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

Percentage Accurate: 42.2% → 89.8%

Time: 7.1s

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: 42.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot {\left({\left(\frac{1}{re}\right)}^{0.25} \cdot {im}^{0.5}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (pow (* (pow (/ 1.0 re) 0.25) (pow im 0.5)) 2.0))
   (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 = 0.5 * pow((pow((1.0 / re), 0.25) * pow(im, 0.5)), 2.0);
	} 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 = 0.5 * Math.pow((Math.pow((1.0 / re), 0.25) * Math.pow(im, 0.5)), 2.0);
	} 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 = 0.5 * math.pow((math.pow((1.0 / re), 0.25) * math.pow(im, 0.5)), 2.0)
	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(0.5 * (Float64((Float64(1.0 / re) ^ 0.25) * (im ^ 0.5)) ^ 2.0));
	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 = 0.5 * ((((1.0 / re) ^ 0.25) * (im ^ 0.5)) ^ 2.0);
	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[(0.5 * N[Power[N[(N[Power[N[(1.0 / re), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[im, 0.5], $MachinePrecision]), $MachinePrecision], 2.0], $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 2 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 14.4%

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

      1. hypot-udef 14.4%

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

      2. add-sqr-sqrt 14.4%

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

      3. pow2 14.4%

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

      4. pow1/2 14.4%

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

      5. sqrt-pow1 14.4%

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

      6. metadata-eval 14.4%

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

   3. Applied egg-rr 14.4%

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

   4. Taylor expanded in re around inf 47.9%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log \left(\frac{1}{re}\right) + \log \left({im}^{2}\right)\right)}\right)}^{2}} \]
   5. Step-by-step derivation

      1. log-pow 91.8%

         \[\leadsto 0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{1}{re}\right) + \color{blue}{2 \cdot \log im}\right)}\right)}^{2} \]

      2. log-pow 47.9%

         \[\leadsto 0.5 \cdot {\left(e^{0.25 \cdot \left(\log \left(\frac{1}{re}\right) + \color{blue}{\log \left({im}^{2}\right)}\right)}\right)}^{2} \]

      3. distribute-rgt-in 47.9%

         \[\leadsto 0.5 \cdot {\left(e^{\color{blue}{\log \left(\frac{1}{re}\right) \cdot 0.25 + \log \left({im}^{2}\right) \cdot 0.25}}\right)}^{2} \]

      4. exp-sum 47.9%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{\log \left(\frac{1}{re}\right) \cdot 0.25} \cdot e^{\log \left({im}^{2}\right) \cdot 0.25}\right)}}^{2} \]

      5. exp-to-pow 48.2%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{{\left(\frac{1}{re}\right)}^{0.25}} \cdot e^{\log \left({im}^{2}\right) \cdot 0.25}\right)}^{2} \]

      6. log-pow 91.6%

         \[\leadsto 0.5 \cdot {\left({\left(\frac{1}{re}\right)}^{0.25} \cdot e^{\color{blue}{\left(2 \cdot \log im\right)} \cdot 0.25}\right)}^{2} \]

      7. *-commutative 91.6%

         \[\leadsto 0.5 \cdot {\left({\left(\frac{1}{re}\right)}^{0.25} \cdot e^{\color{blue}{\left(\log im \cdot 2\right)} \cdot 0.25}\right)}^{2} \]

      8. associate-*l* 91.6%

         \[\leadsto 0.5 \cdot {\left({\left(\frac{1}{re}\right)}^{0.25} \cdot e^{\color{blue}{\log im \cdot \left(2 \cdot 0.25\right)}}\right)}^{2} \]

      9. metadata-eval 91.6%

         \[\leadsto 0.5 \cdot {\left({\left(\frac{1}{re}\right)}^{0.25} \cdot e^{\log im \cdot \color{blue}{0.5}}\right)}^{2} \]

      10. exp-to-pow 99.1%

          \[\leadsto 0.5 \cdot {\left({\left(\frac{1}{re}\right)}^{0.25} \cdot \color{blue}{{im}^{0.5}}\right)}^{2} \]

   6. Simplified 99.1%

      \[\leadsto 0.5 \cdot \color{blue}{{\left({\left(\frac{1}{re}\right)}^{0.25} \cdot {im}^{0.5}\right)}^{2}} \]

   if 0.0 < (sqrt.f64 (*.f64 2 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 42.3%

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

      1. sub-neg 42.3%

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

      2. sqr-neg 42.3%

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

      3. sub-neg 42.3%

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

      4. sqr-neg 42.3%

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

      5. hypot-def 87.8%

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

   3. Simplified 87.8%

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

      1. add-sqr-sqrt 87.2%

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

      2. sqrt-unprod 87.8%

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

      3. *-commutative 87.8%

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

      4. *-commutative 87.8%

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

      5. swap-sqr 87.8%

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

      6. add-sqr-sqrt 87.8%

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

      7. metadata-eval 87.8%

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

   5. Applied egg-rr 87.8%

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

      1. *-commutative 87.8%

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

      2. associate-*r* 87.8%

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

      3. metadata-eval 87.8%

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

   7. Simplified 87.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \leq 0:\\ \;\;\;\;0.5 \cdot {\left({\left(\frac{1}{re}\right)}^{0.25} \cdot {im}^{0.5}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)}\\ \end{array} \]

Alternative 2: 84.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) 0.0)
   (* 0.5 (sqrt (* 2.0 (* 0.5 (/ (pow im 2.0) 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 = 0.5 * sqrt((2.0 * (0.5 * (pow(im, 2.0) / 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 = 0.5 * Math.sqrt((2.0 * (0.5 * (Math.pow(im, 2.0) / 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 = 0.5 * math.sqrt((2.0 * (0.5 * (math.pow(im, 2.0) / 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(0.5 * sqrt(Float64(2.0 * Float64(0.5 * Float64((im ^ 2.0) / 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 = 0.5 * sqrt((2.0 * (0.5 * ((im ^ 2.0) / 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[(0.5 * N[Sqrt[N[(2.0 * N[(0.5 * N[(N[Power[im, 2.0], $MachinePrecision] / re), $MachinePrecision]), $MachinePrecision]), $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 2 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 14.4%

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

   2. Taylor expanded in re around inf 50.6%

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

   if 0.0 < (sqrt.f64 (*.f64 2 (-.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re)))

   1. Initial program 42.3%

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

      1. sub-neg 42.3%

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

      2. sqr-neg 42.3%

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

      3. sub-neg 42.3%

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

      4. sqr-neg 42.3%

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

      5. hypot-def 87.8%

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

   3. Simplified 87.8%

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

      1. add-sqr-sqrt 87.2%

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

      2. sqrt-unprod 87.8%

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

      3. *-commutative 87.8%

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

      4. *-commutative 87.8%

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

      5. swap-sqr 87.8%

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

      6. add-sqr-sqrt 87.8%

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

      7. metadata-eval 87.8%

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

   5. Applied egg-rr 87.8%

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

      1. *-commutative 87.8%

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

      2. associate-*r* 87.8%

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

      3. metadata-eval 87.8%

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

   7. Simplified 87.8%

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

4. Final simplification 82.6%

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

Alternative 3: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[N[(0.5 * N[(N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision] - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 38.4%

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

   1. sub-neg 38.4%

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

   2. sqr-neg 38.4%

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

   3. sub-neg 38.4%

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

   4. sqr-neg 38.4%

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

   5. hypot-def 77.5%

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

3. Simplified 77.5%

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

   1. add-sqr-sqrt 77.0%

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

   2. sqrt-unprod 77.5%

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

   3. *-commutative 77.5%

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

   4. *-commutative 77.5%

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

   5. swap-sqr 77.5%

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

   6. add-sqr-sqrt 77.5%

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

   7. metadata-eval 77.5%

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

5. Applied egg-rr 77.5%

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

   1. *-commutative 77.5%

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

   2. associate-*r* 77.5%

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

   3. metadata-eval 77.5%

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

7. Simplified 77.5%

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

8. Final simplification 77.5%

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

Alternative 4: 64.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -800:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+237}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -800.0)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (if (<= re 1.12e+237)
     (* 0.5 (sqrt (* 2.0 im)))
     (* 0.5 (sqrt (* 2.0 (- re re)))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -800.0) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.12e+237) {
		tmp = 0.5 * sqrt((2.0 * im));
	} else {
		tmp = 0.5 * sqrt((2.0 * (re - 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 <= (-800.0d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    else if (re <= 1.12d+237) then
        tmp = 0.5d0 * sqrt((2.0d0 * im))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re - re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -800.0) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else if (re <= 1.12e+237) {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re - re)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -800.0:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	elif re <= 1.12e+237:
		tmp = 0.5 * math.sqrt((2.0 * im))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re - re)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -800.0)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	elseif (re <= 1.12e+237)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im)));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -800.0)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	elseif (re <= 1.12e+237)
		tmp = 0.5 * sqrt((2.0 * im));
	else
		tmp = 0.5 * sqrt((2.0 * (re - re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -800.0], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 1.12e+237], N[(0.5 * N[Sqrt[N[(2.0 * im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -800

   1. Initial program 38.1%

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

   2. Taylor expanded in re around -inf 84.5%

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

      1. *-commutative 84.5%

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

   4. Simplified 84.5%

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

   if -800 < re < 1.11999999999999997e237

   1. Initial program 40.7%

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

      1. hypot-udef 71.7%

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

      2. add-sqr-sqrt 71.3%

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

      3. pow2 71.3%

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

      4. pow1/2 71.3%

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

      5. sqrt-pow1 71.2%

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

      6. metadata-eval 71.2%

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

   3. Applied egg-rr 71.2%

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

   4. Taylor expanded in im around inf 57.2%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}\right)}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 57.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}\right)} \]

      2. prod-exp 57.2%

         \[\leadsto 0.5 \cdot \color{blue}{e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}} \]

      3. +-commutative 57.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right) + \log 2\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      4. mul-1-neg 57.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{im}\right)\right)} + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      5. log-rec 57.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\left(-\color{blue}{\left(-\log im\right)}\right) + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      6. remove-double-neg 57.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\color{blue}{\log im} + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      7. +-commutative 57.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\left(\log 2 + \log im\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      8. log-prod 57.3%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\log \left(2 \cdot im\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      9. log-pow 57.3%

         \[\leadsto 0.5 \cdot e^{\color{blue}{\log \left({\left(2 \cdot im\right)}^{0.25}\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      10. metadata-eval 57.3%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\color{blue}{\left(\frac{0.5}{2}\right)}}\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      11. +-commutative 57.3%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right) + \log 2\right)}} \]

      12. mul-1-neg 57.3%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{im}\right)\right)} + \log 2\right)} \]

      13. log-rec 57.3%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\left(-\color{blue}{\left(-\log im\right)}\right) + \log 2\right)} \]

      14. remove-double-neg 57.3%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\color{blue}{\log im} + \log 2\right)} \]

   6. Simplified 62.0%

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

   if 1.11999999999999997e237 < re

   1. Initial program 2.0%

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

   2. Taylor expanded in re around inf 48.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -800:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{elif}\;re \leq 1.12 \cdot 10^{+237}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - re\right)}\\ \end{array} \]

Alternative 5: 63.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -0.075:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -0.075)
   (* 0.5 (sqrt (* 2.0 (* re -2.0))))
   (* 0.5 (sqrt (* 2.0 im)))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -0.075) {
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	} 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 <= (-0.075d0)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re * (-2.0d0))))
    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 <= -0.075) {
		tmp = 0.5 * Math.sqrt((2.0 * (re * -2.0)));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -0.075:
		tmp = 0.5 * math.sqrt((2.0 * (re * -2.0)))
	else:
		tmp = 0.5 * math.sqrt((2.0 * im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -0.075)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re * -2.0))));
	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 <= -0.075)
		tmp = 0.5 * sqrt((2.0 * (re * -2.0)));
	else
		tmp = 0.5 * sqrt((2.0 * im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -0.075], N[(0.5 * N[Sqrt[N[(2.0 * N[(re * -2.0), $MachinePrecision]), $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 < -0.0749999999999999972

   1. Initial program 38.1%

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

   2. Taylor expanded in re around -inf 84.5%

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

      1. *-commutative 84.5%

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

   4. Simplified 84.5%

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

   if -0.0749999999999999972 < re

   1. Initial program 38.4%

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

      1. hypot-udef 70.3%

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

      2. add-sqr-sqrt 69.9%

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

      3. pow2 69.9%

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

      4. pow1/2 69.9%

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

      5. sqrt-pow1 69.9%

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

      6. metadata-eval 69.9%

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

   3. Applied egg-rr 69.9%

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

   4. Taylor expanded in im around inf 54.1%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}\right)}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 54.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}\right)} \]

      2. prod-exp 54.2%

         \[\leadsto 0.5 \cdot \color{blue}{e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}} \]

      3. +-commutative 54.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right) + \log 2\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      4. mul-1-neg 54.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{im}\right)\right)} + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      5. log-rec 54.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\left(-\color{blue}{\left(-\log im\right)}\right) + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      6. remove-double-neg 54.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\color{blue}{\log im} + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      7. +-commutative 54.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\left(\log 2 + \log im\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      8. log-prod 54.2%

         \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\log \left(2 \cdot im\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      9. log-pow 54.2%

         \[\leadsto 0.5 \cdot e^{\color{blue}{\log \left({\left(2 \cdot im\right)}^{0.25}\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      10. metadata-eval 54.2%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\color{blue}{\left(\frac{0.5}{2}\right)}}\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

      11. +-commutative 54.2%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right) + \log 2\right)}} \]

      12. mul-1-neg 54.2%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{im}\right)\right)} + \log 2\right)} \]

      13. log-rec 54.2%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\left(-\color{blue}{\left(-\log im\right)}\right) + \log 2\right)} \]

      14. remove-double-neg 54.2%

          \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\color{blue}{\log im} + \log 2\right)} \]

   6. Simplified 58.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -0.075:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \end{array} \]

Alternative 6: 52.2% 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 38.4%

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

   1. hypot-udef 77.5%

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

   2. add-sqr-sqrt 77.0%

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

   3. pow2 77.0%

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

   4. pow1/2 77.0%

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

   5. sqrt-pow1 77.0%

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

   6. metadata-eval 77.0%

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

3. Applied egg-rr 77.0%

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

4. Taylor expanded in im around inf 44.9%

   \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}\right)}^{2}} \]
5. Step-by-step derivation

   1. unpow2 44.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \cdot e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}\right)} \]

   2. prod-exp 44.9%

      \[\leadsto 0.5 \cdot \color{blue}{e^{0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)}} \]

   3. +-commutative 44.9%

      \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right) + \log 2\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   4. mul-1-neg 44.9%

      \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{im}\right)\right)} + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   5. log-rec 44.9%

      \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\left(-\color{blue}{\left(-\log im\right)}\right) + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   6. remove-double-neg 44.9%

      \[\leadsto 0.5 \cdot e^{0.25 \cdot \left(\color{blue}{\log im} + \log 2\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   7. +-commutative 44.9%

      \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\left(\log 2 + \log im\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   8. log-prod 45.0%

      \[\leadsto 0.5 \cdot e^{0.25 \cdot \color{blue}{\log \left(2 \cdot im\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   9. log-pow 45.0%

      \[\leadsto 0.5 \cdot e^{\color{blue}{\log \left({\left(2 \cdot im\right)}^{0.25}\right)} + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   10. metadata-eval 45.0%

       \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\color{blue}{\left(\frac{0.5}{2}\right)}}\right) + 0.25 \cdot \left(\log 2 + -1 \cdot \log \left(\frac{1}{im}\right)\right)} \]

   11. +-commutative 45.0%

       \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{im}\right) + \log 2\right)}} \]

   12. mul-1-neg 45.0%

       \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\color{blue}{\left(-\log \left(\frac{1}{im}\right)\right)} + \log 2\right)} \]

   13. log-rec 45.0%

       \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\left(-\color{blue}{\left(-\log im\right)}\right) + \log 2\right)} \]

   14. remove-double-neg 45.0%

       \[\leadsto 0.5 \cdot e^{\log \left({\left(2 \cdot im\right)}^{\left(\frac{0.5}{2}\right)}\right) + 0.25 \cdot \left(\color{blue}{\log im} + \log 2\right)} \]

6. Simplified 48.6%

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

7. Final simplification 48.6%

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

Reproduce

herbie shell --seed 2023319 
(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)))))