math.sqrt on complex, real part

Percentage Accurate: 41.7% → 84.0%

Time: 8.1s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\\ \mathbf{if}\;re \leq -1.26 \cdot 10^{+128}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-0.125, \frac{im \cdot im}{re} \cdot \frac{t_0}{re}, t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (pow (* (pow (* im im) 0.25) (pow (/ -1.0 re) 0.25)) 2.0)))
   (if (<= re -1.26e+128)
     (* 0.5 (fma -0.125 (* (/ (* im im) re) (/ t_0 re)) t_0))
     (sqrt (* 0.5 (+ re (hypot re im)))))))

C

double code(double re, double im) {
	double t_0 = pow((pow((im * im), 0.25) * pow((-1.0 / re), 0.25)), 2.0);
	double tmp;
	if (re <= -1.26e+128) {
		tmp = 0.5 * fma(-0.125, (((im * im) / re) * (t_0 / re)), t_0);
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64((Float64(im * im) ^ 0.25) * (Float64(-1.0 / re) ^ 0.25)) ^ 2.0
	tmp = 0.0
	if (re <= -1.26e+128)
		tmp = Float64(0.5 * fma(-0.125, Float64(Float64(Float64(im * im) / re) * Float64(t_0 / re)), t_0));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Power[N[(N[Power[N[(im * im), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(-1.0 / re), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[re, -1.26e+128], N[(0.5 * N[(-0.125 * N[(N[(N[(im * im), $MachinePrecision] / re), $MachinePrecision] * N[(t$95$0 / re), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if re < -1.26000000000000009e128

   1. Initial program 3.8%

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

      1. +-commutative 3.8%

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

      2. hypot-def 28.5%

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

   3. Simplified 28.5%

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

      1. add-sqr-sqrt 28.6%

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

      2. pow2 28.6%

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

      3. pow1/2 28.6%

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

      4. sqrt-pow1 28.6%

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

      5. *-commutative 28.6%

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

      6. metadata-eval 28.6%

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

   5. Applied egg-rr 28.6%

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

   6. Taylor expanded in re around -inf 76.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-0.125 \cdot \frac{{\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot {im}^{2}}{{re}^{2}} + {\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2}\right)} \]
   7. Step-by-step derivation

      1. fma-def 76.3%

         \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{{\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2} \cdot {im}^{2}}{{re}^{2}}, {\left(e^{0.25 \cdot \left(\log \left({im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}^{2}\right)} \]

   8. Simplified 86.2%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{im \cdot im}{re} \cdot \frac{{\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}}{re}, {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\right)} \]

   if -1.26000000000000009e128 < re

   1. Initial program 50.0%

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

      1. +-commutative 50.0%

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

      2. hypot-def 90.9%

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

   3. Simplified 90.9%

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

      1. add-sqr-sqrt 90.2%

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

      2. sqrt-unprod 90.9%

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

      3. *-commutative 90.9%

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

      4. *-commutative 90.9%

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

      5. swap-sqr 90.9%

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

      6. add-sqr-sqrt 90.9%

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

      7. *-commutative 90.9%

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

      8. metadata-eval 90.9%

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

   5. Applied egg-rr 90.9%

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

      1. associate-*l* 90.9%

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

      2. metadata-eval 90.9%

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

   7. Simplified 90.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.26 \cdot 10^{+128}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(-0.125, \frac{im \cdot im}{re} \cdot \frac{{\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}}{re}, {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 2: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -1.95e+101)
   (* 0.5 (pow (* (pow (* im im) 0.25) (pow (/ -1.0 re) 0.25)) 2.0))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -1.95e+101) {
		tmp = 0.5 * pow((pow((im * im), 0.25) * pow((-1.0 / re), 0.25)), 2.0);
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -1.95e+101) {
		tmp = 0.5 * Math.pow((Math.pow((im * im), 0.25) * Math.pow((-1.0 / re), 0.25)), 2.0);
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -1.95e+101:
		tmp = 0.5 * math.pow((math.pow((im * im), 0.25) * math.pow((-1.0 / re), 0.25)), 2.0)
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -1.95e+101)
		tmp = Float64(0.5 * (Float64((Float64(im * im) ^ 0.25) * (Float64(-1.0 / re) ^ 0.25)) ^ 2.0));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -1.95e+101)
		tmp = 0.5 * ((((im * im) ^ 0.25) * ((-1.0 / re) ^ 0.25)) ^ 2.0);
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -1.95e+101], N[(0.5 * N[Power[N[(N[Power[N[(im * im), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(-1.0 / re), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.95e101

   1. Initial program 8.5%

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

      1. +-commutative 8.5%

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

      2. hypot-def 31.4%

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

   3. Simplified 31.4%

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

      1. add-sqr-sqrt 31.5%

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

      2. pow2 31.5%

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

      3. pow1/2 31.5%

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

      4. sqrt-pow1 31.5%

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

      5. *-commutative 31.5%

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

      6. metadata-eval 31.5%

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

   5. Applied egg-rr 31.5%

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

   6. Taylor expanded in re around -inf 80.1%

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

      1. distribute-rgt-in 80.1%

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

      2. exp-sum 80.3%

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

      3. exp-to-pow 80.7%

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

      4. unpow2 80.7%

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

      5. exp-to-pow 84.7%

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

   8. Simplified 84.7%

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

   if -1.95e101 < 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. +-commutative 49.7%

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

      2. hypot-def 91.2%

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

   3. Simplified 91.2%

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

      1. add-sqr-sqrt 90.5%

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

      2. sqrt-unprod 91.2%

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

      3. *-commutative 91.2%

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

      4. *-commutative 91.2%

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

      5. swap-sqr 91.2%

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

      6. add-sqr-sqrt 91.2%

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

      7. *-commutative 91.2%

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

      8. metadata-eval 91.2%

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

   5. Applied egg-rr 91.2%

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

      1. associate-*l* 91.2%

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

      2. metadata-eval 91.2%

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

   7. Simplified 91.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.95 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot {\left({\left(im \cdot im\right)}^{0.25} \cdot {\left(\frac{-1}{re}\right)}^{0.25}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re -5e+99)
   (* 0.5 (sqrt (* (- im) (/ im re))))
   (sqrt (* 0.5 (+ re (hypot re im))))))

C

double code(double re, double im) {
	double tmp;
	if (re <= -5e+99) {
		tmp = 0.5 * sqrt((-im * (im / re)));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= -5e+99) {
		tmp = 0.5 * Math.sqrt((-im * (im / re)));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im))));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= -5e+99:
		tmp = 0.5 * math.sqrt((-im * (im / re)))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im))))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= -5e+99)
		tmp = Float64(0.5 * sqrt(Float64(Float64(-im) * Float64(im / re))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= -5e+99)
		tmp = 0.5 * sqrt((-im * (im / re)));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, -5e+99], N[(0.5 * N[Sqrt[N[((-im) * N[(im / re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.00000000000000008e99

   1. Initial program 8.5%

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

      1. +-commutative 8.5%

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

      2. hypot-def 31.4%

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

   3. Simplified 31.4%

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

   4. Taylor expanded in re around -inf 71.7%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 71.7%

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

      2. neg-mul-1 71.7%

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

      3. unpow2 71.7%

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

      4. distribute-rgt-neg-in 71.7%

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

   6. Simplified 71.7%

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

   7. Taylor expanded in im around 0 71.7%

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

      1. mul-1-neg 71.7%

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

      2. unpow2 71.7%

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

      3. associate-*r/ 72.5%

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

      4. distribute-rgt-neg-in 72.5%

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

      5. distribute-neg-frac 72.5%

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

   9. Simplified 72.5%

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

   if -5.00000000000000008e99 < 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. +-commutative 49.7%

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

      2. hypot-def 91.2%

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

   3. Simplified 91.2%

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

      1. add-sqr-sqrt 90.5%

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

      2. sqrt-unprod 91.2%

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

      3. *-commutative 91.2%

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

      4. *-commutative 91.2%

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

      5. swap-sqr 91.2%

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

      6. add-sqr-sqrt 91.2%

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

      7. *-commutative 91.2%

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

      8. metadata-eval 91.2%

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

   5. Applied egg-rr 91.2%

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

      1. associate-*l* 91.2%

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

      2. metadata-eval 91.2%

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

   7. Simplified 91.2%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5 \cdot 10^{+99}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(-im\right) \cdot \frac{im}{re}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 4: 57.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{if}\;im \leq -4.5 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -1.6 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -8.5 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im -2.0)))))
   (if (<= im -4.5e+16)
     t_0
     (if (<= im -1.55e-38)
       (sqrt re)
       (if (<= im -1.6e-120)
         t_0
         (if (<= im -8.5e-287)
           (sqrt re)
           (if (<= im 1.05e-264)
             (* 0.5 (/ im (sqrt (- re))))
             (if (<= im 5.5e-153) (sqrt re) (* 0.5 (sqrt (* im 2.0)))))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * -2.0));
	double tmp;
	if (im <= -4.5e+16) {
		tmp = t_0;
	} else if (im <= -1.55e-38) {
		tmp = sqrt(re);
	} else if (im <= -1.6e-120) {
		tmp = t_0;
	} else if (im <= -8.5e-287) {
		tmp = sqrt(re);
	} else if (im <= 1.05e-264) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (im <= 5.5e-153) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((im * (-2.0d0)))
    if (im <= (-4.5d+16)) then
        tmp = t_0
    else if (im <= (-1.55d-38)) then
        tmp = sqrt(re)
    else if (im <= (-1.6d-120)) then
        tmp = t_0
    else if (im <= (-8.5d-287)) then
        tmp = sqrt(re)
    else if (im <= 1.05d-264) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (im <= 5.5d-153) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((im * -2.0));
	double tmp;
	if (im <= -4.5e+16) {
		tmp = t_0;
	} else if (im <= -1.55e-38) {
		tmp = Math.sqrt(re);
	} else if (im <= -1.6e-120) {
		tmp = t_0;
	} else if (im <= -8.5e-287) {
		tmp = Math.sqrt(re);
	} else if (im <= 1.05e-264) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (im <= 5.5e-153) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * -2.0))
	tmp = 0
	if im <= -4.5e+16:
		tmp = t_0
	elif im <= -1.55e-38:
		tmp = math.sqrt(re)
	elif im <= -1.6e-120:
		tmp = t_0
	elif im <= -8.5e-287:
		tmp = math.sqrt(re)
	elif im <= 1.05e-264:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif im <= 5.5e-153:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * -2.0)))
	tmp = 0.0
	if (im <= -4.5e+16)
		tmp = t_0;
	elseif (im <= -1.55e-38)
		tmp = sqrt(re);
	elseif (im <= -1.6e-120)
		tmp = t_0;
	elseif (im <= -8.5e-287)
		tmp = sqrt(re);
	elseif (im <= 1.05e-264)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (im <= 5.5e-153)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * -2.0));
	tmp = 0.0;
	if (im <= -4.5e+16)
		tmp = t_0;
	elseif (im <= -1.55e-38)
		tmp = sqrt(re);
	elseif (im <= -1.6e-120)
		tmp = t_0;
	elseif (im <= -8.5e-287)
		tmp = sqrt(re);
	elseif (im <= 1.05e-264)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (im <= 5.5e-153)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -4.5e+16], t$95$0, If[LessEqual[im, -1.55e-38], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, -1.6e-120], t$95$0, If[LessEqual[im, -8.5e-287], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 1.05e-264], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.5e-153], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -4.5e16 or -1.54999999999999991e-38 < im < -1.6e-120

   1. Initial program 38.5%

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

      1. +-commutative 38.5%

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

      2. hypot-def 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in im around -inf 66.3%

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

      1. *-commutative 66.3%

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

   6. Simplified 66.3%

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

   if -4.5e16 < im < -1.54999999999999991e-38 or -1.6e-120 < im < -8.5000000000000003e-287 or 1.0500000000000001e-264 < im < 5.49999999999999962e-153

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      2. hypot-def 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in im around 0 56.9%

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

      1. associate-*r* 56.9%

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

      2. unpow2 56.9%

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

      3. rem-square-sqrt 58.1%

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

      4. metadata-eval 58.1%

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

      5. *-lft-identity 58.1%

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

   6. Simplified 58.1%

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

   if -8.5000000000000003e-287 < im < 1.0500000000000001e-264

   1. Initial program 12.6%

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

      1. +-commutative 12.6%

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

      2. hypot-def 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in re around -inf 70.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

      3. unpow2 70.5%

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

      4. distribute-rgt-neg-in 70.5%

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

   6. Simplified 70.5%

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

      1. frac-2neg 70.5%

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

      2. sqrt-div 70.1%

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

      3. distribute-rgt-neg-out 70.1%

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

      4. remove-double-neg 70.1%

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

      5. sqrt-unprod 79.6%

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

      6. add-sqr-sqrt 80.6%

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

   8. Applied egg-rr 80.6%

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

   if 5.49999999999999962e-153 < im

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. hypot-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in re around 0 65.9%

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

      1. *-commutative 65.9%

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

   6. Simplified 65.9%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -4.5 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -1.55 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -1.6 \cdot 10^{-120}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -8.5 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{-264}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 5: 59.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.16 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -8 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.16e-121)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im -8e-287)
     (sqrt re)
     (if (<= im 1.5e-265)
       (* 0.5 (/ im (sqrt (- re))))
       (if (<= im 5.8e-153) (sqrt re) (* 0.5 (sqrt (* 2.0 (+ re im)))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.16e-121) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= -8e-287) {
		tmp = sqrt(re);
	} else if (im <= 1.5e-265) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (im <= 5.8e-153) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + 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 <= (-1.16d-121)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= (-8d-287)) then
        tmp = sqrt(re)
    else if (im <= 1.5d-265) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (im <= 5.8d-153) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.16e-121) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= -8e-287) {
		tmp = Math.sqrt(re);
	} else if (im <= 1.5e-265) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (im <= 5.8e-153) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im)));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.16e-121:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= -8e-287:
		tmp = math.sqrt(re)
	elif im <= 1.5e-265:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif im <= 5.8e-153:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im)))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.16e-121)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= -8e-287)
		tmp = sqrt(re);
	elseif (im <= 1.5e-265)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (im <= 5.8e-153)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.16e-121)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= -8e-287)
		tmp = sqrt(re);
	elseif (im <= 1.5e-265)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (im <= 5.8e-153)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.16e-121], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -8e-287], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 1.5e-265], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.8e-153], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.1600000000000001e-121

   1. Initial program 42.6%

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

      1. +-commutative 42.6%

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

      2. hypot-def 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in im around -inf 65.0%

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

      1. mul-1-neg 65.0%

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

      2. sub-neg 65.0%

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

   6. Simplified 65.0%

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

   if -1.1600000000000001e-121 < im < -8.00000000000000017e-287 or 1.4999999999999999e-265 < im < 5.80000000000000004e-153

   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. +-commutative 42.3%

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

      2. hypot-def 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in im around 0 55.7%

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

      1. associate-*r* 55.7%

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

      2. unpow2 55.7%

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

      3. rem-square-sqrt 56.8%

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

      4. metadata-eval 56.8%

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

      5. *-lft-identity 56.8%

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

   6. Simplified 56.8%

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

   if -8.00000000000000017e-287 < im < 1.4999999999999999e-265

   1. Initial program 12.6%

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

      1. +-commutative 12.6%

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

      2. hypot-def 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in re around -inf 70.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

      3. unpow2 70.5%

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

      4. distribute-rgt-neg-in 70.5%

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

   6. Simplified 70.5%

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

      1. frac-2neg 70.5%

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

      2. sqrt-div 70.1%

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

      3. distribute-rgt-neg-out 70.1%

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

      4. remove-double-neg 70.1%

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

      5. sqrt-unprod 79.6%

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

      6. add-sqr-sqrt 80.6%

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

   8. Applied egg-rr 80.6%

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

   if 5.80000000000000004e-153 < im

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. hypot-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in re around 0 66.4%

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

      1. distribute-lft-out 66.4%

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

      2. +-commutative 66.4%

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

      3. *-commutative 66.4%

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

      4. +-commutative 66.4%

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

   6. Simplified 66.4%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(im + re\right) \cdot 2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.16 \cdot 10^{-121}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -8 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 1.5 \cdot 10^{-265}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 6: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{if}\;im \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (sqrt (* im -2.0)))))
   (if (<= im -5.2e+17)
     t_0
     (if (<= im -6e-41)
       (sqrt re)
       (if (<= im -5.5e-122)
         t_0
         (if (<= im 5.8e-153) (sqrt re) (* 0.5 (sqrt (* im 2.0)))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * sqrt((im * -2.0));
	double tmp;
	if (im <= -5.2e+17) {
		tmp = t_0;
	} else if (im <= -6e-41) {
		tmp = sqrt(re);
	} else if (im <= -5.5e-122) {
		tmp = t_0;
	} else if (im <= 5.8e-153) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * sqrt((im * (-2.0d0)))
    if (im <= (-5.2d+17)) then
        tmp = t_0
    else if (im <= (-6d-41)) then
        tmp = sqrt(re)
    else if (im <= (-5.5d-122)) then
        tmp = t_0
    else if (im <= 5.8d-153) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.sqrt((im * -2.0));
	double tmp;
	if (im <= -5.2e+17) {
		tmp = t_0;
	} else if (im <= -6e-41) {
		tmp = Math.sqrt(re);
	} else if (im <= -5.5e-122) {
		tmp = t_0;
	} else if (im <= 5.8e-153) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.sqrt((im * -2.0))
	tmp = 0
	if im <= -5.2e+17:
		tmp = t_0
	elif im <= -6e-41:
		tmp = math.sqrt(re)
	elif im <= -5.5e-122:
		tmp = t_0
	elif im <= 5.8e-153:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * sqrt(Float64(im * -2.0)))
	tmp = 0.0
	if (im <= -5.2e+17)
		tmp = t_0;
	elseif (im <= -6e-41)
		tmp = sqrt(re);
	elseif (im <= -5.5e-122)
		tmp = t_0;
	elseif (im <= 5.8e-153)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * sqrt((im * -2.0));
	tmp = 0.0;
	if (im <= -5.2e+17)
		tmp = t_0;
	elseif (im <= -6e-41)
		tmp = sqrt(re);
	elseif (im <= -5.5e-122)
		tmp = t_0;
	elseif (im <= 5.8e-153)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -5.2e+17], t$95$0, If[LessEqual[im, -6e-41], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, -5.5e-122], t$95$0, If[LessEqual[im, 5.8e-153], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.2e17 or -5.99999999999999978e-41 < im < -5.50000000000000053e-122

   1. Initial program 38.5%

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

      1. +-commutative 38.5%

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

      2. hypot-def 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in im around -inf 66.3%

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

      1. *-commutative 66.3%

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

   6. Simplified 66.3%

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

   if -5.2e17 < im < -5.99999999999999978e-41 or -5.50000000000000053e-122 < im < 5.80000000000000004e-153

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. hypot-def 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in im around 0 50.7%

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

      1. associate-*r* 50.7%

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

      2. unpow2 50.7%

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

      3. rem-square-sqrt 51.7%

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

      4. metadata-eval 51.7%

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

      5. *-lft-identity 51.7%

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

   6. Simplified 51.7%

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

   if 5.80000000000000004e-153 < im

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. hypot-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in re around 0 65.9%

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

      1. *-commutative 65.9%

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

   6. Simplified 65.9%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq -6 \cdot 10^{-41}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq -5.5 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 7: 59.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -8 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2.35e-122)
   (* 0.5 (sqrt (* 2.0 (- re im))))
   (if (<= im -8e-287)
     (sqrt re)
     (if (<= im 2.3e-263)
       (* 0.5 (/ im (sqrt (- re))))
       (if (<= im 5.8e-153) (sqrt re) (* 0.5 (sqrt (* im 2.0))))))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2.35e-122) {
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	} else if (im <= -8e-287) {
		tmp = sqrt(re);
	} else if (im <= 2.3e-263) {
		tmp = 0.5 * (im / sqrt(-re));
	} else if (im <= 5.8e-153) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((im * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-2.35d-122)) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re - im)))
    else if (im <= (-8d-287)) then
        tmp = sqrt(re)
    else if (im <= 2.3d-263) then
        tmp = 0.5d0 * (im / sqrt(-re))
    else if (im <= 5.8d-153) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((im * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2.35e-122) {
		tmp = 0.5 * Math.sqrt((2.0 * (re - im)));
	} else if (im <= -8e-287) {
		tmp = Math.sqrt(re);
	} else if (im <= 2.3e-263) {
		tmp = 0.5 * (im / Math.sqrt(-re));
	} else if (im <= 5.8e-153) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((im * 2.0));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2.35e-122:
		tmp = 0.5 * math.sqrt((2.0 * (re - im)))
	elif im <= -8e-287:
		tmp = math.sqrt(re)
	elif im <= 2.3e-263:
		tmp = 0.5 * (im / math.sqrt(-re))
	elif im <= 5.8e-153:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((im * 2.0))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2.35e-122)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re - im))));
	elseif (im <= -8e-287)
		tmp = sqrt(re);
	elseif (im <= 2.3e-263)
		tmp = Float64(0.5 * Float64(im / sqrt(Float64(-re))));
	elseif (im <= 5.8e-153)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(im * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2.35e-122)
		tmp = 0.5 * sqrt((2.0 * (re - im)));
	elseif (im <= -8e-287)
		tmp = sqrt(re);
	elseif (im <= 2.3e-263)
		tmp = 0.5 * (im / sqrt(-re));
	elseif (im <= 5.8e-153)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((im * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2.35e-122], N[(0.5 * N[Sqrt[N[(2.0 * N[(re - im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -8e-287], N[Sqrt[re], $MachinePrecision], If[LessEqual[im, 2.3e-263], N[(0.5 * N[(im / N[Sqrt[(-re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 5.8e-153], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(im * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.35e-122

   1. Initial program 42.6%

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

      1. +-commutative 42.6%

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

      2. hypot-def 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in im around -inf 65.0%

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

      1. mul-1-neg 65.0%

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

      2. sub-neg 65.0%

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

   6. Simplified 65.0%

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

   if -2.35e-122 < im < -8.00000000000000017e-287 or 2.30000000000000003e-263 < im < 5.80000000000000004e-153

   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. +-commutative 42.3%

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

      2. hypot-def 73.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in im around 0 55.7%

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

      1. associate-*r* 55.7%

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

      2. unpow2 55.7%

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

      3. rem-square-sqrt 56.8%

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

      4. metadata-eval 56.8%

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

      5. *-lft-identity 56.8%

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

   6. Simplified 56.8%

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

   if -8.00000000000000017e-287 < im < 2.30000000000000003e-263

   1. Initial program 12.6%

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

      1. +-commutative 12.6%

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

      2. hypot-def 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in re around -inf 70.5%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \frac{{im}^{2}}{re}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

      3. unpow2 70.5%

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

      4. distribute-rgt-neg-in 70.5%

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

   6. Simplified 70.5%

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

      1. frac-2neg 70.5%

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

      2. sqrt-div 70.1%

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

      3. distribute-rgt-neg-out 70.1%

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

      4. remove-double-neg 70.1%

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

      5. sqrt-unprod 79.6%

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

      6. add-sqr-sqrt 80.6%

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

   8. Applied egg-rr 80.6%

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

   if 5.80000000000000004e-153 < im

   1. Initial program 47.0%

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

      1. +-commutative 47.0%

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

      2. hypot-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in re around 0 65.9%

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

      1. *-commutative 65.9%

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

   6. Simplified 65.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.35 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re - im\right)}\\ \mathbf{elif}\;im \leq -8 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{elif}\;im \leq 2.3 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot \frac{im}{\sqrt{-re}}\\ \mathbf{elif}\;im \leq 5.8 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot 2}\\ \end{array} \]

Alternative 8: 42.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq 4.5 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= re 4.5e-96) (* 0.5 (sqrt (* im -2.0))) (sqrt re)))

C

double code(double re, double im) {
	double tmp;
	if (re <= 4.5e-96) {
		tmp = 0.5 * sqrt((im * -2.0));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (re <= 4.5d-96) then
        tmp = 0.5d0 * sqrt((im * (-2.0d0)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (re <= 4.5e-96) {
		tmp = 0.5 * Math.sqrt((im * -2.0));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if re <= 4.5e-96:
		tmp = 0.5 * math.sqrt((im * -2.0))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (re <= 4.5e-96)
		tmp = Float64(0.5 * sqrt(Float64(im * -2.0)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (re <= 4.5e-96)
		tmp = 0.5 * sqrt((im * -2.0));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[re, 4.5e-96], N[(0.5 * N[Sqrt[N[(im * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.5e-96

   1. Initial program 42.4%

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

      1. +-commutative 42.4%

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

      2. hypot-def 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in im around -inf 28.0%

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

      1. *-commutative 28.0%

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

   6. Simplified 28.0%

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

   if 4.5e-96 < re

   1. Initial program 44.4%

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

      1. +-commutative 44.4%

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

      2. hypot-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in im around 0 71.4%

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

      1. associate-*r* 71.4%

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

      2. unpow2 71.4%

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

      3. rem-square-sqrt 72.8%

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

      4. metadata-eval 72.8%

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

      5. *-lft-identity 72.8%

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

   6. Simplified 72.8%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.5 \cdot 10^{-96}:\\ \;\;\;\;0.5 \cdot \sqrt{im \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 9: 26.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{re} \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (sqrt re))

C

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

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = sqrt(re)
end function

Java

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

Python

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

Julia

function code(re, im)
	return sqrt(re)
end

MATLAB

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

Wolfram

code[re_, im_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 43.1%

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

   1. +-commutative 43.1%

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

   2. hypot-def 81.7%

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

3. Simplified 81.7%

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

4. Taylor expanded in im around 0 28.1%

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

   1. associate-*r* 28.1%

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

   2. unpow2 28.1%

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

   3. rem-square-sqrt 28.6%

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

   4. metadata-eval 28.6%

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

   5. *-lft-identity 28.6%

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

6. Simplified 28.6%

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

7. Final simplification 28.6%

   \[\leadsto \sqrt{re} \]

Developer target: 48.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{re \cdot re + im \cdot im}\\ \mathbf{if}\;re < 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{t_0 - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(t_0 + re\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* re re) (* im im)))))
   (if (< re 0.0)
     (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- t_0 re)))))
     (* 0.5 (sqrt (* 2.0 (+ t_0 re)))))))

C

double code(double re, double im) {
	double t_0 = sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((re * re) + (im * im)))
    if (re < 0.0d0) then
        tmp = 0.5d0 * (sqrt(2.0d0) * sqrt(((im * im) / (t_0 - re))))
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (t_0 + re)))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = Math.sqrt(((re * re) + (im * im)));
	double tmp;
	if (re < 0.0) {
		tmp = 0.5 * (Math.sqrt(2.0) * Math.sqrt(((im * im) / (t_0 - re))));
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (t_0 + re)));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.sqrt(((re * re) + (im * im)))
	tmp = 0
	if re < 0.0:
		tmp = 0.5 * (math.sqrt(2.0) * math.sqrt(((im * im) / (t_0 - re))))
	else:
		tmp = 0.5 * math.sqrt((2.0 * (t_0 + re)))
	return tmp

Julia

function code(re, im)
	t_0 = sqrt(Float64(Float64(re * re) + Float64(im * im)))
	tmp = 0.0
	if (re < 0.0)
		tmp = Float64(0.5 * Float64(sqrt(2.0) * sqrt(Float64(Float64(im * im) / Float64(t_0 - re)))));
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(t_0 + re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = sqrt(((re * re) + (im * im)));
	tmp = 0.0;
	if (re < 0.0)
		tmp = 0.5 * (sqrt(2.0) * sqrt(((im * im) / (t_0 - re))));
	else
		tmp = 0.5 * sqrt((2.0 * (t_0 + re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im * im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[re, 0.0], N[(0.5 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(im * im), $MachinePrecision] / N[(t$95$0 - re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(t$95$0 + re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023178 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))