow1/2 12.9%

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

      4. sqrt-pow1 12.9%

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

      5. *-commutative 12.9%

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

      6. hypot-define 35.3%

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

      7. metadata-eval 35.3%

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

   4. Applied egg-rr 35.3%

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

   5. Taylor expanded in re around inf 79.4%

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

      1. unpow1/2 79.4%

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

      2. exp-to-pow 75.0%

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

      3. log-rec 75.0%

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

      4. distribute-lft-neg-out 75.0%

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

      5. distribute-rgt-neg-in 75.0%

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

      6. metadata-eval 75.0%

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

      7. exp-to-pow 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.32 \cdot 10^{-52}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 6.2e-44) (* 0.5 (sqrt (* 2.0 im))) (* 0.5 (* im (pow re -0.5)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= 6.2e-44) {
		tmp = 0.5 * sqrt((2.0 * 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 <= 6.2d-44) then
        tmp = 0.5d0 * sqrt((2.0d0 * 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 <= 6.2e-44) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * (im * Math.pow(re, -0.5));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 6.2e-44:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * (im * math.pow(re, -0.5))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 6.2e-44)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * 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 <= 6.2e-44)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * (im * (re ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 6.2e-44], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(im * N[Power[re, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 6.19999999999999968e-44

   1. Initial program 55.2%

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

   3. Taylor expanded in re around 0 56.9%

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

      1. sqrt-unprod 57.2%

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

      2. pow1 57.2%

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

   5. Applied egg-rr 57.2%

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

      1. unpow1 57.2%

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

   7. Simplified 57.2%

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

   if 6.19999999999999968e-44 < re

   1. Initial program 12.9%

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

      1. add-sqr-sqrt 12.9%

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

      2. pow2 12.9%

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

      3. pow1/2 12.9%

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

      4. sqrt-pow1 12.9%

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

      5. *-commutative 12.9%

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

      6. hypot-define 35.3%

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

      7. metadata-eval 35.3%

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

   4. Applied egg-rr 35.3%

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

   5. Taylor expanded in re around inf 79.4%

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

      1. unpow1/2 79.4%

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

      2. exp-to-pow 75.0%

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

      3. log-rec 75.0%

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

      4. distribute-lft-neg-out 75.0%

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

      5. distribute-rgt-neg-in 75.0%

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

      6. metadata-eval 75.0%

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

      7. exp-to-pow 79.5%

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

   7. Simplified 79.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left(im \cdot {re}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 6.2 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot {re}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{2 \cdot im} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.2%

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

3. Taylor expanded in re around 0 47.8%

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

   1. sqrt-unprod 47.8%

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

   2. pow1 47.8%

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

5. Applied egg-rr 47.8%

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

   1. unpow1 47.8%

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

7. Simplified 47.8%

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

8. Final simplification 47.8%

   \[\leadsto 0.5 \cdot \sqrt{2 \cdot im} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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)))))