math.sqrt on complex, real part

Percentage Accurate: 41.8% → 86.8%

Time: 8.4s

Alternatives: 8

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: 41.8% 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: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{-0.5}{re}}\\ \mathbf{if}\;re \leq -3.5 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \left({2}^{0.25} \cdot \left(\left({2}^{0.25} \cdot im_m\right) \cdot t_0\right)\right)\\ \mathbf{elif}\;re \leq -8 \cdot 10^{-6} \lor \neg \left(re \leq -3.5 \cdot 10^{-50}\right):\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im_m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(im_m \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ -0.5 re))))
   (if (<= re -3.5e+105)
     (* 0.5 (* (pow 2.0 0.25) (* (* (pow 2.0 0.25) im_m) t_0)))
     (if (or (<= re -8e-6) (not (<= re -3.5e-50)))
       (sqrt (* 0.5 (+ re (hypot re im_m))))
       (* 0.5 (* (pow 2.0 0.25) (* (pow 2.0 0.25) (* im_m t_0))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double t_0 = sqrt((-0.5 / re));
	double tmp;
	if (re <= -3.5e+105) {
		tmp = 0.5 * (pow(2.0, 0.25) * ((pow(2.0, 0.25) * im_m) * t_0));
	} else if ((re <= -8e-6) || !(re <= -3.5e-50)) {
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	} else {
		tmp = 0.5 * (pow(2.0, 0.25) * (pow(2.0, 0.25) * (im_m * t_0)));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double t_0 = Math.sqrt((-0.5 / re));
	double tmp;
	if (re <= -3.5e+105) {
		tmp = 0.5 * (Math.pow(2.0, 0.25) * ((Math.pow(2.0, 0.25) * im_m) * t_0));
	} else if ((re <= -8e-6) || !(re <= -3.5e-50)) {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im_m))));
	} else {
		tmp = 0.5 * (Math.pow(2.0, 0.25) * (Math.pow(2.0, 0.25) * (im_m * t_0)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	t_0 = math.sqrt((-0.5 / re))
	tmp = 0
	if re <= -3.5e+105:
		tmp = 0.5 * (math.pow(2.0, 0.25) * ((math.pow(2.0, 0.25) * im_m) * t_0))
	elif (re <= -8e-6) or not (re <= -3.5e-50):
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im_m))))
	else:
		tmp = 0.5 * (math.pow(2.0, 0.25) * (math.pow(2.0, 0.25) * (im_m * t_0)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	t_0 = sqrt(Float64(-0.5 / re))
	tmp = 0.0
	if (re <= -3.5e+105)
		tmp = Float64(0.5 * Float64((2.0 ^ 0.25) * Float64(Float64((2.0 ^ 0.25) * im_m) * t_0)));
	elseif ((re <= -8e-6) || !(re <= -3.5e-50))
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	else
		tmp = Float64(0.5 * Float64((2.0 ^ 0.25) * Float64((2.0 ^ 0.25) * Float64(im_m * t_0))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	t_0 = sqrt((-0.5 / re));
	tmp = 0.0;
	if (re <= -3.5e+105)
		tmp = 0.5 * ((2.0 ^ 0.25) * (((2.0 ^ 0.25) * im_m) * t_0));
	elseif ((re <= -8e-6) || ~((re <= -3.5e-50)))
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	else
		tmp = 0.5 * ((2.0 ^ 0.25) * ((2.0 ^ 0.25) * (im_m * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := Block[{t$95$0 = N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[re, -3.5e+105], N[(0.5 * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[(N[Power[2.0, 0.25], $MachinePrecision] * im$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[re, -8e-6], N[Not[LessEqual[re, -3.5e-50]], $MachinePrecision]], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(im$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if re < -3.49999999999999991e105

   1. Initial program 2.7%

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

      1. sqr-neg 2.7%

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

      2. +-commutative 2.7%

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

      3. sqr-neg 2.7%

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

      4. +-commutative 2.7%

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

      5. distribute-rgt-in 2.7%

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

      6. cancel-sign-sub 2.7%

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

      7. distribute-rgt-out-- 2.7%

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

      8. sub-neg 2.7%

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

      9. remove-double-neg 2.7%

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

      10. +-commutative 2.7%

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

      11. hypot-def 16.2%

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

   3. Simplified 16.2%

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

   4. Taylor expanded in re around -inf 41.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}} \]

   5. Taylor expanded in im around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. associate-*l/ 44.7%

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

      3. *-commutative 44.7%

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

   7. Simplified 44.7%

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

      1. sqrt-prod 44.6%

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

      2. add-sqr-sqrt 44.5%

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

      3. *-commutative 44.5%

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

      4. associate-*l/ 44.5%

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

      5. associate-*r/ 44.5%

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

      6. associate-*l* 44.7%

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

      7. pow1/2 44.7%

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

      8. sqrt-pow1 44.7%

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

      9. metadata-eval 44.7%

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

      10. pow1/2 44.7%

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

      11. sqrt-pow1 44.7%

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

      12. metadata-eval 44.7%

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

      13. associate-*r/ 44.7%

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

      14. associate-*l/ 44.5%

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

      15. *-commutative 44.5%

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

      16. sqrt-prod 63.6%

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

      17. unpow2 63.6%

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

      18. sqrt-prod 56.4%

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

      19. add-sqr-sqrt 60.9%

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

   9. Applied egg-rr 60.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(im \cdot \sqrt{\frac{-0.5}{re}}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 61.0%

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

   11. Simplified 61.0%

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

   if -3.49999999999999991e105 < re < -7.99999999999999964e-6 or -3.49999999999999997e-50 < re

   1. Initial program 56.7%

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

      1. sqr-neg 56.7%

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

      2. +-commutative 56.7%

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

      3. sqr-neg 56.7%

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

      4. +-commutative 56.7%

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

      5. distribute-rgt-in 56.7%

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

      6. cancel-sign-sub 56.7%

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

      7. distribute-rgt-out-- 56.7%

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

      8. sub-neg 56.7%

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

      9. remove-double-neg 56.7%

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

      10. +-commutative 56.7%

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

      11. hypot-def 94.8%

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

   3. Simplified 94.8%

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

      1. add-sqr-sqrt 94.1%

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

      2. sqrt-unprod 94.8%

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

      3. *-commutative 94.8%

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

      4. *-commutative 94.8%

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

      5. swap-sqr 94.8%

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

      6. add-sqr-sqrt 94.8%

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

      7. metadata-eval 94.8%

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

   5. Applied egg-rr 94.8%

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

      1. *-commutative 94.8%

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

      2. associate-*r* 94.8%

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

      3. metadata-eval 94.8%

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

   7. Simplified 94.8%

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

   if -7.99999999999999964e-6 < re < -3.49999999999999997e-50

   1. Initial program 21.2%

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

      1. sqr-neg 21.2%

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

      2. +-commutative 21.2%

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

      3. sqr-neg 21.2%

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

      4. +-commutative 21.2%

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

      5. distribute-rgt-in 21.2%

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

      6. cancel-sign-sub 21.2%

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

      7. distribute-rgt-out-- 21.2%

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

      8. sub-neg 21.2%

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

      9. remove-double-neg 21.2%

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

      10. +-commutative 21.2%

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

      11. hypot-def 34.0%

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

   3. Simplified 34.0%

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

   4. Taylor expanded in re around -inf 36.1%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}} \]

   5. Taylor expanded in im around 0 35.9%

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

      1. associate-*r/ 35.9%

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

      2. associate-*l/ 35.9%

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

      3. *-commutative 35.9%

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

   7. Simplified 35.9%

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

      1. sqrt-prod 35.9%

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

      2. add-sqr-sqrt 35.9%

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

      3. *-commutative 35.9%

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

      4. associate-*l/ 35.7%

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

      5. associate-*r/ 35.7%

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

      6. associate-*l* 35.8%

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

      7. pow1/2 35.8%

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

      8. sqrt-pow1 35.8%

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

      9. metadata-eval 35.8%

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

      10. pow1/2 35.8%

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

      11. sqrt-pow1 35.8%

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

      12. metadata-eval 35.8%

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

      13. associate-*r/ 35.8%

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

      14. associate-*l/ 35.9%

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

      15. *-commutative 35.9%

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

      16. sqrt-prod 36.0%

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

      17. unpow2 36.0%

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

      18. sqrt-prod 33.8%

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

      19. add-sqr-sqrt 35.5%

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

   9. Applied egg-rr 35.5%

      \[\leadsto 0.5 \cdot \color{blue}{\left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(im \cdot \sqrt{\frac{-0.5}{re}}\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -3.5 \cdot 10^{+105}:\\ \;\;\;\;0.5 \cdot \left({2}^{0.25} \cdot \left(\left({2}^{0.25} \cdot im\right) \cdot \sqrt{\frac{-0.5}{re}}\right)\right)\\ \mathbf{elif}\;re \leq -8 \cdot 10^{-6} \lor \neg \left(re \leq -3.5 \cdot 10^{-50}\right):\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(im \cdot \sqrt{\frac{-0.5}{re}}\right)\right)\right)\\ \end{array} \]

Alternative 2: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+103} \lor \neg \left(re \leq -7.6 \cdot 10^{-7}\right) \land re \leq -1.1 \cdot 10^{-47}:\\ \;\;\;\;0.5 \cdot \left({2}^{0.25} \cdot \left(\left({2}^{0.25} \cdot im_m\right) \cdot \sqrt{\frac{-0.5}{re}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (or (<= re -5.5e+103) (and (not (<= re -7.6e-7)) (<= re -1.1e-47)))
   (* 0.5 (* (pow 2.0 0.25) (* (* (pow 2.0 0.25) im_m) (sqrt (/ -0.5 re)))))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re <= -5.5e+103) || (!(re <= -7.6e-7) && (re <= -1.1e-47))) {
		tmp = 0.5 * (pow(2.0, 0.25) * ((pow(2.0, 0.25) * im_m) * sqrt((-0.5 / re))));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((re <= -5.5e+103) || (!(re <= -7.6e-7) && (re <= -1.1e-47))) {
		tmp = 0.5 * (Math.pow(2.0, 0.25) * ((Math.pow(2.0, 0.25) * im_m) * Math.sqrt((-0.5 / re))));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re <= -5.5e+103) or (not (re <= -7.6e-7) and (re <= -1.1e-47)):
		tmp = 0.5 * (math.pow(2.0, 0.25) * ((math.pow(2.0, 0.25) * im_m) * math.sqrt((-0.5 / re))))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if ((re <= -5.5e+103) || (!(re <= -7.6e-7) && (re <= -1.1e-47)))
		tmp = Float64(0.5 * Float64((2.0 ^ 0.25) * Float64(Float64((2.0 ^ 0.25) * im_m) * sqrt(Float64(-0.5 / re)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((re <= -5.5e+103) || (~((re <= -7.6e-7)) && (re <= -1.1e-47)))
		tmp = 0.5 * ((2.0 ^ 0.25) * (((2.0 ^ 0.25) * im_m) * sqrt((-0.5 / re))));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[Or[LessEqual[re, -5.5e+103], And[N[Not[LessEqual[re, -7.6e-7]], $MachinePrecision], LessEqual[re, -1.1e-47]]], N[(0.5 * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[(N[Power[2.0, 0.25], $MachinePrecision] * im$95$m), $MachinePrecision] * N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.50000000000000001e103 or -7.60000000000000029e-7 < re < -1.10000000000000009e-47

   1. Initial program 8.0%

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

      1. sqr-neg 8.0%

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

      2. +-commutative 8.0%

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

      3. sqr-neg 8.0%

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

      4. +-commutative 8.0%

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

      5. distribute-rgt-in 8.0%

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

      6. cancel-sign-sub 8.0%

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

      7. distribute-rgt-out-- 8.0%

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

      8. sub-neg 8.0%

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

      9. remove-double-neg 8.0%

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

      10. +-commutative 8.0%

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

      11. hypot-def 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in re around -inf 40.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}} \]

   5. Taylor expanded in im around 0 42.2%

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

      1. associate-*r/ 42.2%

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

      2. associate-*l/ 42.2%

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

      3. *-commutative 42.2%

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

   7. Simplified 42.2%

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

      1. sqrt-prod 42.1%

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

      2. add-sqr-sqrt 42.0%

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

      3. *-commutative 42.0%

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

      4. associate-*l/ 42.0%

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

      5. associate-*r/ 42.0%

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

      6. associate-*l* 42.1%

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

      7. pow1/2 42.1%

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

      8. sqrt-pow1 42.1%

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

      9. metadata-eval 42.1%

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

      10. pow1/2 42.1%

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

      11. sqrt-pow1 42.1%

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

      12. metadata-eval 42.1%

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

      13. associate-*r/ 42.1%

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

      14. associate-*l/ 42.0%

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

      15. *-commutative 42.0%

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

      16. sqrt-prod 55.7%

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

      17. unpow2 55.7%

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

      18. sqrt-prod 49.8%

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

      19. add-sqr-sqrt 53.6%

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

   9. Applied egg-rr 53.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \left(im \cdot \sqrt{\frac{-0.5}{re}}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 53.6%

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

   11. Simplified 53.6%

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

   if -5.50000000000000001e103 < re < -7.60000000000000029e-7 or -1.10000000000000009e-47 < re

   1. Initial program 56.7%

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

      1. sqr-neg 56.7%

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

      2. +-commutative 56.7%

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

      3. sqr-neg 56.7%

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

      4. +-commutative 56.7%

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

      5. distribute-rgt-in 56.7%

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

      6. cancel-sign-sub 56.7%

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

      7. distribute-rgt-out-- 56.7%

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

      8. sub-neg 56.7%

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

      9. remove-double-neg 56.7%

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

      10. +-commutative 56.7%

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

      11. hypot-def 94.8%

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

   3. Simplified 94.8%

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

      1. add-sqr-sqrt 94.1%

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

      2. sqrt-unprod 94.8%

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

      3. *-commutative 94.8%

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

      4. *-commutative 94.8%

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

      5. swap-sqr 94.8%

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

      6. add-sqr-sqrt 94.8%

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

      7. metadata-eval 94.8%

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

   5. Applied egg-rr 94.8%

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

      1. *-commutative 94.8%

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

      2. associate-*r* 94.8%

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

      3. metadata-eval 94.8%

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

   7. Simplified 94.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.5 \cdot 10^{+103} \lor \neg \left(re \leq -7.6 \cdot 10^{-7}\right) \land re \leq -1.1 \cdot 10^{-47}:\\ \;\;\;\;0.5 \cdot \left({2}^{0.25} \cdot \left(\left({2}^{0.25} \cdot im\right) \cdot \sqrt{\frac{-0.5}{re}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 3: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{+104} \lor \neg \left(re \leq -9.4 \cdot 10^{-7}\right) \land re \leq -2.45 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{-0.5}{re}} \cdot \left(im_m \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (or (<= re -4.4e+104) (and (not (<= re -9.4e-7)) (<= re -2.45e-49)))
   (* 0.5 (* (sqrt (/ -0.5 re)) (* im_m (sqrt 2.0))))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re <= -4.4e+104) || (!(re <= -9.4e-7) && (re <= -2.45e-49))) {
		tmp = 0.5 * (sqrt((-0.5 / re)) * (im_m * sqrt(2.0)));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if ((re <= -4.4e+104) || (!(re <= -9.4e-7) && (re <= -2.45e-49))) {
		tmp = 0.5 * (Math.sqrt((-0.5 / re)) * (im_m * Math.sqrt(2.0)));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re <= -4.4e+104) or (not (re <= -9.4e-7) and (re <= -2.45e-49)):
		tmp = 0.5 * (math.sqrt((-0.5 / re)) * (im_m * math.sqrt(2.0)))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if ((re <= -4.4e+104) || (!(re <= -9.4e-7) && (re <= -2.45e-49)))
		tmp = Float64(0.5 * Float64(sqrt(Float64(-0.5 / re)) * Float64(im_m * sqrt(2.0))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if ((re <= -4.4e+104) || (~((re <= -9.4e-7)) && (re <= -2.45e-49)))
		tmp = 0.5 * (sqrt((-0.5 / re)) * (im_m * sqrt(2.0)));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[Or[LessEqual[re, -4.4e+104], And[N[Not[LessEqual[re, -9.4e-7]], $MachinePrecision], LessEqual[re, -2.45e-49]]], N[(0.5 * N[(N[Sqrt[N[(-0.5 / re), $MachinePrecision]], $MachinePrecision] * N[(im$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -4.40000000000000001e104 or -9.4e-7 < re < -2.4500000000000001e-49

   1. Initial program 8.0%

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

      1. sqr-neg 8.0%

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

      2. +-commutative 8.0%

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

      3. sqr-neg 8.0%

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

      4. +-commutative 8.0%

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

      5. distribute-rgt-in 8.0%

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

      6. cancel-sign-sub 8.0%

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

      7. distribute-rgt-out-- 8.0%

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

      8. sub-neg 8.0%

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

      9. remove-double-neg 8.0%

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

      10. +-commutative 8.0%

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

      11. hypot-def 21.3%

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

   3. Simplified 21.3%

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

   4. Taylor expanded in re around -inf 40.0%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}} \]

   5. Taylor expanded in im around 0 42.2%

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

      1. associate-*r/ 42.2%

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

      2. associate-*l/ 42.2%

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

      3. *-commutative 42.2%

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

   7. Simplified 42.2%

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

      1. *-commutative 42.2%

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

      2. *-commutative 42.2%

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

      3. associate-*l/ 42.2%

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

      4. associate-*r/ 42.2%

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

      5. sqrt-unprod 41.9%

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

      6. associate-*r/ 41.9%

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

      7. associate-*l/ 42.1%

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

      8. *-commutative 42.1%

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

      9. sqrt-prod 55.6%

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

      10. unpow2 55.6%

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

      11. sqrt-prod 49.8%

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

      12. add-sqr-sqrt 53.6%

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

   9. Applied egg-rr 53.6%

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

       1. *-commutative 53.6%

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

       2. associate-*r* 53.6%

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

   11. Simplified 53.6%

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

   if -4.40000000000000001e104 < re < -9.4e-7 or -2.4500000000000001e-49 < re

   1. Initial program 56.7%

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

      1. sqr-neg 56.7%

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

      2. +-commutative 56.7%

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

      3. sqr-neg 56.7%

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

      4. +-commutative 56.7%

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

      5. distribute-rgt-in 56.7%

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

      6. cancel-sign-sub 56.7%

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

      7. distribute-rgt-out-- 56.7%

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

      8. sub-neg 56.7%

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

      9. remove-double-neg 56.7%

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

      10. +-commutative 56.7%

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

      11. hypot-def 94.8%

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

   3. Simplified 94.8%

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

      1. add-sqr-sqrt 94.1%

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

      2. sqrt-unprod 94.8%

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

      3. *-commutative 94.8%

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

      4. *-commutative 94.8%

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

      5. swap-sqr 94.8%

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

      6. add-sqr-sqrt 94.8%

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

      7. metadata-eval 94.8%

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

   5. Applied egg-rr 94.8%

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

      1. *-commutative 94.8%

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

      2. associate-*r* 94.8%

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

      3. metadata-eval 94.8%

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

   7. Simplified 94.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.4 \cdot 10^{+104} \lor \neg \left(re \leq -9.4 \cdot 10^{-7}\right) \land re \leq -2.45 \cdot 10^{-49}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{-0.5}{re}} \cdot \left(im \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 4: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im_m \cdot -0.5}{\frac{re}{im_m}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -1.7e+119)
   (* 0.5 (sqrt (* 2.0 (/ (* im_m -0.5) (/ re im_m)))))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -1.7e+119) {
		tmp = 0.5 * sqrt((2.0 * ((im_m * -0.5) / (re / im_m))));
	} else {
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	}
	return tmp;
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -1.7e+119) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im_m * -0.5) / (re / im_m))));
	} else {
		tmp = Math.sqrt((0.5 * (re + Math.hypot(re, im_m))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -1.7e+119:
		tmp = 0.5 * math.sqrt((2.0 * ((im_m * -0.5) / (re / im_m))))
	else:
		tmp = math.sqrt((0.5 * (re + math.hypot(re, im_m))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -1.7e+119)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im_m * -0.5) / Float64(re / im_m)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -1.7e+119)
		tmp = 0.5 * sqrt((2.0 * ((im_m * -0.5) / (re / im_m))));
	else
		tmp = sqrt((0.5 * (re + hypot(re, im_m))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -1.7e+119], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im$95$m * -0.5), $MachinePrecision] / N[(re / im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -1.70000000000000007e119

   1. Initial program 2.6%

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

      1. sqr-neg 2.6%

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

      2. +-commutative 2.6%

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

      3. sqr-neg 2.6%

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

      4. +-commutative 2.6%

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

      5. distribute-rgt-in 2.6%

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

      6. cancel-sign-sub 2.6%

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

      7. distribute-rgt-out-- 2.6%

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

      8. sub-neg 2.6%

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

      9. remove-double-neg 2.6%

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

      10. +-commutative 2.6%

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

      11. hypot-def 16.4%

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

   3. Simplified 16.4%

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

   4. Taylor expanded in re around -inf 42.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}} \]

   5. Taylor expanded in im around 0 45.8%

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

      1. associate-*r/ 45.8%

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

      2. associate-*l/ 45.8%

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

      3. *-commutative 45.8%

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

   7. Simplified 45.8%

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

      1. *-commutative 45.8%

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

      2. associate-*l/ 45.8%

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

      3. unpow2 45.8%

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

      4. associate-*r* 45.8%

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

      5. associate-*r/ 53.9%

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

      6. clear-num 53.9%

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

      7. un-div-inv 54.0%

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

      8. *-commutative 54.0%

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

   9. Applied egg-rr 54.0%

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

   if -1.70000000000000007e119 < re

   1. Initial program 54.1%

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

      1. sqr-neg 54.1%

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

      2. +-commutative 54.1%

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

      3. sqr-neg 54.1%

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

      4. +-commutative 54.1%

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

      5. distribute-rgt-in 54.1%

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

      6. cancel-sign-sub 54.1%

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

      7. distribute-rgt-out-- 54.1%

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

      8. sub-neg 54.1%

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

      9. remove-double-neg 54.1%

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

      10. +-commutative 54.1%

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

      11. hypot-def 90.2%

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

   3. Simplified 90.2%

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

      1. add-sqr-sqrt 89.6%

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

      2. sqrt-unprod 90.2%

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

      3. *-commutative 90.2%

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

      4. *-commutative 90.2%

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

      5. swap-sqr 90.2%

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

      6. add-sqr-sqrt 90.2%

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

      7. metadata-eval 90.2%

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

   5. Applied egg-rr 90.2%

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

      1. *-commutative 90.2%

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

      2. associate-*r* 90.2%

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

      3. metadata-eval 90.2%

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

   7. Simplified 90.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot -0.5}{\frac{re}{im}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]

Alternative 5: 70.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2.55 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im_m \cdot -0.5}{\frac{re}{im_m}}}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2.55e+107)
   (* 0.5 (sqrt (* 2.0 (/ (* im_m -0.5) (/ re im_m)))))
   (if (<= re 9.5e-15) (* 0.5 (sqrt (* 2.0 (+ re im_m)))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2.55e+107) {
		tmp = 0.5 * sqrt((2.0 * ((im_m * -0.5) / (re / im_m))));
	} else if (re <= 9.5e-15) {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= (-2.55d+107)) then
        tmp = 0.5d0 * sqrt((2.0d0 * ((im_m * (-0.5d0)) / (re / im_m))))
    else if (re <= 9.5d-15) then
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= -2.55e+107) {
		tmp = 0.5 * Math.sqrt((2.0 * ((im_m * -0.5) / (re / im_m))));
	} else if (re <= 9.5e-15) {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -2.55e+107:
		tmp = 0.5 * math.sqrt((2.0 * ((im_m * -0.5) / (re / im_m))))
	elif re <= 9.5e-15:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -2.55e+107)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(Float64(im_m * -0.5) / Float64(re / im_m)))));
	elseif (re <= 9.5e-15)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -2.55e+107)
		tmp = 0.5 * sqrt((2.0 * ((im_m * -0.5) / (re / im_m))));
	elseif (re <= 9.5e-15)
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -2.55e+107], N[(0.5 * N[Sqrt[N[(2.0 * N[(N[(im$95$m * -0.5), $MachinePrecision] / N[(re / im$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9.5e-15], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.5500000000000001e107

   1. Initial program 2.7%

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

      1. sqr-neg 2.7%

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

      2. +-commutative 2.7%

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

      3. sqr-neg 2.7%

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

      4. +-commutative 2.7%

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

      5. distribute-rgt-in 2.7%

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

      6. cancel-sign-sub 2.7%

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

      7. distribute-rgt-out-- 2.7%

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

      8. sub-neg 2.7%

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

      9. remove-double-neg 2.7%

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

      10. +-commutative 2.7%

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

      11. hypot-def 16.2%

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

   3. Simplified 16.2%

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

   4. Taylor expanded in re around -inf 41.5%

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{im}^{2}}{re} + 0.125 \cdot \frac{{im}^{4}}{{re}^{3}}\right)}} \]

   5. Taylor expanded in im around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. associate-*l/ 44.7%

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

      3. *-commutative 44.7%

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

   7. Simplified 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-*l/ 44.8%

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

      3. unpow2 44.8%

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

      4. associate-*r* 44.8%

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

      5. associate-*r/ 52.7%

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

      6. clear-num 52.7%

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

      7. un-div-inv 52.8%

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

      8. *-commutative 52.8%

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

   9. Applied egg-rr 52.8%

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

   if -2.5500000000000001e107 < re < 9.5000000000000005e-15

   1. Initial program 54.3%

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

      1. sqr-neg 54.3%

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

      2. +-commutative 54.3%

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

      3. sqr-neg 54.3%

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

      4. +-commutative 54.3%

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

      5. distribute-rgt-in 54.3%

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

      6. cancel-sign-sub 54.3%

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

      7. distribute-rgt-out-- 54.3%

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

      8. sub-neg 54.3%

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

      9. remove-double-neg 54.3%

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

      10. +-commutative 54.3%

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

      11. hypot-def 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in re around 0 42.3%

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

   if 9.5000000000000005e-15 < re

   1. Initial program 54.3%

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

      1. sqr-neg 54.3%

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

      2. +-commutative 54.3%

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

      3. sqr-neg 54.3%

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

      4. +-commutative 54.3%

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

      5. distribute-rgt-in 54.3%

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

      6. cancel-sign-sub 54.3%

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

      7. distribute-rgt-out-- 54.3%

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

      8. sub-neg 54.3%

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

      9. remove-double-neg 54.3%

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

      10. +-commutative 54.3%

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

      11. hypot-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in im around 0 73.9%

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

      1. *-commutative 73.9%

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

      2. unpow2 73.9%

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

      3. rem-square-sqrt 75.3%

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

      4. associate-*r* 75.3%

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

      5. metadata-eval 75.3%

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

      6. *-lft-identity 75.3%

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

   6. Simplified 75.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2.55 \cdot 10^{+107}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot -0.5}{\frac{re}{im}}}\\ \mathbf{elif}\;re \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 6: 59.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;im_m \leq 1.38 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im_m\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= im_m 1.38e-42) (sqrt re) (* 0.5 (sqrt (* 2.0 (+ re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.38e-42) {
		tmp = sqrt(re);
	} else {
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (im_m <= 1.38d-42) then
        tmp = sqrt(re)
    else
        tmp = 0.5d0 * sqrt((2.0d0 * (re + im_m)))
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (im_m <= 1.38e-42) {
		tmp = Math.sqrt(re);
	} else {
		tmp = 0.5 * Math.sqrt((2.0 * (re + im_m)));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if im_m <= 1.38e-42:
		tmp = math.sqrt(re)
	else:
		tmp = 0.5 * math.sqrt((2.0 * (re + im_m)))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (im_m <= 1.38e-42)
		tmp = sqrt(re);
	else
		tmp = Float64(0.5 * sqrt(Float64(2.0 * Float64(re + im_m))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (im_m <= 1.38e-42)
		tmp = sqrt(re);
	else
		tmp = 0.5 * sqrt((2.0 * (re + im_m)));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[im$95$m, 1.38e-42], N[Sqrt[re], $MachinePrecision], N[(0.5 * N[Sqrt[N[(2.0 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < 1.37999999999999993e-42

   1. Initial program 45.1%

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

      1. sqr-neg 45.1%

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

      2. +-commutative 45.1%

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

      3. sqr-neg 45.1%

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

      4. +-commutative 45.1%

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

      5. distribute-rgt-in 45.1%

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

      6. cancel-sign-sub 45.1%

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

      7. distribute-rgt-out-- 45.1%

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

      8. sub-neg 45.1%

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

      9. remove-double-neg 45.1%

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

      10. +-commutative 45.1%

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

      11. hypot-def 74.9%

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

   3. Simplified 74.9%

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

   4. Taylor expanded in im around 0 31.3%

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

      1. *-commutative 31.3%

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

      2. unpow2 31.3%

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

      3. rem-square-sqrt 31.9%

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

      4. associate-*r* 31.9%

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

      5. metadata-eval 31.9%

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

      6. *-lft-identity 31.9%

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

   6. Simplified 31.9%

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

   if 1.37999999999999993e-42 < im

   1. Initial program 50.3%

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

      1. sqr-neg 50.3%

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

      2. +-commutative 50.3%

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

      3. sqr-neg 50.3%

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

      4. +-commutative 50.3%

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

      5. distribute-rgt-in 50.3%

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

      6. cancel-sign-sub 50.3%

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

      7. distribute-rgt-out-- 50.3%

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

      8. sub-neg 50.3%

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

      9. remove-double-neg 50.3%

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

      10. +-commutative 50.3%

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

      11. hypot-def 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in re around 0 77.3%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq 1.38 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{re}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \end{array} \]

Alternative 7: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re 4.4e-15) (* 0.5 (sqrt (* 2.0 im_m))) (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= 4.4e-15) {
		tmp = 0.5 * sqrt((2.0 * im_m));
	} else {
		tmp = sqrt(re);
	}
	return tmp;
}

Fortran

im_m = abs(im)
real(8) function code(re, im_m)
    real(8), intent (in) :: re
    real(8), intent (in) :: im_m
    real(8) :: tmp
    if (re <= 4.4d-15) then
        tmp = 0.5d0 * sqrt((2.0d0 * im_m))
    else
        tmp = sqrt(re)
    end if
    code = tmp
end function

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	double tmp;
	if (re <= 4.4e-15) {
		tmp = 0.5 * Math.sqrt((2.0 * im_m));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= 4.4e-15:
		tmp = 0.5 * math.sqrt((2.0 * im_m))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= 4.4e-15)
		tmp = Float64(0.5 * sqrt(Float64(2.0 * im_m)));
	else
		tmp = sqrt(re);
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= 4.4e-15)
		tmp = 0.5 * sqrt((2.0 * im_m));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, 4.4e-15], N[(0.5 * N[Sqrt[N[(2.0 * im$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 4.39999999999999971e-15

   1. Initial program 44.2%

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

      1. sqr-neg 44.2%

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

      2. +-commutative 44.2%

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

      3. sqr-neg 44.2%

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

      4. +-commutative 44.2%

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

      5. distribute-rgt-in 44.2%

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

      6. cancel-sign-sub 44.2%

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

      7. distribute-rgt-out-- 44.2%

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

      8. sub-neg 44.2%

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

      9. remove-double-neg 44.2%

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

      10. +-commutative 44.2%

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

      11. hypot-def 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in re around 0 33.4%

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

   if 4.39999999999999971e-15 < re

   1. Initial program 54.3%

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

      1. sqr-neg 54.3%

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

      2. +-commutative 54.3%

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

      3. sqr-neg 54.3%

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

      4. +-commutative 54.3%

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

      5. distribute-rgt-in 54.3%

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

      6. cancel-sign-sub 54.3%

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

      7. distribute-rgt-out-- 54.3%

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

      8. sub-neg 54.3%

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

      9. remove-double-neg 54.3%

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

      10. +-commutative 54.3%

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

      11. hypot-def 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in im around 0 73.9%

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

      1. *-commutative 73.9%

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

      2. unpow2 73.9%

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

      3. rem-square-sqrt 75.3%

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

      4. associate-*r* 75.3%

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

      5. metadata-eval 75.3%

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

      6. *-lft-identity 75.3%

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

   6. Simplified 75.3%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 4.4 \cdot 10^{-15}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot im}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]

Alternative 8: 25.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{re} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (sqrt re))

C

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

Fortran

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

Java

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

Python

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

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(re)
end

MATLAB

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

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[re], $MachinePrecision]

Derivation

1. Initial program 46.8%

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

   1. sqr-neg 46.8%

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

   2. +-commutative 46.8%

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

   3. sqr-neg 46.8%

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

   4. +-commutative 46.8%

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

   5. distribute-rgt-in 46.8%

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

   6. cancel-sign-sub 46.8%

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

   7. distribute-rgt-out-- 46.8%

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

   8. sub-neg 46.8%

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

   9. remove-double-neg 46.8%

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

   10. +-commutative 46.8%

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

   11. hypot-def 79.9%

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

3. Simplified 79.9%

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

4. Taylor expanded in im around 0 26.3%

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

   1. *-commutative 26.3%

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

   2. unpow2 26.3%

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

   3. rem-square-sqrt 26.7%

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

   4. associate-*r* 26.7%

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

   5. metadata-eval 26.7%

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

   6. *-lft-identity 26.7%

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

6. Simplified 26.7%

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

7. Final simplification 26.7%

   \[\leadsto \sqrt{re} \]

Developer target: 49.0% 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 2023333 
(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)))))