Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.7%

Time: 12.7s

Alternatives: 5

Speedup: 2.3×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)}} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (* 2.0 (* (/ l Om) (sin ky)))))))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, (2.0 * ((l / Om) * sin(ky))))))));
}

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, (2.0 * ((l / Om) * Math.sin(ky))))))));
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, (2.0 * ((l / Om) * math.sin(ky))))))))

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(2.0 * Float64(Float64(l / Om) * sin(ky))))))))
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, (2.0 * ((l / Om) * sin(ky))))))));
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l / Om), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in 98.4%

      \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

   2. metadata-eval 98.4%

      \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

   3. metadata-eval 98.4%

      \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

   4. associate-/l* 98.4%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

   5. metadata-eval 98.4%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

4. Taylor expanded in kx around 0 76.3%

   \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
5. Step-by-step derivation

   1. associate-/l* 75.6%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{\sin ky}^{2}}}}}} \cdot 0.5} \]

   2. associate-/r/ 76.6%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]

   3. unpow2 76.6%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

   4. unpow2 76.6%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

   5. times-frac 89.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

6. Simplified 89.0%

   \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 89.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)} \cdot \sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}}} \cdot 0.5} \]

   2. hypot-1-def 89.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)}} \cdot 0.5} \]

   3. sqrt-prod 89.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{4} \cdot \sqrt{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}}}\right)} \cdot 0.5} \]

   4. metadata-eval 89.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{2} \cdot \sqrt{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}}\right)} \cdot 0.5} \]

   5. sqrt-prod 89.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \color{blue}{\left(\sqrt{\frac{\ell}{Om} \cdot \frac{\ell}{Om}} \cdot \sqrt{{\sin ky}^{2}}\right)}\right)} \cdot 0.5} \]

   6. sqrt-prod 48.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\color{blue}{\left(\sqrt{\frac{\ell}{Om}} \cdot \sqrt{\frac{\ell}{Om}}\right)} \cdot \sqrt{{\sin ky}^{2}}\right)\right)} \cdot 0.5} \]

   7. add-sqr-sqrt 89.7%

      \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\color{blue}{\frac{\ell}{Om}} \cdot \sqrt{{\sin ky}^{2}}\right)\right)} \cdot 0.5} \]

   8. unpow2 89.7%

      \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky}}\right)\right)} \cdot 0.5} \]

   9. sqrt-prod 47.3%

      \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\sqrt{\sin ky} \cdot \sqrt{\sin ky}\right)}\right)\right)} \cdot 0.5} \]

   10. add-sqr-sqrt 94.6%

       \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\sin ky}\right)\right)} \cdot 0.5} \]

8. Applied egg-rr 94.6%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)}} \cdot 0.5} \]

9. Final simplification 94.6%

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)}} \]

Alternative 2: 72.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 4.8 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 3.6 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 4.8e-154)
   (sqrt 0.5)
   (if (<= Om 3.6e+149)
     (sqrt
      (+
       0.5
       (*
        0.5
        (/
         1.0
         (+
          1.0
          (*
           2.0
           (/ (+ 0.5 (* (cos (* 2.0 ky)) -0.5)) (/ (* Om Om) (* l l)))))))))
     1.0)))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 4.8e-154) {
		tmp = sqrt(0.5);
	} else if (Om <= 3.6e+149) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((0.5 + (cos((2.0 * ky)) * -0.5)) / ((Om * Om) / (l * l)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 4.8d-154) then
        tmp = sqrt(0.5d0)
    else if (om <= 3.6d+149) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((0.5d0 + (cos((2.0d0 * ky)) * (-0.5d0))) / ((om * om) / (l * l)))))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 4.8e-154) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 3.6e+149) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((0.5 + (Math.cos((2.0 * ky)) * -0.5)) / ((Om * Om) / (l * l)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 4.8e-154:
		tmp = math.sqrt(0.5)
	elif Om <= 3.6e+149:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((0.5 + (math.cos((2.0 * ky)) * -0.5)) / ((Om * Om) / (l * l)))))))))
	else:
		tmp = 1.0
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 4.8e-154)
		tmp = sqrt(0.5);
	elseif (Om <= 3.6e+149)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(0.5 + Float64(cos(Float64(2.0 * ky)) * -0.5)) / Float64(Float64(Om * Om) / Float64(l * l)))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 4.8e-154)
		tmp = sqrt(0.5);
	elseif (Om <= 3.6e+149)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((0.5 + (cos((2.0 * ky)) * -0.5)) / ((Om * Om) / (l * l)))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 4.8e-154], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 3.6e+149], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(0.5 + N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 4.79999999999999974e-154

   1. Initial program 98.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in 98.1%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

      2. metadata-eval 98.1%

         \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      3. metadata-eval 98.1%

         \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      4. associate-/l* 98.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      5. metadata-eval 98.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

   4. Taylor expanded in Om around 0 51.3%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-*r* 51.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(2 \cdot \sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right) \cdot \frac{\ell}{Om}}} \cdot 0.5} \]

      2. *-commutative 51.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot 2\right)} \cdot \frac{\ell}{Om}} \cdot 0.5} \]

      3. associate-*r* 51.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

      4. unpow2 51.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

      5. unpow2 51.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

      6. hypot-def 53.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

   6. Simplified 53.3%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

   7. Taylor expanded in l around inf 60.8%

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

   if 4.79999999999999974e-154 < Om < 3.59999999999999995e149

   1. Initial program 98.6%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in 98.6%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

      2. metadata-eval 98.6%

         \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      3. metadata-eval 98.6%

         \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      4. associate-/l* 98.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      5. metadata-eval 98.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

   4. Taylor expanded in kx around 0 92.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-/l* 90.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{\sin ky}^{2}}}}}} \cdot 0.5} \]

      2. associate-/r/ 92.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]

      3. unpow2 92.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

      4. unpow2 92.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

      5. times-frac 92.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

   6. Simplified 92.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. unpow2 92.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\left(\sin ky \cdot \sin ky\right)}\right)}} \cdot 0.5} \]

      2. sin-mult 82.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}}\right)}} \cdot 0.5} \]

      3. cos-sum 82.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \frac{\cos \left(ky - ky\right) - \color{blue}{\left(\cos ky \cdot \cos ky - \sin ky \cdot \sin ky\right)}}{2}\right)}} \cdot 0.5} \]

      4. cos-2 82.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \frac{\cos \left(ky - ky\right) - \color{blue}{\cos \left(2 \cdot ky\right)}}{2}\right)}} \cdot 0.5} \]

   8. Applied egg-rr 82.2%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\frac{\cos \left(ky - ky\right) - \cos \left(2 \cdot ky\right)}{2}}\right)}} \cdot 0.5} \]
   9. Step-by-step derivation

      1. div-sub 82.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\left(\frac{\cos \left(ky - ky\right)}{2} - \frac{\cos \left(2 \cdot ky\right)}{2}\right)}\right)}} \cdot 0.5} \]

      2. +-inverses 82.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(2 \cdot ky\right)}{2}\right)\right)}} \cdot 0.5} \]

      3. cos-0 82.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(2 \cdot ky\right)}{2}\right)\right)}} \cdot 0.5} \]

      4. metadata-eval 82.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(\color{blue}{0.5} - \frac{\cos \left(2 \cdot ky\right)}{2}\right)\right)}} \cdot 0.5} \]

   10. Simplified 82.2%

       \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\left(0.5 - \frac{\cos \left(2 \cdot ky\right)}{2}\right)}\right)}} \cdot 0.5} \]

   11. Taylor expanded in l around 0 81.8%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{1 + 2 \cdot \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) \cdot {\ell}^{2}}{{Om}^{2}}}} \cdot 0.5} \]
   12. Step-by-step derivation

       1. associate-/l* 81.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}{\frac{{Om}^{2}}{{\ell}^{2}}}}} \cdot 0.5} \]

       2. sub-neg 81.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{\color{blue}{0.5 + \left(-0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}{\frac{{Om}^{2}}{{\ell}^{2}}}} \cdot 0.5} \]

       3. *-commutative 81.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{0.5 + \left(-\color{blue}{\cos \left(2 \cdot ky\right) \cdot 0.5}\right)}{\frac{{Om}^{2}}{{\ell}^{2}}}} \cdot 0.5} \]

       4. distribute-rgt-neg-in 81.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{0.5 + \color{blue}{\cos \left(2 \cdot ky\right) \cdot \left(-0.5\right)}}{\frac{{Om}^{2}}{{\ell}^{2}}}} \cdot 0.5} \]

       5. metadata-eval 81.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{0.5 + \cos \left(2 \cdot ky\right) \cdot \color{blue}{-0.5}}{\frac{{Om}^{2}}{{\ell}^{2}}}} \cdot 0.5} \]

       6. unpow2 81.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5}{\frac{\color{blue}{Om \cdot Om}}{{\ell}^{2}}}} \cdot 0.5} \]

       7. unpow2 81.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5}{\frac{Om \cdot Om}{\color{blue}{\ell \cdot \ell}}}} \cdot 0.5} \]

   13. Simplified 81.8%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{1 + 2 \cdot \frac{0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5}{\frac{Om \cdot Om}{\ell \cdot \ell}}}} \cdot 0.5} \]

   if 3.59999999999999995e149 < Om

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

      2. metadata-eval 100.0%

         \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      3. metadata-eval 100.0%

         \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      4. associate-/l* 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      5. metadata-eval 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

   4. Taylor expanded in kx around 0 69.7%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-/l* 69.7%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{\sin ky}^{2}}}}}} \cdot 0.5} \]

      2. associate-/r/ 69.7%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]

      3. unpow2 69.7%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

      4. unpow2 69.7%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

      5. times-frac 97.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

   6. Simplified 97.6%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 97.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)} \cdot \sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}}} \cdot 0.5} \]

      2. hypot-1-def 97.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)}} \cdot 0.5} \]

      3. sqrt-prod 97.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{4} \cdot \sqrt{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}}}\right)} \cdot 0.5} \]

      4. metadata-eval 97.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{2} \cdot \sqrt{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}}\right)} \cdot 0.5} \]

      5. sqrt-prod 97.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \color{blue}{\left(\sqrt{\frac{\ell}{Om} \cdot \frac{\ell}{Om}} \cdot \sqrt{{\sin ky}^{2}}\right)}\right)} \cdot 0.5} \]

      6. sqrt-prod 58.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\color{blue}{\left(\sqrt{\frac{\ell}{Om}} \cdot \sqrt{\frac{\ell}{Om}}\right)} \cdot \sqrt{{\sin ky}^{2}}\right)\right)} \cdot 0.5} \]

      7. add-sqr-sqrt 97.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\color{blue}{\frac{\ell}{Om}} \cdot \sqrt{{\sin ky}^{2}}\right)\right)} \cdot 0.5} \]

      8. unpow2 97.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky}}\right)\right)} \cdot 0.5} \]

      9. sqrt-prod 57.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\sqrt{\sin ky} \cdot \sqrt{\sin ky}\right)}\right)\right)} \cdot 0.5} \]

      10. add-sqr-sqrt 97.6%

          \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\sin ky}\right)\right)} \cdot 0.5} \]

   8. Applied egg-rr 97.6%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)}} \cdot 0.5} \]

   9. Taylor expanded in l around 0 90.3%

      \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 4.8 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 3.6 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 69.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 28000000:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot ky}}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+119}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 9.5e-31)
   1.0
   (if (<= l 28000000.0)
     (sqrt (+ 0.5 (* 0.25 (/ Om (* l ky)))))
     (if (<= l 9.8e+119) 1.0 (sqrt 0.5)))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 9.5e-31) {
		tmp = 1.0;
	} else if (l <= 28000000.0) {
		tmp = sqrt((0.5 + (0.25 * (Om / (l * ky)))));
	} else if (l <= 9.8e+119) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 9.5d-31) then
        tmp = 1.0d0
    else if (l <= 28000000.0d0) then
        tmp = sqrt((0.5d0 + (0.25d0 * (om / (l * ky)))))
    else if (l <= 9.8d+119) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 9.5e-31) {
		tmp = 1.0;
	} else if (l <= 28000000.0) {
		tmp = Math.sqrt((0.5 + (0.25 * (Om / (l * ky)))));
	} else if (l <= 9.8e+119) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if l <= 9.5e-31:
		tmp = 1.0
	elif l <= 28000000.0:
		tmp = math.sqrt((0.5 + (0.25 * (Om / (l * ky)))))
	elif l <= 9.8e+119:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 9.5e-31)
		tmp = 1.0;
	elseif (l <= 28000000.0)
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(Om / Float64(l * ky)))));
	elseif (l <= 9.8e+119)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 9.5e-31)
		tmp = 1.0;
	elseif (l <= 28000000.0)
		tmp = sqrt((0.5 + (0.25 * (Om / (l * ky)))));
	elseif (l <= 9.8e+119)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[l, 9.5e-31], 1.0, If[LessEqual[l, 28000000.0], N[Sqrt[N[(0.5 + N[(0.25 * N[(Om / N[(l * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 9.8e+119], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 9.5000000000000008e-31 or 2.8e7 < l < 9.79999999999999992e119

   1. Initial program 98.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in 98.1%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

      2. metadata-eval 98.1%

         \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      3. metadata-eval 98.1%

         \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      4. associate-/l* 98.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      5. metadata-eval 98.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

   4. Taylor expanded in kx around 0 77.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-/l* 78.4%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{\sin ky}^{2}}}}}} \cdot 0.5} \]

      2. associate-/r/ 78.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]

      3. unpow2 78.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

      4. unpow2 78.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

      5. times-frac 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

   6. Simplified 90.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)} \cdot \sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}}} \cdot 0.5} \]

      2. hypot-1-def 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)}} \cdot 0.5} \]

      3. sqrt-prod 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{4} \cdot \sqrt{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}}}\right)} \cdot 0.5} \]

      4. metadata-eval 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{2} \cdot \sqrt{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}}\right)} \cdot 0.5} \]

      5. sqrt-prod 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \color{blue}{\left(\sqrt{\frac{\ell}{Om} \cdot \frac{\ell}{Om}} \cdot \sqrt{{\sin ky}^{2}}\right)}\right)} \cdot 0.5} \]

      6. sqrt-prod 50.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\color{blue}{\left(\sqrt{\frac{\ell}{Om}} \cdot \sqrt{\frac{\ell}{Om}}\right)} \cdot \sqrt{{\sin ky}^{2}}\right)\right)} \cdot 0.5} \]

      7. add-sqr-sqrt 90.7%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\color{blue}{\frac{\ell}{Om}} \cdot \sqrt{{\sin ky}^{2}}\right)\right)} \cdot 0.5} \]

      8. unpow2 90.7%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky}}\right)\right)} \cdot 0.5} \]

      9. sqrt-prod 46.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\sqrt{\sin ky} \cdot \sqrt{\sin ky}\right)}\right)\right)} \cdot 0.5} \]

      10. add-sqr-sqrt 95.1%

          \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\sin ky}\right)\right)} \cdot 0.5} \]

   8. Applied egg-rr 95.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)}} \cdot 0.5} \]

   9. Taylor expanded in l around 0 70.8%

      \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]

   if 9.5000000000000008e-31 < l < 2.8e7

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

      2. metadata-eval 100.0%

         \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      3. metadata-eval 100.0%

         \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      4. associate-/l* 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      5. metadata-eval 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

   4. Taylor expanded in Om around 0 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(2 \cdot \sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right) \cdot \frac{\ell}{Om}}} \cdot 0.5} \]

      2. *-commutative 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot 2\right)} \cdot \frac{\ell}{Om}} \cdot 0.5} \]

      3. associate-*r* 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

      4. unpow2 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

      5. unpow2 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

      6. hypot-def 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

   6. Simplified 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

   7. Taylor expanded in kx around 0 98.4%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   8. Step-by-step derivation

      1. +-commutative 98.4%

         \[\leadsto \sqrt{\color{blue}{0.25 \cdot \frac{Om}{\ell \cdot \sin ky} + 0.5}} \]

   9. Simplified 98.4%

      \[\leadsto \color{blue}{\sqrt{0.25 \cdot \frac{Om}{\ell \cdot \sin ky} + 0.5}} \]

   10. Taylor expanded in ky around 0 98.4%

       \[\leadsto \sqrt{0.25 \cdot \frac{Om}{\color{blue}{\ell \cdot ky}} + 0.5} \]

   if 9.79999999999999992e119 < l

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

      2. metadata-eval 100.0%

         \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      3. metadata-eval 100.0%

         \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      4. associate-/l* 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      5. metadata-eval 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

   4. Taylor expanded in Om around 0 81.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-*r* 81.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(2 \cdot \sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right) \cdot \frac{\ell}{Om}}} \cdot 0.5} \]

      2. *-commutative 81.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot 2\right)} \cdot \frac{\ell}{Om}} \cdot 0.5} \]

      3. associate-*r* 81.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

      4. unpow2 81.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

      5. unpow2 81.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

      6. hypot-def 81.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

   6. Simplified 81.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

   7. Taylor expanded in l around inf 84.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9.5 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 28000000:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot ky}}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+119}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4: 69.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 50000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+119}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 9e-31)
   1.0
   (if (<= l 50000000.0) (sqrt 0.5) (if (<= l 9.8e+119) 1.0 (sqrt 0.5)))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 9e-31) {
		tmp = 1.0;
	} else if (l <= 50000000.0) {
		tmp = sqrt(0.5);
	} else if (l <= 9.8e+119) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 9d-31) then
        tmp = 1.0d0
    else if (l <= 50000000.0d0) then
        tmp = sqrt(0.5d0)
    else if (l <= 9.8d+119) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 9e-31) {
		tmp = 1.0;
	} else if (l <= 50000000.0) {
		tmp = Math.sqrt(0.5);
	} else if (l <= 9.8e+119) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if l <= 9e-31:
		tmp = 1.0
	elif l <= 50000000.0:
		tmp = math.sqrt(0.5)
	elif l <= 9.8e+119:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 9e-31)
		tmp = 1.0;
	elseif (l <= 50000000.0)
		tmp = sqrt(0.5);
	elseif (l <= 9.8e+119)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 9e-31)
		tmp = 1.0;
	elseif (l <= 50000000.0)
		tmp = sqrt(0.5);
	elseif (l <= 9.8e+119)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[l, 9e-31], 1.0, If[LessEqual[l, 50000000.0], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, 9.8e+119], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 9.0000000000000008e-31 or 5e7 < l < 9.79999999999999992e119

   1. Initial program 98.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in 98.1%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

      2. metadata-eval 98.1%

         \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      3. metadata-eval 98.1%

         \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      4. associate-/l* 98.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      5. metadata-eval 98.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

   4. Taylor expanded in kx around 0 77.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-/l* 78.4%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{\sin ky}^{2}}}}}} \cdot 0.5} \]

      2. associate-/r/ 78.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]

      3. unpow2 78.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{Om}^{2}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

      4. unpow2 78.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{Om \cdot Om}} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

      5. times-frac 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \left(\color{blue}{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)} \cdot {\sin ky}^{2}\right)}} \cdot 0.5} \]

   6. Simplified 90.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)} \cdot \sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}}}} \cdot 0.5} \]

      2. hypot-1-def 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)}} \cdot 0.5} \]

      3. sqrt-prod 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{4} \cdot \sqrt{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}}}\right)} \cdot 0.5} \]

      4. metadata-eval 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{2} \cdot \sqrt{\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}}\right)} \cdot 0.5} \]

      5. sqrt-prod 90.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \color{blue}{\left(\sqrt{\frac{\ell}{Om} \cdot \frac{\ell}{Om}} \cdot \sqrt{{\sin ky}^{2}}\right)}\right)} \cdot 0.5} \]

      6. sqrt-prod 50.6%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\color{blue}{\left(\sqrt{\frac{\ell}{Om}} \cdot \sqrt{\frac{\ell}{Om}}\right)} \cdot \sqrt{{\sin ky}^{2}}\right)\right)} \cdot 0.5} \]

      7. add-sqr-sqrt 90.7%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\color{blue}{\frac{\ell}{Om}} \cdot \sqrt{{\sin ky}^{2}}\right)\right)} \cdot 0.5} \]

      8. unpow2 90.7%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sqrt{\color{blue}{\sin ky \cdot \sin ky}}\right)\right)} \cdot 0.5} \]

      9. sqrt-prod 46.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\left(\sqrt{\sin ky} \cdot \sqrt{\sin ky}\right)}\right)\right)} \cdot 0.5} \]

      10. add-sqr-sqrt 95.1%

          \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \color{blue}{\sin ky}\right)\right)} \cdot 0.5} \]

   8. Applied egg-rr 95.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)}} \cdot 0.5} \]

   9. Taylor expanded in l around 0 70.8%

      \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]

   if 9.0000000000000008e-31 < l < 5e7 or 9.79999999999999992e119 < l

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

      2. metadata-eval 100.0%

         \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      3. metadata-eval 100.0%

         \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      4. associate-/l* 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

      5. metadata-eval 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

   4. Taylor expanded in Om around 0 84.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-*r* 84.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(2 \cdot \sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right) \cdot \frac{\ell}{Om}}} \cdot 0.5} \]

      2. *-commutative 84.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot 2\right)} \cdot \frac{\ell}{Om}} \cdot 0.5} \]

      3. associate-*r* 84.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

      4. unpow2 84.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

      5. unpow2 84.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

      6. hypot-def 84.1%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

   6. Simplified 84.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

   7. Taylor expanded in l around inf 86.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 50000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 9.8 \cdot 10^{+119}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 56.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \sqrt{0.5} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky) :precision binary64 (sqrt 0.5))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(0.5d0)
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(0.5);
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	return sqrt(0.5)
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp = code(l, Om, kx, ky)
	tmp = sqrt(0.5);
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in 98.4%

      \[\leadsto \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]

   2. metadata-eval 98.4%

      \[\leadsto \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

   3. metadata-eval 98.4%

      \[\leadsto \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

   4. associate-/l* 98.4%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]

   5. metadata-eval 98.4%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

3. Simplified 98.4%

   \[\leadsto \color{blue}{\sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5}} \]

4. Taylor expanded in Om around 0 44.1%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]
5. Step-by-step derivation

   1. associate-*r* 44.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(2 \cdot \sqrt{{\sin ky}^{2} + {\sin kx}^{2}}\right) \cdot \frac{\ell}{Om}}} \cdot 0.5} \]

   2. *-commutative 44.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\left(\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot 2\right)} \cdot \frac{\ell}{Om}} \cdot 0.5} \]

   3. associate-*r* 44.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{{\sin ky}^{2} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

   4. unpow2 44.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

   5. unpow2 44.1%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

   6. hypot-def 45.7%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(2 \cdot \frac{\ell}{Om}\right)} \cdot 0.5} \]

6. Simplified 45.7%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)}} \cdot 0.5} \]

7. Taylor expanded in l around inf 54.6%

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

8. Final simplification 54.6%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023240 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))