Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 99.3%

Time: 5.7s

Alternatives: 6

Speedup: 0.6×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx, ky)
use fmin_fmax_functions
    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: 98.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx, ky)
use fmin_fmax_functions
    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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)}}\right)} \leq 0.72:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \left(4 \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \mathsf{fma}\left(ky\_m, ky\_m, 0.5 - 0.5 \cdot \cos \left(kx\_m + kx\_m\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(t\_0, t\_0 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\_m\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\_m\right)\right)\right), 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l l) Om)))
   (if (<=
        (sqrt
         (*
          (/ 1.0 2.0)
          (+
           1.0
           (/
            1.0
            (sqrt
             (+
              1.0
              (*
               (pow (/ (* 2.0 l) Om) 2.0)
               (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))))))
        0.72)
     (sqrt
      (*
       0.5
       (+
        1.0
        (/
         1.0
         (sqrt
          (+
           1.0
           (*
            (* 4.0 (* (/ l Om) (/ l Om)))
            (fma ky_m ky_m (- 0.5 (* 0.5 (cos (+ kx_m kx_m))))))))))))
     (sqrt
      (*
       0.5
       (+
        1.0
        (/
         1.0
         (sqrt
          (fma
           t_0
           (*
            t_0
            (+
             (- 0.5 (* 0.5 (cos (* 2.0 ky_m))))
             (- 0.5 (* 0.5 (cos (* 2.0 kx_m))))))
           1.0)))))))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double t_0 = (l + l) / Om;
	double tmp;
	if (sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))))))) <= 0.72) {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt((1.0 + ((4.0 * ((l / Om) * (l / Om))) * fma(ky_m, ky_m, (0.5 - (0.5 * cos((kx_m + kx_m))))))))))));
	} else {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(t_0, (t_0 * ((0.5 - (0.5 * cos((2.0 * ky_m)))) + (0.5 - (0.5 * cos((2.0 * kx_m)))))), 1.0))))));
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	t_0 = Float64(Float64(l + l) / Om)
	tmp = 0.0
	if (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_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))))))) <= 0.72)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(l / Om) * Float64(l / Om))) * fma(ky_m, ky_m, Float64(0.5 - Float64(0.5 * cos(Float64(kx_m + kx_m))))))))))));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(t_0, Float64(t_0 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky_m)))) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx_m)))))), 1.0))))));
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[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$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.72], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(4.0 * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(ky$95$m * ky$95$m + N[(0.5 - N[(0.5 * N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(t$95$0 * N[(t$95$0 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))))))) < 0.71999999999999997

   1. Initial program 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 98.5

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

   3. Applied rewrites 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 93.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 93.5

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a-rev N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 74.0

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. cos-2 N/A

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

      12. cos-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-+.f64 74.0

          \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \left(4 \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \mathsf{fma}\left(ky, ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}}\right)} \]

   10. Applied rewrites 74.0%

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

   if 0.71999999999999997 < (sqrt.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))))))

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 92.6

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

   5. Applied rewrites 92.6%

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

Alternative 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)} \leq 5000000000000:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(ky\_m + ky\_m\right) \cdot 0.5\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\_m\right)\right), 4 \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right), 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (sqrt
       (+
        1.0
        (*
         (pow (/ (* 2.0 l) Om) 2.0)
         (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))
      5000000000000.0)
   (sqrt
    (+
     (/
      0.5
      (sqrt
       (fma
        (+
         (- 0.5 (* (cos (+ ky_m ky_m)) 0.5))
         (- 0.5 (* 0.5 (cos (* 2.0 kx_m)))))
        (* 4.0 (* (/ l Om) (/ l Om)))
        1.0)))
     0.5))
   (sqrt 0.5)))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))) <= 5000000000000.0) {
		tmp = sqrt(((0.5 / sqrt(fma(((0.5 - (cos((ky_m + ky_m)) * 0.5)) + (0.5 - (0.5 * cos((2.0 * kx_m))))), (4.0 * ((l / Om) * (l / Om))), 1.0))) + 0.5));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 5000000000000.0)
		tmp = sqrt(Float64(Float64(0.5 / sqrt(fma(Float64(Float64(0.5 - Float64(cos(Float64(ky_m + ky_m)) * 0.5)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx_m))))), Float64(4.0 * Float64(Float64(l / Om) * Float64(l / Om))), 1.0))) + 0.5));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[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$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5000000000000.0], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[(N[(0.5 - N[(N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 5e12

   1. Initial program 98.5%

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

   3. Applied rewrites 91.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.6

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

      4. lift-*.f64 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f64 91.6

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

   5. Applied rewrites 91.6%

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

   if 5e12 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 98.5

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

   3. Applied rewrites 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 98.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.5

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 56.1

         \[\leadsto \sqrt{0.5} \]

   10. Applied rewrites 56.1%

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

Alternative 3: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)} \leq 5000000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \left(4 \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(ky\_m + ky\_m\right)\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (sqrt
       (+
        1.0
        (*
         (pow (/ (* 2.0 l) Om) 2.0)
         (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))
      5000000000000.0)
   (sqrt
    (*
     0.5
     (+
      1.0
      (/
       1.0
       (sqrt
        (+
         1.0
         (*
          (* 4.0 (* (/ l Om) (/ l Om)))
          (- 0.5 (* 0.5 (cos (+ ky_m ky_m)))))))))))
   (sqrt 0.5)))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))) <= 5000000000000.0) {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt((1.0 + ((4.0 * ((l / Om) * (l / Om))) * (0.5 - (0.5 * cos((ky_m + ky_m)))))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

kx_m =     private
ky_m =     private
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))))) <= 5000000000000.0d0) then
        tmp = sqrt((0.5d0 * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((4.0d0 * ((l / om) * (l / om))) * (0.5d0 - (0.5d0 * cos((ky_m + ky_m)))))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))))) <= 5000000000000.0) {
		tmp = Math.sqrt((0.5 * (1.0 + (1.0 / Math.sqrt((1.0 + ((4.0 * ((l / Om) * (l / Om))) * (0.5 - (0.5 * Math.cos((ky_m + ky_m)))))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))))) <= 5000000000000.0:
		tmp = math.sqrt((0.5 * (1.0 + (1.0 / math.sqrt((1.0 + ((4.0 * ((l / Om) * (l / Om))) * (0.5 - (0.5 * math.cos((ky_m + ky_m)))))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 5000000000000.0)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(l / Om) * Float64(l / Om))) * Float64(0.5 - Float64(0.5 * cos(Float64(ky_m + ky_m)))))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 5000000000000.0)
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt((1.0 + ((4.0 * ((l / Om) * (l / Om))) * (0.5 - (0.5 * cos((ky_m + ky_m)))))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[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$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5000000000000.0], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(4.0 * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 5e12

   1. Initial program 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 98.5

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

   3. Applied rewrites 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 98.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.5

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. sqr-sin-a-rev N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 91.2

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. cos-2 N/A

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

      10. cos-sum N/A

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

      11. lift-cos.f64 N/A

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

      12. lift-+.f64 91.2

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

   10. Applied rewrites 91.2%

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

   if 5e12 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 98.5

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

   3. Applied rewrites 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 98.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.5

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 56.1

         \[\leadsto \sqrt{0.5} \]

   10. Applied rewrites 56.1%

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

Alternative 4: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{1 + \left(4 \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(0.5 - \cos \left(kx\_m + kx\_m\right) \cdot 0.5\right) + {\sin ky\_m}^{2}\right)}}\right)} \end{array} \]

FPCore

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

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return sqrt((0.5 * (1.0 + (1.0 / sqrt((1.0 + ((4.0 * ((l / Om) * (l / Om))) * ((0.5 - (cos((kx_m + kx_m)) * 0.5)) + pow(sin(ky_m), 2.0)))))))));
}

Fortran

kx_m =     private
ky_m =     private
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = sqrt((0.5d0 * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((4.0d0 * ((l / om) * (l / om))) * ((0.5d0 - (cos((kx_m + kx_m)) * 0.5d0)) + (sin(ky_m) ** 2.0d0)))))))))
end function

Java

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

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	return math.sqrt((0.5 * (1.0 + (1.0 / math.sqrt((1.0 + ((4.0 * ((l / Om) * (l / Om))) * ((0.5 - (math.cos((kx_m + kx_m)) * 0.5)) + math.pow(math.sin(ky_m), 2.0)))))))))

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(l / Om) * Float64(l / Om))) * Float64(Float64(0.5 - Float64(cos(Float64(kx_m + kx_m)) * 0.5)) + (sin(ky_m) ^ 2.0)))))))))
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp = code(l, Om, kx_m, ky_m)
	tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt((1.0 + ((4.0 * ((l / Om) * (l / Om))) * ((0.5 - (cos((kx_m + kx_m)) * 0.5)) + (sin(ky_m) ^ 2.0)))))))));
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(4.0 * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. swap-sqr N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 98.5

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

3. Applied rewrites 98.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 98.5

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 98.5

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

   13. lift-*.f64 N/A

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

   14. count-2-rev N/A

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

   15. lower-+.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 98.5

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

7. Applied rewrites 98.5%

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

Alternative 5: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

kx_m =     private
ky_m =     private
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))))) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[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$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 2

   1. Initial program 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 98.5

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

   3. Applied rewrites 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 98.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.5

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in l around 0

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 62.4%

       \[\leadsto \color{blue}{1} \]

   if 2 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. swap-sqr N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 98.5

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

   3. Applied rewrites 98.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 98.5

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.5

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

      13. lift-*.f64 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 56.1

         \[\leadsto \sqrt{0.5} \]

   10. Applied rewrites 56.1%

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

Alternative 6: 62.4% accurate, 142.7× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ 1 \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m) :precision binary64 1.0)

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return 1.0;
}

Fortran

kx_m =     private
ky_m =     private
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = 1.0d0
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	return 1.0;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	return 1.0

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	return 1.0
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp = code(l, Om, kx_m, ky_m)
	tmp = 1.0;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := 1.0

Derivation

1. Initial program 98.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. swap-sqr N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 98.5

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

3. Applied rewrites 98.5%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a-rev N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 98.5

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 98.5

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

   13. lift-*.f64 N/A

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

   14. count-2-rev N/A

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

   15. lower-+.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in l around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 62.4%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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))))))))))