Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.2%

Time: 18.0s

Alternatives: 10

Speedup: 1.7×

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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{2 \cdot l\_m}{Om\_m}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {t\_0}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot {\left(l\_m \cdot ky\_m\right)}^{2}}{Om\_m \cdot Om\_m}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l_m) Om_m)))
   (if (<= t_0 2e+125)
     (sqrt
      (*
       (/ 1.0 2.0)
       (+
        1.0
        (/
         1.0
         (sqrt
          (+
           1.0
           (*
            (pow t_0 2.0)
            (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))))))
     (sqrt
      (+
       0.5
       (/ 0.5 (+ 1.0 (/ (* 2.0 (pow (* l_m ky_m) 2.0)) (* Om_m Om_m)))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (2.0 * l_m) / Om_m;
	double tmp;
	if (t_0 <= 2e+125) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(t_0, 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0)))))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 * l_m) / om_m
    if (t_0 <= 2d+125) then
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((t_0 ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0)))))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + ((2.0d0 * ((l_m * ky_m) ** 2.0d0)) / (om_m * om_m))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (2.0 * l_m) / Om_m;
	double tmp;
	if (t_0 <= 2e+125) {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(t_0, 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0)))))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * Math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	t_0 = (2.0 * l_m) / Om_m
	tmp = 0
	if t_0 <= 2e+125:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(t_0, 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0)))))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(2.0 * l_m) / Om_m)
	tmp = 0.0
	if (t_0 <= 2e+125)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((t_0 ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(2.0 * (Float64(l_m * ky_m) ^ 2.0)) / Float64(Om_m * Om_m))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	t_0 = (2.0 * l_m) / Om_m;
	tmp = 0.0;
	if (t_0 <= 2e+125)
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((t_0 ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))));
	else
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l_m * ky_m) ^ 2.0)) / (Om_m * Om_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+125], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(2.0 * N[Power[N[(l$95$m * ky$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.9999999999999998e125

   1. Initial program 97.7%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   if 1.9999999999999998e125 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 97.3%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 81.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 81.1%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + 2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 81.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   10. Simplified 81.1%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. pow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot {\ell}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\left(\sin ky \cdot \ell\right)}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\sin ky \cdot \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin ky, \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       6. sin-lowering-sin.f64 94.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 94.6%

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

   13. Taylor expanded in ky around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\color{blue}{\left(ky \cdot \ell\right)}, 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\ell \cdot ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 94.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   15. Simplified 94.6%

       \[\leadsto \sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot {\color{blue}{\left(\ell \cdot ky\right)}}^{2}}{Om \cdot Om}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{\frac{l\_m \cdot 4}{Om\_m}}{\frac{Om\_m}{l\_m}}\\ t_1 := {\sin ky\_m}^{2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot {\left(l\_m \cdot ky\_m\right)}^{2}}{Om\_m \cdot Om\_m}}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{t\_0 \cdot \left(2 \cdot \left(ky\_m \cdot ky\_m\right)\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{t\_0 \cdot \left(1 - \cos \left(2 \cdot ky\_m\right)\right)}{2}}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (/ (* l_m 4.0) Om_m) (/ Om_m l_m))) (t_1 (pow (sin ky_m) 2.0)))
   (if (<= t_1 0.0)
     (sqrt
      (+ 0.5 (/ 0.5 (+ 1.0 (/ (* 2.0 (pow (* l_m ky_m) 2.0)) (* Om_m Om_m))))))
     (if (<= t_1 5e-34)
       (sqrt
        (+ 0.5 (/ 0.5 (sqrt (+ 1.0 (/ (* t_0 (* 2.0 (* ky_m ky_m))) 2.0))))))
       (sqrt
        (+
         0.5
         (/
          0.5
          (sqrt (+ 1.0 (/ (* t_0 (- 1.0 (cos (* 2.0 ky_m)))) 2.0))))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = ((l_m * 4.0) / Om_m) / (Om_m / l_m);
	double t_1 = pow(sin(ky_m), 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	} else if (t_1 <= 5e-34) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((t_0 * (2.0 * (ky_m * ky_m))) / 2.0))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((t_0 * (1.0 - cos((2.0 * ky_m)))) / 2.0))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((l_m * 4.0d0) / om_m) / (om_m / l_m)
    t_1 = sin(ky_m) ** 2.0d0
    if (t_1 <= 0.0d0) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + ((2.0d0 * ((l_m * ky_m) ** 2.0d0)) / (om_m * om_m))))))
    else if (t_1 <= 5d-34) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((t_0 * (2.0d0 * (ky_m * ky_m))) / 2.0d0))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((t_0 * (1.0d0 - cos((2.0d0 * ky_m)))) / 2.0d0))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = ((l_m * 4.0) / Om_m) / (Om_m / l_m);
	double t_1 = Math.pow(Math.sin(ky_m), 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * Math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	} else if (t_1 <= 5e-34) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((t_0 * (2.0 * (ky_m * ky_m))) / 2.0))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((t_0 * (1.0 - Math.cos((2.0 * ky_m)))) / 2.0))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	t_0 = ((l_m * 4.0) / Om_m) / (Om_m / l_m)
	t_1 = math.pow(math.sin(ky_m), 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))))
	elif t_1 <= 5e-34:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((t_0 * (2.0 * (ky_m * ky_m))) / 2.0))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((t_0 * (1.0 - math.cos((2.0 * ky_m)))) / 2.0))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(Float64(l_m * 4.0) / Om_m) / Float64(Om_m / l_m))
	t_1 = sin(ky_m) ^ 2.0
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(2.0 * (Float64(l_m * ky_m) ^ 2.0)) / Float64(Om_m * Om_m))))));
	elseif (t_1 <= 5e-34)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(t_0 * Float64(2.0 * Float64(ky_m * ky_m))) / 2.0))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(t_0 * Float64(1.0 - cos(Float64(2.0 * ky_m)))) / 2.0))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	t_0 = ((l_m * 4.0) / Om_m) / (Om_m / l_m);
	t_1 = sin(ky_m) ^ 2.0;
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l_m * ky_m) ^ 2.0)) / (Om_m * Om_m))))));
	elseif (t_1 <= 5e-34)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((t_0 * (2.0 * (ky_m * ky_m))) / 2.0))))));
	else
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((t_0 * (1.0 - cos((2.0 * ky_m)))) / 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision] / N[(Om$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(2.0 * N[Power[N[(l$95$m * ky$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 5e-34], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(t$95$0 * N[(2.0 * N[(ky$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(t$95$0 * N[(1.0 - N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 0.0

   1. Initial program 90.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 76.5%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 49.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 49.3%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + 2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 49.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   10. Simplified 49.3%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. pow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot {\ell}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\left(\sin ky \cdot \ell\right)}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\sin ky \cdot \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin ky, \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       6. sin-lowering-sin.f64 72.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 72.2%

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

   13. Taylor expanded in ky around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\color{blue}{\left(ky \cdot \ell\right)}, 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\ell \cdot ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 72.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   15. Simplified 72.2%

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

   if 0.0 < (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 5.0000000000000003e-34

   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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 85.7%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 81.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 81.4%

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

      1. pow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\ell \cdot \left(\ell \cdot 4\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\sin ky \cdot \sin ky\right)\right)\right)\right)\right)\right)\right) \]

      5. sin-mult N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)}{2}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 43.9%

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

   10. Taylor expanded in ky around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \color{blue}{\left(2 \cdot {ky}^{2}\right)}\right), 2\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \left({ky}^{2}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \left(ky \cdot ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 92.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(ky, ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   12. Simplified 92.5%

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

   if 5.0000000000000003e-34 < (pow.f64 (sin.f64 ky) #s(literal 2 binary64))

   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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 91.4%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 90.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 90.4%

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

      1. pow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\ell \cdot \left(\ell \cdot 4\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\sin ky \cdot \sin ky\right)\right)\right)\right)\right)\right)\right) \]

      5. sin-mult N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)}{2}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 95.1%

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\sqrt{1 + \color{blue}{\frac{\frac{\frac{\ell \cdot 4}{Om}}{\frac{Om}{\ell}} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{2}}}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{l\_m \cdot 4}{Om\_m}\\ \mathbf{if}\;{\sin ky\_m}^{2} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + t\_0 \cdot \frac{l\_m \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\_m\right)\right) + ky\_m \cdot ky\_m\right)}{Om\_m}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{t\_0}{\frac{Om\_m}{l\_m}} \cdot \left(1 - \cos \left(2 \cdot ky\_m\right)\right)}{2}}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (* l_m 4.0) Om_m)))
   (if (<= (pow (sin ky_m) 2.0) 5e-34)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (*
           t_0
           (/
            (* l_m (+ (- 0.5 (* 0.5 (cos (* 2.0 kx_m)))) (* ky_m ky_m)))
            Om_m)))))))
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (/ (* (/ t_0 (/ Om_m l_m)) (- 1.0 (cos (* 2.0 ky_m)))) 2.0)))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (l_m * 4.0) / Om_m;
	double tmp;
	if (pow(sin(ky_m), 2.0) <= 5e-34) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (t_0 * ((l_m * ((0.5 - (0.5 * cos((2.0 * kx_m)))) + (ky_m * ky_m))) / Om_m)))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((t_0 / (Om_m / l_m)) * (1.0 - cos((2.0 * ky_m)))) / 2.0))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l_m * 4.0d0) / om_m
    if ((sin(ky_m) ** 2.0d0) <= 5d-34) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (t_0 * ((l_m * ((0.5d0 - (0.5d0 * cos((2.0d0 * kx_m)))) + (ky_m * ky_m))) / om_m)))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((t_0 / (om_m / l_m)) * (1.0d0 - cos((2.0d0 * ky_m)))) / 2.0d0))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (l_m * 4.0) / Om_m;
	double tmp;
	if (Math.pow(Math.sin(ky_m), 2.0) <= 5e-34) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (t_0 * ((l_m * ((0.5 - (0.5 * Math.cos((2.0 * kx_m)))) + (ky_m * ky_m))) / Om_m)))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((t_0 / (Om_m / l_m)) * (1.0 - Math.cos((2.0 * ky_m)))) / 2.0))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	t_0 = (l_m * 4.0) / Om_m
	tmp = 0
	if math.pow(math.sin(ky_m), 2.0) <= 5e-34:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (t_0 * ((l_m * ((0.5 - (0.5 * math.cos((2.0 * kx_m)))) + (ky_m * ky_m))) / Om_m)))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((t_0 / (Om_m / l_m)) * (1.0 - math.cos((2.0 * ky_m)))) / 2.0))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(l_m * 4.0) / Om_m)
	tmp = 0.0
	if ((sin(ky_m) ^ 2.0) <= 5e-34)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(t_0 * Float64(Float64(l_m * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx_m)))) + Float64(ky_m * ky_m))) / Om_m)))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(t_0 / Float64(Om_m / l_m)) * Float64(1.0 - cos(Float64(2.0 * ky_m)))) / 2.0))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	t_0 = (l_m * 4.0) / Om_m;
	tmp = 0.0;
	if ((sin(ky_m) ^ 2.0) <= 5e-34)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (t_0 * ((l_m * ((0.5 - (0.5 * cos((2.0 * kx_m)))) + (ky_m * ky_m))) / Om_m)))))));
	else
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((t_0 / (Om_m / l_m)) * (1.0 - cos((2.0 * ky_m)))) / 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision], 5e-34], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(t$95$0 * N[(N[(l$95$m * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(ky$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(t$95$0 / N[(Om$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 5.0000000000000003e-34

   1. Initial program 95.3%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 81.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell\right) \cdot \left(\ell \cdot 4\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om} \cdot \frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om}\right), \left(\frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 83.9%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \color{blue}{\left({ky}^{2}\right)}\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \left(ky \cdot ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 93.2%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \mathsf{*.f64}\left(ky, ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 93.2%

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

   if 5.0000000000000003e-34 < (pow.f64 (sin.f64 ky) #s(literal 2 binary64))

   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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 91.4%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 90.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 90.4%

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

      1. pow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\ell \cdot \left(\ell \cdot 4\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\sin ky \cdot \sin ky\right)\right)\right)\right)\right)\right)\right) \]

      5. sin-mult N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)}{2}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 95.1%

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\sqrt{1 + \color{blue}{\frac{\frac{\frac{\ell \cdot 4}{Om}}{\frac{Om}{\ell}} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{2}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 5 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot 4}{Om} \cdot \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky\right)}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\frac{\ell \cdot 4}{Om}}{\frac{Om}{\ell}} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \cos \left(2 \cdot kx\_m\right)\\ t_1 := \frac{l\_m \cdot 4}{Om\_m}\\ \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{l\_m \cdot \left(1 - 0.5 \cdot \left(t\_0 + \cos \left(2 \cdot ky\_m\right)\right)\right)}{Om\_m} \cdot t\_1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + t\_1 \cdot \frac{l\_m \cdot \left(\left(0.5 - 0.5 \cdot t\_0\right) + ky\_m \cdot ky\_m\right)}{Om\_m}}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 kx_m))) (t_1 (/ (* l_m 4.0) Om_m)))
   (if (<= (/ (* 2.0 l_m) Om_m) 2e+15)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (*
           (/ (* l_m (- 1.0 (* 0.5 (+ t_0 (cos (* 2.0 ky_m)))))) Om_m)
           t_1))))))
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (*
           t_1
           (/ (* l_m (+ (- 0.5 (* 0.5 t_0)) (* ky_m ky_m))) Om_m))))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = cos((2.0 * kx_m));
	double t_1 = (l_m * 4.0) / Om_m;
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 2e+15) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l_m * (1.0 - (0.5 * (t_0 + cos((2.0 * ky_m)))))) / Om_m) * t_1))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (t_1 * ((l_m * ((0.5 - (0.5 * t_0)) + (ky_m * ky_m))) / Om_m)))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((2.0d0 * kx_m))
    t_1 = (l_m * 4.0d0) / om_m
    if (((2.0d0 * l_m) / om_m) <= 2d+15) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((l_m * (1.0d0 - (0.5d0 * (t_0 + cos((2.0d0 * ky_m)))))) / om_m) * t_1))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (t_1 * ((l_m * ((0.5d0 - (0.5d0 * t_0)) + (ky_m * ky_m))) / om_m)))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = Math.cos((2.0 * kx_m));
	double t_1 = (l_m * 4.0) / Om_m;
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 2e+15) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((l_m * (1.0 - (0.5 * (t_0 + Math.cos((2.0 * ky_m)))))) / Om_m) * t_1))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (t_1 * ((l_m * ((0.5 - (0.5 * t_0)) + (ky_m * ky_m))) / Om_m)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	t_0 = math.cos((2.0 * kx_m))
	t_1 = (l_m * 4.0) / Om_m
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 2e+15:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((l_m * (1.0 - (0.5 * (t_0 + math.cos((2.0 * ky_m)))))) / Om_m) * t_1))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (t_1 * ((l_m * ((0.5 - (0.5 * t_0)) + (ky_m * ky_m))) / Om_m)))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = cos(Float64(2.0 * kx_m))
	t_1 = Float64(Float64(l_m * 4.0) / Om_m)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 2e+15)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(l_m * Float64(1.0 - Float64(0.5 * Float64(t_0 + cos(Float64(2.0 * ky_m)))))) / Om_m) * t_1))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(t_1 * Float64(Float64(l_m * Float64(Float64(0.5 - Float64(0.5 * t_0)) + Float64(ky_m * ky_m))) / Om_m)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	t_0 = cos((2.0 * kx_m));
	t_1 = (l_m * 4.0) / Om_m;
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 2e+15)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l_m * (1.0 - (0.5 * (t_0 + cos((2.0 * ky_m)))))) / Om_m) * t_1))))));
	else
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (t_1 * ((l_m * ((0.5 - (0.5 * t_0)) + (ky_m * ky_m))) / Om_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[Cos[N[(2.0 * kx$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2e+15], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(l$95$m * N[(1.0 - N[(0.5 * N[(t$95$0 + N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(t$95$1 * N[(N[(l$95$m * N[(N[(0.5 - N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(ky$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 2e15

   1. Initial program 97.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 86.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell\right) \cdot \left(\ell \cdot 4\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om} \cdot \frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om}\right), \left(\frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 94.3%

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

   7. Taylor expanded in kx around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(1 - \left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right)}\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(1, \left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(1, \left(\frac{1}{2} \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\cos \left(2 \cdot kx\right), \cos \left(2 \cdot ky\right)\right)\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot kx\right)\right), \cos \left(2 \cdot ky\right)\right)\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \cos \left(2 \cdot ky\right)\right)\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \mathsf{cos.f64}\left(\left(2 \cdot ky\right)\right)\right)\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 94.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 94.3%

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

   if 2e15 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 85.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell\right) \cdot \left(\ell \cdot 4\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om} \cdot \frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om}\right), \left(\frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \color{blue}{\left({ky}^{2}\right)}\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \left(ky \cdot ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 95.9%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \mathsf{*.f64}\left(ky, ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 95.9%

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \color{blue}{ky \cdot ky}\right)}{Om} \cdot \frac{\ell \cdot 4}{Om}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(1 - 0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)}{Om} \cdot \frac{\ell \cdot 4}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot 4}{Om} \cdot \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + ky \cdot ky\right)}{Om}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;ky\_m \leq 4.8 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot {\left(l\_m \cdot ky\_m\right)}^{2}}{Om\_m \cdot Om\_m}}}\\ \mathbf{elif}\;ky\_m \leq 10^{+70}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\frac{l\_m \cdot 4}{Om\_m}}{\frac{Om\_m}{l\_m}} \cdot \left(2 \cdot \left(ky\_m \cdot ky\_m\right)\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\_m\right)\right) \cdot \left(l\_m \cdot \frac{l\_m}{Om\_m \cdot Om\_m}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<= ky_m 4.8e-169)
   (sqrt
    (+ 0.5 (/ 0.5 (+ 1.0 (/ (* 2.0 (pow (* l_m ky_m) 2.0)) (* Om_m Om_m))))))
   (if (<= ky_m 1e+70)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (/
           (* (/ (/ (* l_m 4.0) Om_m) (/ Om_m l_m)) (* 2.0 (* ky_m ky_m)))
           2.0))))))
     (sqrt
      (+
       0.5
       (/
        0.5
        (+
         1.0
         (*
          2.0
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 ky_m))))
           (* l_m (/ l_m (* Om_m Om_m))))))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (ky_m <= 4.8e-169) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	} else if (ky_m <= 1e+70) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((((l_m * 4.0) / Om_m) / (Om_m / l_m)) * (2.0 * (ky_m * ky_m))) / 2.0))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((0.5 - (0.5 * cos((2.0 * ky_m)))) * (l_m * (l_m / (Om_m * Om_m)))))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (ky_m <= 4.8d-169) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + ((2.0d0 * ((l_m * ky_m) ** 2.0d0)) / (om_m * om_m))))))
    else if (ky_m <= 1d+70) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((((l_m * 4.0d0) / om_m) / (om_m / l_m)) * (2.0d0 * (ky_m * ky_m))) / 2.0d0))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + (2.0d0 * ((0.5d0 - (0.5d0 * cos((2.0d0 * ky_m)))) * (l_m * (l_m / (om_m * om_m)))))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (ky_m <= 4.8e-169) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * Math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	} else if (ky_m <= 1e+70) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((((l_m * 4.0) / Om_m) / (Om_m / l_m)) * (2.0 * (ky_m * ky_m))) / 2.0))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((0.5 - (0.5 * Math.cos((2.0 * ky_m)))) * (l_m * (l_m / (Om_m * Om_m)))))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	tmp = 0
	if ky_m <= 4.8e-169:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))))
	elif ky_m <= 1e+70:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((((l_m * 4.0) / Om_m) / (Om_m / l_m)) * (2.0 * (ky_m * ky_m))) / 2.0))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((0.5 - (0.5 * math.cos((2.0 * ky_m)))) * (l_m * (l_m / (Om_m * Om_m)))))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (ky_m <= 4.8e-169)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(2.0 * (Float64(l_m * ky_m) ^ 2.0)) / Float64(Om_m * Om_m))))));
	elseif (ky_m <= 1e+70)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(Float64(l_m * 4.0) / Om_m) / Float64(Om_m / l_m)) * Float64(2.0 * Float64(ky_m * ky_m))) / 2.0))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(2.0 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky_m)))) * Float64(l_m * Float64(l_m / Float64(Om_m * Om_m)))))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0;
	if (ky_m <= 4.8e-169)
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l_m * ky_m) ^ 2.0)) / (Om_m * Om_m))))));
	elseif (ky_m <= 1e+70)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((((l_m * 4.0) / Om_m) / (Om_m / l_m)) * (2.0 * (ky_m * ky_m))) / 2.0))))));
	else
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((0.5 - (0.5 * cos((2.0 * ky_m)))) * (l_m * (l_m / (Om_m * Om_m)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[ky$95$m, 4.8e-169], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(2.0 * N[Power[N[(l$95$m * ky$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ky$95$m, 1e+70], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision] / N[(Om$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(ky$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(2.0 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * N[(l$95$m / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 4.80000000000000021e-169

   1. Initial program 96.4%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 86.1%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 73.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 73.2%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + 2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   10. Simplified 73.4%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. pow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot {\ell}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\left(\sin ky \cdot \ell\right)}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\sin ky \cdot \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin ky, \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       6. sin-lowering-sin.f64 82.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 82.9%

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

   13. Taylor expanded in ky around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\color{blue}{\left(ky \cdot \ell\right)}, 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\ell \cdot ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   15. Simplified 73.5%

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

   if 4.80000000000000021e-169 < ky < 1.00000000000000007e70

   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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 90.5%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 90.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 90.5%

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

      1. pow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\ell \cdot \left(\ell \cdot 4\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\sin ky \cdot \sin ky\right)\right)\right)\right)\right)\right)\right) \]

      5. sin-mult N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)}{2}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 65.7%

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

   10. Taylor expanded in ky around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \color{blue}{\left(2 \cdot {ky}^{2}\right)}\right), 2\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \left({ky}^{2}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \left(ky \cdot ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(ky, ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   12. Simplified 98.1%

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

   if 1.00000000000000007e70 < ky

   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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 83.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 82.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 82.4%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + 2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   10. Simplified 82.3%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot {\sin ky}^{2}}{Om \cdot Om}\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{\left(\ell \cdot \ell\right) \cdot {\sin ky}^{2}}{Om \cdot Om} \cdot 2\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left(\ell \cdot \ell\right) \cdot {\sin ky}^{2}}{Om \cdot Om}\right), 2\right)\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{{\sin ky}^{2} \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}\right), 2\right)\right)\right)\right)\right) \]

       5. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\sin ky}^{2} \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right), 2\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\sin ky}^{2}\right), \left(\frac{\ell \cdot \ell}{Om \cdot Om}\right)\right), 2\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sin ky \cdot \sin ky\right), \left(\frac{\ell \cdot \ell}{Om \cdot Om}\right)\right), 2\right)\right)\right)\right)\right) \]

       8. sqr-sin-a N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right), \left(\frac{\ell \cdot \ell}{Om \cdot Om}\right)\right), 2\right)\right)\right)\right)\right) \]

       9. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right), \left(\frac{\ell \cdot \ell}{Om \cdot Om}\right)\right), 2\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \cos \left(2 \cdot ky\right)\right)\right), \left(\frac{\ell \cdot \ell}{Om \cdot Om}\right)\right), 2\right)\right)\right)\right)\right) \]

       11. cos-lowering-cos.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot ky\right)\right)\right)\right), \left(\frac{\ell \cdot \ell}{Om \cdot Om}\right)\right), 2\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \left(\frac{\ell \cdot \ell}{Om \cdot Om}\right)\right), 2\right)\right)\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(Om \cdot Om\right)\right)\right), 2\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(Om \cdot Om\right)\right)\right), 2\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 80.2%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right), 2\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 80.2%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right), 2\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \left(\frac{\ell}{Om \cdot Om} \cdot \ell\right)\right), 2\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{\ell}{Om \cdot Om}\right), \ell\right)\right), 2\right)\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(Om \cdot Om\right)\right), \ell\right)\right), 2\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 91.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(Om, Om\right)\right), \ell\right)\right), 2\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 91.3%

       \[\leadsto \sqrt{0.5 + \frac{0.5}{1 + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) \cdot \color{blue}{\left(\frac{\ell}{Om \cdot Om} \cdot \ell\right)}\right) \cdot 2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 4.8 \cdot 10^{-169}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot {\left(\ell \cdot ky\right)}^{2}}{Om \cdot Om}}}\\ \mathbf{elif}\;ky \leq 10^{+70}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\frac{\ell \cdot 4}{Om}}{\frac{Om}{\ell}} \cdot \left(2 \cdot \left(ky \cdot ky\right)\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) \cdot \left(\ell \cdot \frac{\ell}{Om \cdot Om}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;ky\_m \leq 10^{-167}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot {\left(l\_m \cdot ky\_m\right)}^{2}}{Om\_m \cdot Om\_m}}}\\ \mathbf{elif}\;ky\_m \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\frac{l\_m \cdot 4}{Om\_m}}{\frac{Om\_m}{l\_m}} \cdot \left(2 \cdot \left(ky\_m \cdot ky\_m\right)\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{\left(1 - \cos \left(2 \cdot ky\_m\right)\right) \cdot \left(l\_m \cdot l\_m\right)}{Om\_m \cdot Om\_m}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<= ky_m 1e-167)
   (sqrt
    (+ 0.5 (/ 0.5 (+ 1.0 (/ (* 2.0 (pow (* l_m ky_m) 2.0)) (* Om_m Om_m))))))
   (if (<= ky_m 2e+73)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (/
           (* (/ (/ (* l_m 4.0) Om_m) (/ Om_m l_m)) (* 2.0 (* ky_m ky_m)))
           2.0))))))
     (sqrt
      (+
       0.5
       (/
        0.5
        (+
         1.0
         (/ (* (- 1.0 (cos (* 2.0 ky_m))) (* l_m l_m)) (* Om_m Om_m)))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (ky_m <= 1e-167) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	} else if (ky_m <= 2e+73) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((((l_m * 4.0) / Om_m) / (Om_m / l_m)) * (2.0 * (ky_m * ky_m))) / 2.0))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / (1.0 + (((1.0 - cos((2.0 * ky_m))) * (l_m * l_m)) / (Om_m * Om_m))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (ky_m <= 1d-167) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + ((2.0d0 * ((l_m * ky_m) ** 2.0d0)) / (om_m * om_m))))))
    else if (ky_m <= 2d+73) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((((l_m * 4.0d0) / om_m) / (om_m / l_m)) * (2.0d0 * (ky_m * ky_m))) / 2.0d0))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + (((1.0d0 - cos((2.0d0 * ky_m))) * (l_m * l_m)) / (om_m * om_m))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (ky_m <= 1e-167) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * Math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	} else if (ky_m <= 2e+73) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((((l_m * 4.0) / Om_m) / (Om_m / l_m)) * (2.0 * (ky_m * ky_m))) / 2.0))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + (((1.0 - Math.cos((2.0 * ky_m))) * (l_m * l_m)) / (Om_m * Om_m))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	tmp = 0
	if ky_m <= 1e-167:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))))
	elif ky_m <= 2e+73:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((((l_m * 4.0) / Om_m) / (Om_m / l_m)) * (2.0 * (ky_m * ky_m))) / 2.0))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + (((1.0 - math.cos((2.0 * ky_m))) * (l_m * l_m)) / (Om_m * Om_m))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (ky_m <= 1e-167)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(2.0 * (Float64(l_m * ky_m) ^ 2.0)) / Float64(Om_m * Om_m))))));
	elseif (ky_m <= 2e+73)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(Float64(l_m * 4.0) / Om_m) / Float64(Om_m / l_m)) * Float64(2.0 * Float64(ky_m * ky_m))) / 2.0))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky_m))) * Float64(l_m * l_m)) / Float64(Om_m * Om_m))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0;
	if (ky_m <= 1e-167)
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l_m * ky_m) ^ 2.0)) / (Om_m * Om_m))))));
	elseif (ky_m <= 2e+73)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((((l_m * 4.0) / Om_m) / (Om_m / l_m)) * (2.0 * (ky_m * ky_m))) / 2.0))))));
	else
		tmp = sqrt((0.5 + (0.5 / (1.0 + (((1.0 - cos((2.0 * ky_m))) * (l_m * l_m)) / (Om_m * Om_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[ky$95$m, 1e-167], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(2.0 * N[Power[N[(l$95$m * ky$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ky$95$m, 2e+73], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision] / N[(Om$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(ky$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 1e-167

   1. Initial program 96.4%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 86.1%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 73.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 73.2%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + 2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 73.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   10. Simplified 73.4%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. pow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot {\ell}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\left(\sin ky \cdot \ell\right)}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\sin ky \cdot \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin ky, \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       6. sin-lowering-sin.f64 82.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 82.9%

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

   13. Taylor expanded in ky around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\color{blue}{\left(ky \cdot \ell\right)}, 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\ell \cdot ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 73.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   15. Simplified 73.5%

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

   if 1e-167 < ky < 1.99999999999999997e73

   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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 88.4%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 88.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 88.4%

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

      1. pow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\ell \cdot \left(\ell \cdot 4\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\sin ky \cdot \sin ky\right)\right)\right)\right)\right)\right)\right) \]

      5. sin-mult N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)}{2}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 66.5%

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

   10. Taylor expanded in ky around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \color{blue}{\left(2 \cdot {ky}^{2}\right)}\right), 2\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \left({ky}^{2}\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \left(ky \cdot ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 98.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(ky, ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   12. Simplified 98.1%

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

   if 1.99999999999999997e73 < ky

   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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 84.8%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 84.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 84.2%

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

      1. pow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\ell \cdot \left(\ell \cdot 4\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\sin ky \cdot \sin ky\right)\right)\right)\right)\right)\right)\right) \]

      5. sin-mult N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)}{2}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 96.8%

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

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + \frac{{\ell}^{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{{Om}^{2}}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{{\ell}^{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{{Om}^{2}}\right)\right)\right)\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(1 - \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(1 - \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(1 - \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       7. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(\left(2 \cdot ky\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 82.0%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   12. Simplified 82.0%

       \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{1 + \frac{\left(\ell \cdot \ell\right) \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{Om \cdot Om}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 10^{-167}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot {\left(\ell \cdot ky\right)}^{2}}{Om \cdot Om}}}\\ \mathbf{elif}\;ky \leq 2 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\frac{\ell \cdot 4}{Om}}{\frac{Om}{\ell}} \cdot \left(2 \cdot \left(ky \cdot ky\right)\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 1.55 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om\_m \leq 1.9 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{\left(1 - \cos \left(2 \cdot ky\_m\right)\right) \cdot \left(l\_m \cdot l\_m\right)}{Om\_m \cdot Om\_m}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<= Om_m 1.55e-161)
   (sqrt 0.5)
   (if (<= Om_m 1.9e+127)
     (sqrt
      (+
       0.5
       (/
        0.5
        (+ 1.0 (/ (* (- 1.0 (cos (* 2.0 ky_m))) (* l_m l_m)) (* Om_m Om_m))))))
     1.0)))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (Om_m <= 1.55e-161) {
		tmp = sqrt(0.5);
	} else if (Om_m <= 1.9e+127) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + (((1.0 - cos((2.0 * ky_m))) * (l_m * l_m)) / (Om_m * Om_m))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (om_m <= 1.55d-161) then
        tmp = sqrt(0.5d0)
    else if (om_m <= 1.9d+127) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + (((1.0d0 - cos((2.0d0 * ky_m))) * (l_m * l_m)) / (om_m * om_m))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (Om_m <= 1.55e-161) {
		tmp = Math.sqrt(0.5);
	} else if (Om_m <= 1.9e+127) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + (((1.0 - Math.cos((2.0 * ky_m))) * (l_m * l_m)) / (Om_m * Om_m))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	tmp = 0
	if Om_m <= 1.55e-161:
		tmp = math.sqrt(0.5)
	elif Om_m <= 1.9e+127:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + (((1.0 - math.cos((2.0 * ky_m))) * (l_m * l_m)) / (Om_m * Om_m))))))
	else:
		tmp = 1.0
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Om_m <= 1.55e-161)
		tmp = sqrt(0.5);
	elseif (Om_m <= 1.9e+127)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky_m))) * Float64(l_m * l_m)) / Float64(Om_m * Om_m))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0;
	if (Om_m <= 1.55e-161)
		tmp = sqrt(0.5);
	elseif (Om_m <= 1.9e+127)
		tmp = sqrt((0.5 + (0.5 / (1.0 + (((1.0 - cos((2.0 * ky_m))) * (l_m * l_m)) / (Om_m * Om_m))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[Om$95$m, 1.55e-161], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om$95$m, 1.9e+127], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 1.5499999999999999e-161

   1. Initial program 97.5%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right) \]
   4. Step-by-step derivation

   5. Simplified 61.4%

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

   if 1.5499999999999999e-161 < Om < 1.8999999999999999e127

   1. Initial program 96.9%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 96.1%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 88.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 88.2%

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

      1. pow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\ell \cdot \left(\ell \cdot 4\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\sin ky \cdot \sin ky\right)\right)\right)\right)\right)\right)\right) \]

      5. sin-mult N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \frac{\cos \left(ky - ky\right) - \cos \left(ky + ky\right)}{2}\right)\right)\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)}{2}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{\ell \cdot \left(\ell \cdot 4\right)}{Om \cdot Om} \cdot \left(\cos \left(ky - ky\right) - \cos \left(ky + ky\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 76.0%

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

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + \frac{{\ell}^{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{{Om}^{2}}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{{\ell}^{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{{Om}^{2}}\right)\right)\right)\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(1 - \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(1 - \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(1 - \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \cos \left(2 \cdot ky\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       7. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(\left(2 \cdot ky\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 76.6%

           \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   12. Simplified 76.6%

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

   if 1.8999999999999999e127 < 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 94.1%

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

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 100.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.55 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.9 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om\_m \leq 9.5 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot {\left(l\_m \cdot ky\_m\right)}^{2}}{Om\_m \cdot Om\_m}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<= Om_m 2e-186)
   (sqrt 0.5)
   (if (<= Om_m 9.5e+126)
     (sqrt
      (+ 0.5 (/ 0.5 (+ 1.0 (/ (* 2.0 (pow (* l_m ky_m) 2.0)) (* Om_m Om_m))))))
     1.0)))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (Om_m <= 2e-186) {
		tmp = sqrt(0.5);
	} else if (Om_m <= 9.5e+126) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (om_m <= 2d-186) then
        tmp = sqrt(0.5d0)
    else if (om_m <= 9.5d+126) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + ((2.0d0 * ((l_m * ky_m) ** 2.0d0)) / (om_m * om_m))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (Om_m <= 2e-186) {
		tmp = Math.sqrt(0.5);
	} else if (Om_m <= 9.5e+126) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * Math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	tmp = 0
	if Om_m <= 2e-186:
		tmp = math.sqrt(0.5)
	elif Om_m <= 9.5e+126:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * math.pow((l_m * ky_m), 2.0)) / (Om_m * Om_m))))))
	else:
		tmp = 1.0
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Om_m <= 2e-186)
		tmp = sqrt(0.5);
	elseif (Om_m <= 9.5e+126)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(2.0 * (Float64(l_m * ky_m) ^ 2.0)) / Float64(Om_m * Om_m))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0;
	if (Om_m <= 2e-186)
		tmp = sqrt(0.5);
	elseif (Om_m <= 9.5e+126)
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l_m * ky_m) ^ 2.0)) / (Om_m * Om_m))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[Om$95$m, 2e-186], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om$95$m, 9.5e+126], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(2.0 * N[Power[N[(l$95$m * ky$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 1.9999999999999998e-186

   1. Initial program 97.4%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right) \]
   4. Step-by-step derivation

   5. Simplified 60.6%

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

   if 1.9999999999999998e-186 < Om < 9.49999999999999951e126

   1. Initial program 97.0%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 96.3%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 87.2%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 87.2%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(1 + 2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(2 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({\sin ky}^{2}\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 87.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   10. Simplified 87.5%

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

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot \left(\ell \cdot \ell\right)\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. pow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\sin ky}^{2} \cdot {\ell}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       3. pow-prod-down N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left({\left(\sin ky \cdot \ell\right)}^{2}\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\sin ky \cdot \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\sin ky, \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       6. sin-lowering-sin.f64 93.8%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(ky\right), \ell\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 93.8%

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

   13. Taylor expanded in ky around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\color{blue}{\left(ky \cdot \ell\right)}, 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\left(\ell \cdot ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

       2. *-lowering-*.f64 85.4%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\ell, ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right) \]

   15. Simplified 85.4%

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

   if 9.49999999999999951e126 < 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 94.1%

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

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 100.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<= Om_m 7.8e-100) (sqrt 0.5) 1.0))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (Om_m <= 7.8e-100) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (om_m <= 7.8d-100) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (Om_m <= 7.8e-100) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	tmp = 0
	if Om_m <= 7.8e-100:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Om_m <= 7.8e-100)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0;
	if (Om_m <= 7.8e-100)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[Om$95$m, 7.8e-100], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 7.79999999999999955e-100

   1. Initial program 97.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right) \]
   4. Step-by-step derivation

   5. Simplified 62.2%

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

   if 7.79999999999999955e-100 < Om

   1. Initial program 98.8%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 96.5%

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

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 80.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.8% accurate, 722.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m) :precision binary64 1.0)

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return 1.0;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = 1.0d0
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return 1.0;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	return 1.0

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	return 1.0
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp = code(l_m, Om_m, kx_m, ky_m)
	tmp = 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := 1.0

Derivation

1. Initial program 97.7%

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

   4. distribute-rgt1-in N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

3. Simplified 86.2%

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

5. Taylor expanded in l around 0

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Simplified 63.8%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))))))))