math.sqrt on complex, real part

Percentage Accurate: 41.4% → 89.6%

Time: 17.7s

Alternatives: 11

Speedup: 1.9×

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: 41.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im\_m \cdot im\_m} \leq 0:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im\_m}\right)\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (*
    0.5
    (pow
     (*
      (cbrt (sqrt 2.0))
      (*
       (expm1 (log1p (* (pow 0.5 0.16666666666666666) (cbrt im_m))))
       (pow (/ -1.0 re) 0.16666666666666666)))
     3.0))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * pow((cbrt(sqrt(2.0)) * (expm1(log1p((pow(0.5, 0.16666666666666666) * cbrt(im_m)))) * pow((-1.0 / re), 0.16666666666666666))), 3.0);
	} 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 + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * Math.pow((Math.cbrt(Math.sqrt(2.0)) * (Math.expm1(Math.log1p((Math.pow(0.5, 0.16666666666666666) * Math.cbrt(im_m)))) * Math.pow((-1.0 / re), 0.16666666666666666))), 3.0);
	} 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 (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(0.5 * (Float64(cbrt(sqrt(2.0)) * Float64(expm1(log1p(Float64((0.5 ^ 0.16666666666666666) * cbrt(im_m)))) * (Float64(-1.0 / re) ^ 0.16666666666666666))) ^ 3.0));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(Exp[N[Log[1 + N[(N[Power[0.5, 0.16666666666666666], $MachinePrecision] * N[Power[im$95$m, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[Power[N[(-1.0 / re), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $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 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. +-commutative 7.7%

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

      5. distribute-rgt-in 7.7%

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

      6. cancel-sign-sub 7.7%

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

      7. distribute-rgt-out-- 7.7%

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

      8. sub-neg 7.7%

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

      9. remove-double-neg 7.7%

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

      10. +-commutative 7.7%

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

   3. Simplified 13.6%

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

      1. *-commutative 13.6%

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

      2. hypot-define 7.7%

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

      3. +-commutative 7.7%

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

      4. *-commutative 7.7%

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

      5. add-cube-cbrt 7.7%

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

      6. pow3 7.7%

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

      7. +-commutative 7.7%

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

      8. hypot-define 13.5%

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

   6. Applied egg-rr 13.5%

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

   7. Taylor expanded in re around -inf 51.5%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)} \cdot \sqrt[3]{\sqrt{2}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. *-commutative 51.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}}^{3} \]

      2. distribute-rgt-in 51.4%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot e^{\color{blue}{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666 + \log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}}\right)}^{3} \]

      3. exp-sum 52.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{\left(e^{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)}\right)}^{3} \]

      4. exp-to-pow 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{{\left(0.5 \cdot {im}^{2}\right)}^{0.16666666666666666}} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      5. *-commutative 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      6. exp-to-pow 51.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(\frac{-1}{re}\right)}^{0.16666666666666666}}\right)\right)}^{3} \]

   9. Simplified 51.8%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}}^{3} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 51.7%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)\right)} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

       2. expm1-undefine 19.0%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)} - 1\right)} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

       3. *-commutative 19.0%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\left(e^{\mathsf{log1p}\left({\color{blue}{\left(0.5 \cdot {im}^{2}\right)}}^{0.16666666666666666}\right)} - 1\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

       4. unpow-prod-down 19.0%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{{0.5}^{0.16666666666666666} \cdot {\left({im}^{2}\right)}^{0.16666666666666666}}\right)} - 1\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

       5. pow-pow 10.3%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\left(e^{\mathsf{log1p}\left({0.5}^{0.16666666666666666} \cdot \color{blue}{{im}^{\left(2 \cdot 0.16666666666666666\right)}}\right)} - 1\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

       6. metadata-eval 10.3%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\left(e^{\mathsf{log1p}\left({0.5}^{0.16666666666666666} \cdot {im}^{\color{blue}{0.3333333333333333}}\right)} - 1\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

       7. pow1/3 13.1%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\left(e^{\mathsf{log1p}\left({0.5}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{im}}\right)} - 1\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

   11. Applied egg-rr 13.1%

       \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)} - 1\right)} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]
   12. Step-by-step derivation

       1. expm1-define 51.5%

          \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)\right)} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

   13. Simplified 51.5%

       \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)\right)} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

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

   1. Initial program 45.9%

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

      1. sqr-neg 45.9%

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

      2. +-commutative 45.9%

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

      3. sqr-neg 45.9%

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

      4. +-commutative 45.9%

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

      5. distribute-rgt-in 45.9%

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

      6. cancel-sign-sub 45.9%

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

      7. distribute-rgt-out-- 45.9%

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

      8. sub-neg 45.9%

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

      9. remove-double-neg 45.9%

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

      10. +-commutative 45.9%

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

   3. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. hypot-define 45.9%

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

      3. +-commutative 45.9%

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

      4. *-commutative 45.9%

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

      5. add-sqr-sqrt 45.6%

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

      6. sqrt-unprod 45.9%

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

      7. *-commutative 45.9%

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

      8. *-commutative 45.9%

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

      9. swap-sqr 45.9%

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

   6. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. metadata-eval 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im\_m \cdot im\_m} \leq 0:\\ \;\;\;\;0.5 \cdot \left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt[3]{im\_m} \cdot \left({0.5}^{0.16666666666666666} \cdot {2}^{0.16666666666666666}\right)\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (*
    0.5
    (*
     (pow (pow (/ -1.0 re) 0.16666666666666666) 3.0)
     (pow
      (*
       (cbrt im_m)
       (* (pow 0.5 0.16666666666666666) (pow 2.0 0.16666666666666666)))
      3.0)))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (pow(pow((-1.0 / re), 0.16666666666666666), 3.0) * pow((cbrt(im_m) * (pow(0.5, 0.16666666666666666) * pow(2.0, 0.16666666666666666))), 3.0));
	} 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 + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (Math.pow(Math.pow((-1.0 / re), 0.16666666666666666), 3.0) * Math.pow((Math.cbrt(im_m) * (Math.pow(0.5, 0.16666666666666666) * Math.pow(2.0, 0.16666666666666666))), 3.0));
	} 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 (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(0.5 * Float64(((Float64(-1.0 / re) ^ 0.16666666666666666) ^ 3.0) * (Float64(cbrt(im_m) * Float64((0.5 ^ 0.16666666666666666) * (2.0 ^ 0.16666666666666666))) ^ 3.0)));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[Power[N[Power[N[(-1.0 / re), $MachinePrecision], 0.16666666666666666], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[(N[Power[im$95$m, 1/3], $MachinePrecision] * N[(N[Power[0.5, 0.16666666666666666], $MachinePrecision] * N[Power[2.0, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $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 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. +-commutative 7.7%

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

      5. distribute-rgt-in 7.7%

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

      6. cancel-sign-sub 7.7%

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

      7. distribute-rgt-out-- 7.7%

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

      8. sub-neg 7.7%

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

      9. remove-double-neg 7.7%

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

      10. +-commutative 7.7%

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

   3. Simplified 13.6%

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

      1. *-commutative 13.6%

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

      2. hypot-define 7.7%

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

      3. +-commutative 7.7%

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

      4. *-commutative 7.7%

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

      5. add-cube-cbrt 7.7%

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

      6. pow3 7.7%

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

      7. +-commutative 7.7%

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

      8. hypot-define 13.5%

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

   6. Applied egg-rr 13.5%

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

   7. Taylor expanded in re around -inf 51.5%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)} \cdot \sqrt[3]{\sqrt{2}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. *-commutative 51.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}}^{3} \]

      2. distribute-rgt-in 51.4%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot e^{\color{blue}{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666 + \log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}}\right)}^{3} \]

      3. exp-sum 52.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{\left(e^{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)}\right)}^{3} \]

      4. exp-to-pow 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{{\left(0.5 \cdot {im}^{2}\right)}^{0.16666666666666666}} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      5. *-commutative 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      6. exp-to-pow 51.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(\frac{-1}{re}\right)}^{0.16666666666666666}}\right)\right)}^{3} \]

   9. Simplified 51.8%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}}^{3} \]
   10. Step-by-step derivation

       1. associate-*r* 51.8%

          \[\leadsto 0.5 \cdot {\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}}^{3} \]

       2. unpow-prod-down 51.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{2}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right)} \]

       3. pow1/3 51.8%

          \[\leadsto 0.5 \cdot \left({\left(\color{blue}{{\left(\sqrt{2}\right)}^{0.3333333333333333}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       4. sqrt-pow2 51.8%

          \[\leadsto 0.5 \cdot \left({\left(\color{blue}{{2}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       5. metadata-eval 51.8%

          \[\leadsto 0.5 \cdot \left({\left({2}^{\color{blue}{0.16666666666666666}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       6. *-commutative 51.8%

          \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot {\color{blue}{\left(0.5 \cdot {im}^{2}\right)}}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       7. unpow-prod-down 51.8%

          \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot \color{blue}{\left({0.5}^{0.16666666666666666} \cdot {\left({im}^{2}\right)}^{0.16666666666666666}\right)}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       8. pow-pow 48.0%

          \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot \color{blue}{{im}^{\left(2 \cdot 0.16666666666666666\right)}}\right)\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       9. metadata-eval 48.0%

          \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot {im}^{\color{blue}{0.3333333333333333}}\right)\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       10. pow1/3 51.5%

           \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{im}}\right)\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

   11. Applied egg-rr 51.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left({\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 51.5%

          \[\leadsto 0.5 \cdot \color{blue}{\left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)\right)}^{3}\right)} \]

       2. associate-*r* 51.5%

          \[\leadsto 0.5 \cdot \left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\color{blue}{\left(\left({2}^{0.16666666666666666} \cdot {0.5}^{0.16666666666666666}\right) \cdot \sqrt[3]{im}\right)}}^{3}\right) \]

   13. Simplified 51.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\left({2}^{0.16666666666666666} \cdot {0.5}^{0.16666666666666666}\right) \cdot \sqrt[3]{im}\right)}^{3}\right)} \]

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

   1. Initial program 45.9%

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

      1. sqr-neg 45.9%

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

      2. +-commutative 45.9%

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

      3. sqr-neg 45.9%

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

      4. +-commutative 45.9%

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

      5. distribute-rgt-in 45.9%

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

      6. cancel-sign-sub 45.9%

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

      7. distribute-rgt-out-- 45.9%

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

      8. sub-neg 45.9%

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

      9. remove-double-neg 45.9%

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

      10. +-commutative 45.9%

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

   3. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. hypot-define 45.9%

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

      3. +-commutative 45.9%

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

      4. *-commutative 45.9%

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

      5. add-sqr-sqrt 45.6%

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

      6. sqrt-unprod 45.9%

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

      7. *-commutative 45.9%

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

      8. *-commutative 45.9%

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

      9. swap-sqr 45.9%

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

   6. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. metadata-eval 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt[3]{im} \cdot \left({0.5}^{0.16666666666666666} \cdot {2}^{0.16666666666666666}\right)\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im\_m \cdot im\_m} \leq 0:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im\_m}\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (*
    0.5
    (pow
     (*
      (cbrt (sqrt 2.0))
      (*
       (* (pow 0.5 0.16666666666666666) (cbrt im_m))
       (pow (/ -1.0 re) 0.16666666666666666)))
     3.0))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * pow((cbrt(sqrt(2.0)) * ((pow(0.5, 0.16666666666666666) * cbrt(im_m)) * pow((-1.0 / re), 0.16666666666666666))), 3.0);
	} 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 + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * Math.pow((Math.cbrt(Math.sqrt(2.0)) * ((Math.pow(0.5, 0.16666666666666666) * Math.cbrt(im_m)) * Math.pow((-1.0 / re), 0.16666666666666666))), 3.0);
	} 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 (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(0.5 * (Float64(cbrt(sqrt(2.0)) * Float64(Float64((0.5 ^ 0.16666666666666666) * cbrt(im_m)) * (Float64(-1.0 / re) ^ 0.16666666666666666))) ^ 3.0));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(N[Power[0.5, 0.16666666666666666], $MachinePrecision] * N[Power[im$95$m, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[(-1.0 / re), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $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 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. +-commutative 7.7%

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

      5. distribute-rgt-in 7.7%

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

      6. cancel-sign-sub 7.7%

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

      7. distribute-rgt-out-- 7.7%

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

      8. sub-neg 7.7%

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

      9. remove-double-neg 7.7%

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

      10. +-commutative 7.7%

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

   3. Simplified 13.6%

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

      1. *-commutative 13.6%

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

      2. hypot-define 7.7%

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

      3. +-commutative 7.7%

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

      4. *-commutative 7.7%

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

      5. add-cube-cbrt 7.7%

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

      6. pow3 7.7%

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

      7. +-commutative 7.7%

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

      8. hypot-define 13.5%

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

   6. Applied egg-rr 13.5%

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

   7. Taylor expanded in re around -inf 51.5%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)} \cdot \sqrt[3]{\sqrt{2}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. *-commutative 51.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}}^{3} \]

      2. distribute-rgt-in 51.4%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot e^{\color{blue}{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666 + \log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}}\right)}^{3} \]

      3. exp-sum 52.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{\left(e^{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)}\right)}^{3} \]

      4. exp-to-pow 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{{\left(0.5 \cdot {im}^{2}\right)}^{0.16666666666666666}} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      5. *-commutative 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      6. exp-to-pow 51.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(\frac{-1}{re}\right)}^{0.16666666666666666}}\right)\right)}^{3} \]

   9. Simplified 51.8%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}}^{3} \]

   10. Taylor expanded in im around 0 51.5%

       \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3} \]

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

   1. Initial program 45.9%

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

      1. sqr-neg 45.9%

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

      2. +-commutative 45.9%

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

      3. sqr-neg 45.9%

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

      4. +-commutative 45.9%

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

      5. distribute-rgt-in 45.9%

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

      6. cancel-sign-sub 45.9%

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

      7. distribute-rgt-out-- 45.9%

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

      8. sub-neg 45.9%

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

      9. remove-double-neg 45.9%

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

      10. +-commutative 45.9%

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

   3. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. hypot-define 45.9%

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

      3. +-commutative 45.9%

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

      4. *-commutative 45.9%

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

      5. add-sqr-sqrt 45.6%

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

      6. sqrt-unprod 45.9%

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

      7. *-commutative 45.9%

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

      8. *-commutative 45.9%

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

      9. swap-sqr 45.9%

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

   6. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. metadata-eval 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im\_m \cdot im\_m} \leq 0:\\ \;\;\;\;0.5 \cdot \left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot \left(im\_m \cdot {\left({0.5}^{0.16666666666666666} \cdot {2}^{0.16666666666666666}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (*
    0.5
    (*
     (pow (pow (/ -1.0 re) 0.16666666666666666) 3.0)
     (*
      im_m
      (pow
       (* (pow 0.5 0.16666666666666666) (pow 2.0 0.16666666666666666))
       3.0))))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (pow(pow((-1.0 / re), 0.16666666666666666), 3.0) * (im_m * pow((pow(0.5, 0.16666666666666666) * pow(2.0, 0.16666666666666666)), 3.0)));
	} 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 + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * (Math.pow(Math.pow((-1.0 / re), 0.16666666666666666), 3.0) * (im_m * Math.pow((Math.pow(0.5, 0.16666666666666666) * Math.pow(2.0, 0.16666666666666666)), 3.0)));
	} 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 + math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0:
		tmp = 0.5 * (math.pow(math.pow((-1.0 / re), 0.16666666666666666), 3.0) * (im_m * math.pow((math.pow(0.5, 0.16666666666666666) * math.pow(2.0, 0.16666666666666666)), 3.0)))
	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 (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(0.5 * Float64(((Float64(-1.0 / re) ^ 0.16666666666666666) ^ 3.0) * Float64(im_m * (Float64((0.5 ^ 0.16666666666666666) * (2.0 ^ 0.16666666666666666)) ^ 3.0))));
	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 + sqrt(((re * re) + (im_m * im_m)))) <= 0.0)
		tmp = 0.5 * ((((-1.0 / re) ^ 0.16666666666666666) ^ 3.0) * (im_m * (((0.5 ^ 0.16666666666666666) * (2.0 ^ 0.16666666666666666)) ^ 3.0)));
	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[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[Power[N[Power[N[(-1.0 / re), $MachinePrecision], 0.16666666666666666], $MachinePrecision], 3.0], $MachinePrecision] * N[(im$95$m * N[Power[N[(N[Power[0.5, 0.16666666666666666], $MachinePrecision] * N[Power[2.0, 0.16666666666666666], $MachinePrecision]), $MachinePrecision], 3.0], $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 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. +-commutative 7.7%

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

      5. distribute-rgt-in 7.7%

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

      6. cancel-sign-sub 7.7%

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

      7. distribute-rgt-out-- 7.7%

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

      8. sub-neg 7.7%

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

      9. remove-double-neg 7.7%

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

      10. +-commutative 7.7%

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

   3. Simplified 13.6%

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

      1. *-commutative 13.6%

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

      2. hypot-define 7.7%

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

      3. +-commutative 7.7%

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

      4. *-commutative 7.7%

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

      5. add-cube-cbrt 7.7%

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

      6. pow3 7.7%

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

      7. +-commutative 7.7%

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

      8. hypot-define 13.5%

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

   6. Applied egg-rr 13.5%

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

   7. Taylor expanded in re around -inf 51.5%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)} \cdot \sqrt[3]{\sqrt{2}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. *-commutative 51.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}}^{3} \]

      2. distribute-rgt-in 51.4%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot e^{\color{blue}{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666 + \log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}}\right)}^{3} \]

      3. exp-sum 52.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{\left(e^{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)}\right)}^{3} \]

      4. exp-to-pow 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{{\left(0.5 \cdot {im}^{2}\right)}^{0.16666666666666666}} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      5. *-commutative 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      6. exp-to-pow 51.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(\frac{-1}{re}\right)}^{0.16666666666666666}}\right)\right)}^{3} \]

   9. Simplified 51.8%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}}^{3} \]
   10. Step-by-step derivation

       1. associate-*r* 51.8%

          \[\leadsto 0.5 \cdot {\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right) \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}}^{3} \]

       2. unpow-prod-down 51.8%

          \[\leadsto 0.5 \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{2}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right)} \]

       3. pow1/3 51.8%

          \[\leadsto 0.5 \cdot \left({\left(\color{blue}{{\left(\sqrt{2}\right)}^{0.3333333333333333}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       4. sqrt-pow2 51.8%

          \[\leadsto 0.5 \cdot \left({\left(\color{blue}{{2}^{\left(\frac{0.3333333333333333}{2}\right)}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       5. metadata-eval 51.8%

          \[\leadsto 0.5 \cdot \left({\left({2}^{\color{blue}{0.16666666666666666}} \cdot {\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       6. *-commutative 51.8%

          \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot {\color{blue}{\left(0.5 \cdot {im}^{2}\right)}}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       7. unpow-prod-down 51.8%

          \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot \color{blue}{\left({0.5}^{0.16666666666666666} \cdot {\left({im}^{2}\right)}^{0.16666666666666666}\right)}\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       8. pow-pow 48.0%

          \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot \color{blue}{{im}^{\left(2 \cdot 0.16666666666666666\right)}}\right)\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       9. metadata-eval 48.0%

          \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot {im}^{\color{blue}{0.3333333333333333}}\right)\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

       10. pow1/3 51.5%

           \[\leadsto 0.5 \cdot \left({\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{im}}\right)\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right) \]

   11. Applied egg-rr 51.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left({\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)\right)}^{3} \cdot {\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 51.5%

          \[\leadsto 0.5 \cdot \color{blue}{\left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({2}^{0.16666666666666666} \cdot \left({0.5}^{0.16666666666666666} \cdot \sqrt[3]{im}\right)\right)}^{3}\right)} \]

       2. associate-*r* 51.5%

          \[\leadsto 0.5 \cdot \left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\color{blue}{\left(\left({2}^{0.16666666666666666} \cdot {0.5}^{0.16666666666666666}\right) \cdot \sqrt[3]{im}\right)}}^{3}\right) \]

       3. cube-prod 51.5%

          \[\leadsto 0.5 \cdot \left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot \color{blue}{\left({\left({2}^{0.16666666666666666} \cdot {0.5}^{0.16666666666666666}\right)}^{3} \cdot {\left(\sqrt[3]{im}\right)}^{3}\right)}\right) \]

       4. rem-cube-cbrt 51.4%

          \[\leadsto 0.5 \cdot \left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot \left({\left({2}^{0.16666666666666666} \cdot {0.5}^{0.16666666666666666}\right)}^{3} \cdot \color{blue}{im}\right)\right) \]

   13. Simplified 51.4%

       \[\leadsto 0.5 \cdot \color{blue}{\left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot \left({\left({2}^{0.16666666666666666} \cdot {0.5}^{0.16666666666666666}\right)}^{3} \cdot im\right)\right)} \]

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

   1. Initial program 45.9%

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

      1. sqr-neg 45.9%

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

      2. +-commutative 45.9%

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

      3. sqr-neg 45.9%

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

      4. +-commutative 45.9%

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

      5. distribute-rgt-in 45.9%

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

      6. cancel-sign-sub 45.9%

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

      7. distribute-rgt-out-- 45.9%

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

      8. sub-neg 45.9%

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

      9. remove-double-neg 45.9%

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

      10. +-commutative 45.9%

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

   3. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. hypot-define 45.9%

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

      3. +-commutative 45.9%

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

      4. *-commutative 45.9%

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

      5. add-sqr-sqrt 45.6%

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

      6. sqrt-unprod 45.9%

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

      7. *-commutative 45.9%

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

      8. *-commutative 45.9%

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

      9. swap-sqr 45.9%

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

   6. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. metadata-eval 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot \left({\left({\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)}^{3} \cdot \left(im \cdot {\left({0.5}^{0.16666666666666666} \cdot {2}^{0.16666666666666666}\right)}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im\_m \cdot im\_m} \leq 0:\\ \;\;\;\;0.5 \cdot {\left({\left(\frac{-0.5}{re}\right)}^{0.16666666666666666} \cdot \sqrt[3]{im\_m \cdot \sqrt{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (+ re (sqrt (+ (* re re) (* im_m im_m)))) 0.0)
   (*
    0.5
    (pow
     (* (pow (/ -0.5 re) 0.16666666666666666) (cbrt (* im_m (sqrt 2.0))))
     3.0))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re + sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * pow((pow((-0.5 / re), 0.16666666666666666) * cbrt((im_m * sqrt(2.0)))), 3.0);
	} 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 + Math.sqrt(((re * re) + (im_m * im_m)))) <= 0.0) {
		tmp = 0.5 * Math.pow((Math.pow((-0.5 / re), 0.16666666666666666) * Math.cbrt((im_m * Math.sqrt(2.0)))), 3.0);
	} 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 (Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) <= 0.0)
		tmp = Float64(0.5 * (Float64((Float64(-0.5 / re) ^ 0.16666666666666666) * cbrt(Float64(im_m * sqrt(2.0)))) ^ 3.0));
	else
		tmp = sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))));
	end
	return tmp
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[N[(N[Power[N[(-0.5 / re), $MachinePrecision], 0.16666666666666666], $MachinePrecision] * N[Power[N[(im$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $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 (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re) < 0.0

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. +-commutative 7.7%

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

      5. distribute-rgt-in 7.7%

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

      6. cancel-sign-sub 7.7%

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

      7. distribute-rgt-out-- 7.7%

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

      8. sub-neg 7.7%

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

      9. remove-double-neg 7.7%

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

      10. +-commutative 7.7%

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

   3. Simplified 13.6%

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

      1. *-commutative 13.6%

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

      2. hypot-define 7.7%

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

      3. +-commutative 7.7%

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

      4. *-commutative 7.7%

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

      5. add-cube-cbrt 7.7%

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

      6. pow3 7.7%

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

      7. +-commutative 7.7%

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

      8. hypot-define 13.5%

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

   6. Applied egg-rr 13.5%

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

   7. Taylor expanded in re around -inf 51.5%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)} \cdot \sqrt[3]{\sqrt{2}}\right)}}^{3} \]
   8. Step-by-step derivation

      1. *-commutative 51.5%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot e^{0.16666666666666666 \cdot \left(\log \left(0.5 \cdot {im}^{2}\right) + \log \left(\frac{-1}{re}\right)\right)}\right)}}^{3} \]

      2. distribute-rgt-in 51.4%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot e^{\color{blue}{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666 + \log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}}\right)}^{3} \]

      3. exp-sum 52.0%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \color{blue}{\left(e^{\log \left(0.5 \cdot {im}^{2}\right) \cdot 0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)}\right)}^{3} \]

      4. exp-to-pow 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left(\color{blue}{{\left(0.5 \cdot {im}^{2}\right)}^{0.16666666666666666}} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      5. *-commutative 51.6%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\color{blue}{\left({im}^{2} \cdot 0.5\right)}}^{0.16666666666666666} \cdot e^{\log \left(\frac{-1}{re}\right) \cdot 0.16666666666666666}\right)\right)}^{3} \]

      6. exp-to-pow 51.8%

         \[\leadsto 0.5 \cdot {\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(\frac{-1}{re}\right)}^{0.16666666666666666}}\right)\right)}^{3} \]

   9. Simplified 51.8%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \left({\left({im}^{2} \cdot 0.5\right)}^{0.16666666666666666} \cdot {\left(\frac{-1}{re}\right)}^{0.16666666666666666}\right)\right)}}^{3} \]

   10. Taylor expanded in im around 0 51.4%

       \[\leadsto 0.5 \cdot {\color{blue}{\left({\left(\frac{-0.5}{re}\right)}^{0.16666666666666666} \cdot \sqrt[3]{im \cdot \sqrt{2}}\right)}}^{3} \]

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

   1. Initial program 45.9%

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

      1. sqr-neg 45.9%

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

      2. +-commutative 45.9%

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

      3. sqr-neg 45.9%

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

      4. +-commutative 45.9%

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

      5. distribute-rgt-in 45.9%

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

      6. cancel-sign-sub 45.9%

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

      7. distribute-rgt-out-- 45.9%

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

      8. sub-neg 45.9%

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

      9. remove-double-neg 45.9%

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

      10. +-commutative 45.9%

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

   3. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. hypot-define 45.9%

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

      3. +-commutative 45.9%

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

      4. *-commutative 45.9%

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

      5. add-sqr-sqrt 45.6%

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

      6. sqrt-unprod 45.9%

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

      7. *-commutative 45.9%

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

      8. *-commutative 45.9%

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

      9. swap-sqr 45.9%

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

   6. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. metadata-eval 90.6%

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

   8. Simplified 90.6%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re + \sqrt{re \cdot re + im \cdot im} \leq 0:\\ \;\;\;\;0.5 \cdot {\left({\left(\frac{-0.5}{re}\right)}^{0.16666666666666666} \cdot \sqrt[3]{im \cdot \sqrt{2}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.5× speedup

Math

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

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= (sqrt (* (+ re (sqrt (+ (* re re) (* im_m im_m)))) 2.0)) 0.0)
   (* 0.5 (sqrt (/ (pow im_m 2.0) (- re))))
   (sqrt (* 0.5 (+ re (hypot re im_m))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (sqrt(((re + sqrt(((re * re) + (im_m * im_m)))) * 2.0)) <= 0.0) {
		tmp = 0.5 * sqrt((pow(im_m, 2.0) / -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 (Math.sqrt(((re + Math.sqrt(((re * re) + (im_m * im_m)))) * 2.0)) <= 0.0) {
		tmp = 0.5 * Math.sqrt((Math.pow(im_m, 2.0) / -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 math.sqrt(((re + math.sqrt(((re * re) + (im_m * im_m)))) * 2.0)) <= 0.0:
		tmp = 0.5 * math.sqrt((math.pow(im_m, 2.0) / -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 (sqrt(Float64(Float64(re + sqrt(Float64(Float64(re * re) + Float64(im_m * im_m)))) * 2.0)) <= 0.0)
		tmp = Float64(0.5 * sqrt(Float64((im_m ^ 2.0) / Float64(-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 (sqrt(((re + sqrt(((re * re) + (im_m * im_m)))) * 2.0)) <= 0.0)
		tmp = 0.5 * sqrt(((im_m ^ 2.0) / -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[LessEqual[N[Sqrt[N[(N[(re + N[Sqrt[N[(N[(re * re), $MachinePrecision] + N[(im$95$m * im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], 0.0], N[(0.5 * N[Sqrt[N[(N[Power[im$95$m, 2.0], $MachinePrecision] / (-re)), $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 (sqrt.f64 (*.f64 #s(literal 2 binary64) (+.f64 (sqrt.f64 (+.f64 (*.f64 re re) (*.f64 im im))) re))) < 0.0

   1. Initial program 8.9%

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

      1. sqr-neg 8.9%

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

      2. +-commutative 8.9%

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

      3. sqr-neg 8.9%

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

      4. +-commutative 8.9%

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

      5. distribute-rgt-in 8.9%

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

      6. cancel-sign-sub 8.9%

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

      7. distribute-rgt-out-- 8.9%

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

      8. sub-neg 8.9%

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

      9. remove-double-neg 8.9%

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

      10. +-commutative 8.9%

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

   3. Simplified 8.9%

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

   5. Taylor expanded in re around -inf 58.5%

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

      1. mul-1-neg 58.5%

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

      2. distribute-neg-frac2 58.5%

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

   7. Simplified 58.5%

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

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

   1. Initial program 44.9%

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

      1. sqr-neg 44.9%

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

      2. +-commutative 44.9%

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

      3. sqr-neg 44.9%

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

      4. +-commutative 44.9%

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

      5. distribute-rgt-in 44.9%

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

      6. cancel-sign-sub 44.9%

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

      7. distribute-rgt-out-- 44.9%

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

      8. sub-neg 44.9%

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

      9. remove-double-neg 44.9%

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

      10. +-commutative 44.9%

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

   3. Simplified 89.5%

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

      1. *-commutative 89.5%

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

      2. hypot-define 44.9%

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

      3. +-commutative 44.9%

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

      4. *-commutative 44.9%

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

      5. add-sqr-sqrt 44.6%

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

      6. sqrt-unprod 44.9%

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

      7. *-commutative 44.9%

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

      8. *-commutative 44.9%

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

      9. swap-sqr 44.9%

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

   6. Applied egg-rr 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*r* 89.5%

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

      3. metadata-eval 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 85.8%

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

Alternative 7: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\_m\right)\right)} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m) :precision binary64 (sqrt (* 0.5 (+ re (hypot re im_m)))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	return sqrt((0.5 * (re + hypot(re, im_m))));
}

Java

im_m = Math.abs(im);
public static double code(double re, double im_m) {
	return Math.sqrt((0.5 * (re + Math.hypot(re, im_m))));
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	return math.sqrt((0.5 * (re + math.hypot(re, im_m))))

Julia

im_m = abs(im)
function code(re, im_m)
	return sqrt(Float64(0.5 * Float64(re + hypot(re, im_m))))
end

MATLAB

im_m = abs(im);
function tmp = code(re, im_m)
	tmp = sqrt((0.5 * (re + hypot(re, im_m))));
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := N[Sqrt[N[(0.5 * N[(re + N[Sqrt[re ^ 2 + im$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 40.5%

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

   1. sqr-neg 40.5%

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

   2. +-commutative 40.5%

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

   3. sqr-neg 40.5%

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

   4. +-commutative 40.5%

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

   5. distribute-rgt-in 40.5%

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

   6. cancel-sign-sub 40.5%

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

   7. distribute-rgt-out-- 40.5%

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

   8. sub-neg 40.5%

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

   9. remove-double-neg 40.5%

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

   10. +-commutative 40.5%

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

3. Simplified 79.7%

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

   1. *-commutative 79.7%

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

   2. hypot-define 40.5%

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

   3. +-commutative 40.5%

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

   4. *-commutative 40.5%

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

   5. add-sqr-sqrt 40.3%

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

   6. sqrt-unprod 40.5%

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

   7. *-commutative 40.5%

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

   8. *-commutative 40.5%

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

   9. swap-sqr 40.5%

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

6. Applied egg-rr 79.7%

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

   1. *-commutative 79.7%

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

   2. associate-*r* 79.7%

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

   3. metadata-eval 79.7%

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

8. Simplified 79.7%

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

9. Final simplification 79.7%

   \[\leadsto \sqrt{0.5 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)} \]
10. Add Preprocessing

Alternative 8: 63.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+171}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot 4 + im\_m \cdot \frac{im\_m}{re}}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -4.5e+171)
   (* 0.5 (sqrt 0.0))
   (if (<= re 9e+134)
     (sqrt (* 0.5 (+ re im_m)))
     (* 0.5 (sqrt (+ (* re 4.0) (* im_m (/ im_m re))))))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -4.5e+171) {
		tmp = 0.5 * sqrt(0.0);
	} else if (re <= 9e+134) {
		tmp = sqrt((0.5 * (re + im_m)));
	} else {
		tmp = 0.5 * sqrt(((re * 4.0) + (im_m * (im_m / 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 <= (-4.5d+171)) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else if (re <= 9d+134) then
        tmp = sqrt((0.5d0 * (re + im_m)))
    else
        tmp = 0.5d0 * sqrt(((re * 4.0d0) + (im_m * (im_m / 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 <= -4.5e+171) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else if (re <= 9e+134) {
		tmp = Math.sqrt((0.5 * (re + im_m)));
	} else {
		tmp = 0.5 * Math.sqrt(((re * 4.0) + (im_m * (im_m / re))));
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if re <= -4.5e+171:
		tmp = 0.5 * math.sqrt(0.0)
	elif re <= 9e+134:
		tmp = math.sqrt((0.5 * (re + im_m)))
	else:
		tmp = 0.5 * math.sqrt(((re * 4.0) + (im_m * (im_m / re))))
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if (re <= -4.5e+171)
		tmp = Float64(0.5 * sqrt(0.0));
	elseif (re <= 9e+134)
		tmp = sqrt(Float64(0.5 * Float64(re + im_m)));
	else
		tmp = Float64(0.5 * sqrt(Float64(Float64(re * 4.0) + Float64(im_m * Float64(im_m / re)))));
	end
	return tmp
end

MATLAB

im_m = abs(im);
function tmp_2 = code(re, im_m)
	tmp = 0.0;
	if (re <= -4.5e+171)
		tmp = 0.5 * sqrt(0.0);
	elseif (re <= 9e+134)
		tmp = sqrt((0.5 * (re + im_m)));
	else
		tmp = 0.5 * sqrt(((re * 4.0) + (im_m * (im_m / re))));
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[LessEqual[re, -4.5e+171], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+134], N[Sqrt[N[(0.5 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[Sqrt[N[(N[(re * 4.0), $MachinePrecision] + N[(im$95$m * N[(im$95$m / re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -4.49999999999999969e171

   1. Initial program 2.5%

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

      1. sqr-neg 2.5%

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

      2. +-commutative 2.5%

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

      3. sqr-neg 2.5%

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

      4. +-commutative 2.5%

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

      5. distribute-rgt-in 2.5%

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

      6. cancel-sign-sub 2.5%

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

      7. distribute-rgt-out-- 2.5%

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

      8. sub-neg 2.5%

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

      9. remove-double-neg 2.5%

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

      10. +-commutative 2.5%

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

   3. Simplified 31.3%

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

      1. hypot-define 2.5%

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

      2. distribute-lft-in 2.5%

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

      3. *-commutative 2.5%

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

      4. add-cube-cbrt 2.5%

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

      5. fma-define 2.5%

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

      6. hypot-define 10.2%

         \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{re \cdot 2} \cdot \sqrt[3]{re \cdot 2}, \sqrt[3]{re \cdot 2}, 2 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   6. Applied egg-rr 10.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt[3]{re \cdot 2} \cdot \sqrt[3]{re \cdot 2}, \sqrt[3]{re \cdot 2}, 2 \cdot \mathsf{hypot}\left(re, im\right)\right)}} \]

   7. Taylor expanded in re around -inf 0.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \left(re \cdot \left(2 + {\left(\sqrt[3]{-2}\right)}^{3}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-1 \cdot re\right) \cdot \left(2 + {\left(\sqrt[3]{-2}\right)}^{3}\right)}} \]

      2. rem-cube-cbrt 24.3%

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

      3. metadata-eval 24.3%

         \[\leadsto 0.5 \cdot \sqrt{\left(-1 \cdot re\right) \cdot \color{blue}{0}} \]

      4. mul0-rgt 24.3%

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

   9. Simplified 24.3%

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

   if -4.49999999999999969e171 < re < 8.9999999999999995e134

   1. Initial program 51.9%

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

      1. sqr-neg 51.9%

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

      2. +-commutative 51.9%

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

      3. sqr-neg 51.9%

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

      4. +-commutative 51.9%

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

      5. distribute-rgt-in 51.9%

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

      6. cancel-sign-sub 51.9%

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

      7. distribute-rgt-out-- 51.9%

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

      8. sub-neg 51.9%

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

      9. remove-double-neg 51.9%

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

      10. +-commutative 51.9%

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

   3. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      2. hypot-define 51.9%

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

      3. +-commutative 51.9%

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

      4. *-commutative 51.9%

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

      5. add-sqr-sqrt 51.6%

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

      6. sqrt-unprod 51.9%

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

      7. *-commutative 51.9%

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

      8. *-commutative 51.9%

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

      9. swap-sqr 51.9%

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

   6. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*r* 83.0%

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

      3. metadata-eval 83.0%

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

   8. Simplified 83.0%

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

   9. Taylor expanded in re around 0 29.3%

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

   if 8.9999999999999995e134 < re

   1. Initial program 9.5%

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

      1. sqr-neg 9.5%

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

      2. +-commutative 9.5%

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

      3. sqr-neg 9.5%

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

      4. +-commutative 9.5%

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

      5. distribute-rgt-in 9.5%

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

      6. cancel-sign-sub 9.5%

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

      7. distribute-rgt-out-- 9.5%

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

      8. sub-neg 9.5%

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

      9. remove-double-neg 9.5%

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

      10. +-commutative 9.5%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 91.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re + \frac{{im}^{2}}{re}}} \]
   6. Step-by-step derivation

      1. unpow2 91.3%

         \[\leadsto 0.5 \cdot \sqrt{4 \cdot re + \frac{\color{blue}{im \cdot im}}{re}} \]

      2. associate-/l* 94.2%

         \[\leadsto 0.5 \cdot \sqrt{4 \cdot re + \color{blue}{im \cdot \frac{im}{re}}} \]

   7. Applied egg-rr 94.2%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -4.5 \cdot 10^{+171}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{re \cdot 4 + im \cdot \frac{im}{re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+22} \lor \neg \left(re \leq 1.85 \cdot 10^{+67}\right) \land re \leq 10^{+135}:\\ \;\;\;\;\sqrt{im\_m \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (or (<= re 5.8e+22) (and (not (<= re 1.85e+67)) (<= re 1e+135)))
   (sqrt (* im_m 0.5))
   (sqrt re)))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if ((re <= 5.8e+22) || (!(re <= 1.85e+67) && (re <= 1e+135))) {
		tmp = sqrt((im_m * 0.5));
	} 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 <= 5.8d+22) .or. (.not. (re <= 1.85d+67)) .and. (re <= 1d+135)) then
        tmp = sqrt((im_m * 0.5d0))
    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 <= 5.8e+22) || (!(re <= 1.85e+67) && (re <= 1e+135))) {
		tmp = Math.sqrt((im_m * 0.5));
	} else {
		tmp = Math.sqrt(re);
	}
	return tmp;
}

Python

im_m = math.fabs(im)
def code(re, im_m):
	tmp = 0
	if (re <= 5.8e+22) or (not (re <= 1.85e+67) and (re <= 1e+135)):
		tmp = math.sqrt((im_m * 0.5))
	else:
		tmp = math.sqrt(re)
	return tmp

Julia

im_m = abs(im)
function code(re, im_m)
	tmp = 0.0
	if ((re <= 5.8e+22) || (!(re <= 1.85e+67) && (re <= 1e+135)))
		tmp = sqrt(Float64(im_m * 0.5));
	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 <= 5.8e+22) || (~((re <= 1.85e+67)) && (re <= 1e+135)))
		tmp = sqrt((im_m * 0.5));
	else
		tmp = sqrt(re);
	end
	tmp_2 = tmp;
end

Wolfram

im_m = N[Abs[im], $MachinePrecision]
code[re_, im$95$m_] := If[Or[LessEqual[re, 5.8e+22], And[N[Not[LessEqual[re, 1.85e+67]], $MachinePrecision], LessEqual[re, 1e+135]]], N[Sqrt[N[(im$95$m * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < 5.8e22 or 1.8499999999999999e67 < re < 9.99999999999999962e134

   1. Initial program 44.1%

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

      1. sqr-neg 44.1%

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

      2. +-commutative 44.1%

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

      3. sqr-neg 44.1%

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

      4. +-commutative 44.1%

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

      5. distribute-rgt-in 44.1%

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

      6. cancel-sign-sub 44.1%

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

      7. distribute-rgt-out-- 44.1%

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

      8. sub-neg 44.1%

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

      9. remove-double-neg 44.1%

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

      10. +-commutative 44.1%

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

   3. Simplified 75.8%

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

      1. *-commutative 75.8%

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

      2. hypot-define 44.1%

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

      3. +-commutative 44.1%

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

      4. *-commutative 44.1%

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

      5. add-sqr-sqrt 43.8%

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

      6. sqrt-unprod 44.1%

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

      7. *-commutative 44.1%

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

      8. *-commutative 44.1%

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

      9. swap-sqr 44.1%

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

   6. Applied egg-rr 75.8%

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

      1. *-commutative 75.8%

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

      2. associate-*r* 75.8%

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

      3. metadata-eval 75.8%

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

   8. Simplified 75.8%

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

   9. Taylor expanded in re around 0 25.7%

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

   if 5.8e22 < re < 1.8499999999999999e67 or 9.99999999999999962e134 < re

   1. Initial program 22.4%

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

      1. sqr-neg 22.4%

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

      2. +-commutative 22.4%

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

      3. sqr-neg 22.4%

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

      4. +-commutative 22.4%

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

      5. distribute-rgt-in 22.4%

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

      6. cancel-sign-sub 22.4%

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

      7. distribute-rgt-out-- 22.4%

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

      8. sub-neg 22.4%

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

      9. remove-double-neg 22.4%

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

      10. +-commutative 22.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 92.6%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re + \frac{{im}^{2}}{re}}} \]

   6. Taylor expanded in re around inf 93.7%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq 5.8 \cdot 10^{+22} \lor \neg \left(re \leq 1.85 \cdot 10^{+67}\right) \land re \leq 10^{+135}:\\ \;\;\;\;\sqrt{im \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} im_m = \left|im\right| \\ \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{+172}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \end{array} \]

FPCore

im_m = (fabs.f64 im)
(FPCore (re im_m)
 :precision binary64
 (if (<= re -2e+172)
   (* 0.5 (sqrt 0.0))
   (if (<= re 9e+134) (sqrt (* 0.5 (+ re im_m))) (sqrt re))))

C

im_m = fabs(im);
double code(double re, double im_m) {
	double tmp;
	if (re <= -2e+172) {
		tmp = 0.5 * sqrt(0.0);
	} else if (re <= 9e+134) {
		tmp = sqrt((0.5 * (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 <= (-2d+172)) then
        tmp = 0.5d0 * sqrt(0.0d0)
    else if (re <= 9d+134) then
        tmp = sqrt((0.5d0 * (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 <= -2e+172) {
		tmp = 0.5 * Math.sqrt(0.0);
	} else if (re <= 9e+134) {
		tmp = Math.sqrt((0.5 * (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 <= -2e+172:
		tmp = 0.5 * math.sqrt(0.0)
	elif re <= 9e+134:
		tmp = math.sqrt((0.5 * (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 <= -2e+172)
		tmp = Float64(0.5 * sqrt(0.0));
	elseif (re <= 9e+134)
		tmp = sqrt(Float64(0.5 * 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 <= -2e+172)
		tmp = 0.5 * sqrt(0.0);
	elseif (re <= 9e+134)
		tmp = sqrt((0.5 * (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, -2e+172], N[(0.5 * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[re, 9e+134], N[Sqrt[N[(0.5 * N[(re + im$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if re < -2.0000000000000002e172

   1. Initial program 2.5%

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

      1. sqr-neg 2.5%

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

      2. +-commutative 2.5%

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

      3. sqr-neg 2.5%

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

      4. +-commutative 2.5%

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

      5. distribute-rgt-in 2.5%

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

      6. cancel-sign-sub 2.5%

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

      7. distribute-rgt-out-- 2.5%

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

      8. sub-neg 2.5%

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

      9. remove-double-neg 2.5%

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

      10. +-commutative 2.5%

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

   3. Simplified 31.3%

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

      1. hypot-define 2.5%

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

      2. distribute-lft-in 2.5%

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

      3. *-commutative 2.5%

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

      4. add-cube-cbrt 2.5%

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

      5. fma-define 2.5%

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

      6. hypot-define 10.2%

         \[\leadsto 0.5 \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{re \cdot 2} \cdot \sqrt[3]{re \cdot 2}, \sqrt[3]{re \cdot 2}, 2 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right)} \]

   6. Applied egg-rr 10.2%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt[3]{re \cdot 2} \cdot \sqrt[3]{re \cdot 2}, \sqrt[3]{re \cdot 2}, 2 \cdot \mathsf{hypot}\left(re, im\right)\right)}} \]

   7. Taylor expanded in re around -inf 0.0%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{-1 \cdot \left(re \cdot \left(2 + {\left(\sqrt[3]{-2}\right)}^{3}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\left(-1 \cdot re\right) \cdot \left(2 + {\left(\sqrt[3]{-2}\right)}^{3}\right)}} \]

      2. rem-cube-cbrt 24.3%

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

      3. metadata-eval 24.3%

         \[\leadsto 0.5 \cdot \sqrt{\left(-1 \cdot re\right) \cdot \color{blue}{0}} \]

      4. mul0-rgt 24.3%

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

   9. Simplified 24.3%

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

   if -2.0000000000000002e172 < re < 8.9999999999999995e134

   1. Initial program 51.9%

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

      1. sqr-neg 51.9%

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

      2. +-commutative 51.9%

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

      3. sqr-neg 51.9%

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

      4. +-commutative 51.9%

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

      5. distribute-rgt-in 51.9%

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

      6. cancel-sign-sub 51.9%

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

      7. distribute-rgt-out-- 51.9%

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

      8. sub-neg 51.9%

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

      9. remove-double-neg 51.9%

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

      10. +-commutative 51.9%

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

   3. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      2. hypot-define 51.9%

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

      3. +-commutative 51.9%

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

      4. *-commutative 51.9%

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

      5. add-sqr-sqrt 51.6%

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

      6. sqrt-unprod 51.9%

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

      7. *-commutative 51.9%

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

      8. *-commutative 51.9%

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

      9. swap-sqr 51.9%

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

   6. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*r* 83.0%

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

      3. metadata-eval 83.0%

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

   8. Simplified 83.0%

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

   9. Taylor expanded in re around 0 29.3%

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

   if 8.9999999999999995e134 < re

   1. Initial program 9.5%

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

      1. sqr-neg 9.5%

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

      2. +-commutative 9.5%

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

      3. sqr-neg 9.5%

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

      4. +-commutative 9.5%

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

      5. distribute-rgt-in 9.5%

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

      6. cancel-sign-sub 9.5%

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

      7. distribute-rgt-out-- 9.5%

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

      8. sub-neg 9.5%

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

      9. remove-double-neg 9.5%

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

      10. +-commutative 9.5%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in im around 0 91.3%

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re + \frac{{im}^{2}}{re}}} \]

   6. Taylor expanded in re around inf 92.7%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -2 \cdot 10^{+172}:\\ \;\;\;\;0.5 \cdot \sqrt{0}\\ \mathbf{elif}\;re \leq 9 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(re + im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.2% 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 40.5%

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

   1. sqr-neg 40.5%

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

   2. +-commutative 40.5%

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

   3. sqr-neg 40.5%

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

   4. +-commutative 40.5%

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

   5. distribute-rgt-in 40.5%

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

   6. cancel-sign-sub 40.5%

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

   7. distribute-rgt-out-- 40.5%

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

   8. sub-neg 40.5%

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

   9. remove-double-neg 40.5%

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

   10. +-commutative 40.5%

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

3. Simplified 79.7%

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

5. Taylor expanded in im around 0 26.6%

   \[\leadsto 0.5 \cdot \sqrt{\color{blue}{4 \cdot re + \frac{{im}^{2}}{re}}} \]

6. Taylor expanded in re around inf 26.8%

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

7. Final simplification 26.8%

   \[\leadsto \sqrt{re} \]
8. Add Preprocessing

Developer target: 48.6% 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 2024109 
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :alt
  (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)))))