Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 98.3%

Time: 17.1s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

Alternative 2: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{0.5}{\sqrt{1 + \ell \cdot \left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{\frac{\ell \cdot 4}{Om}}{Om}\right)}}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (/
    0.5
    (sqrt
     (+
      1.0
      (*
       l
       (*
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))
        (/ (/ (* l 4.0) Om) Om)))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 / sqrt((1.0 + (l * ((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * (((l * 4.0) / Om) / Om))))))));
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (l * (((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)) * (((l * 4.0d0) / om) / om))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (l * ((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)) * (((l * 4.0) / Om) / Om))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (l * ((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)) * (((l * 4.0) / Om) / Om))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(l * Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * Float64(Float64(Float64(l * 4.0) / Om) / Om))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (l * (((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (((l * 4.0) / Om) / Om))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(l * N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

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

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt1-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

3. Simplified 98.6%

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

Alternative 3: 94.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}{\frac{Om}{\frac{\ell \cdot 4}{Om}}}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 1e+96)
   (sqrt
    (+
     0.5
     (*
      0.5
      (pow
       (+
        1.0
        (/
         (*
          l
          (+
           (- 0.5 (* 0.5 (cos (* 2.0 kx))))
           (- 0.5 (* 0.5 (cos (* 2.0 ky))))))
         (/ Om (/ (* l 4.0) Om))))
       -0.5))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = sqrt((0.5 + (0.5 * pow((1.0 + ((l * ((0.5 - (0.5 * cos((2.0 * kx)))) + (0.5 - (0.5 * cos((2.0 * ky)))))) / (Om / ((l * 4.0) / Om)))), -0.5))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l) / om) <= 1d+96) then
        tmp = sqrt((0.5d0 + (0.5d0 * ((1.0d0 + ((l * ((0.5d0 - (0.5d0 * cos((2.0d0 * kx)))) + (0.5d0 - (0.5d0 * cos((2.0d0 * ky)))))) / (om / ((l * 4.0d0) / om)))) ** (-0.5d0)))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = Math.sqrt((0.5 + (0.5 * Math.pow((1.0 + ((l * ((0.5 - (0.5 * Math.cos((2.0 * kx)))) + (0.5 - (0.5 * Math.cos((2.0 * ky)))))) / (Om / ((l * 4.0) / Om)))), -0.5))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((2.0 * l) / Om) <= 1e+96:
		tmp = math.sqrt((0.5 + (0.5 * math.pow((1.0 + ((l * ((0.5 - (0.5 * math.cos((2.0 * kx)))) + (0.5 - (0.5 * math.cos((2.0 * ky)))))) / (Om / ((l * 4.0) / Om)))), -0.5))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 1e+96)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * (Float64(1.0 + Float64(Float64(l * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky)))))) / Float64(Om / Float64(Float64(l * 4.0) / Om)))) ^ -0.5))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((2.0 * l) / Om) <= 1e+96)
		tmp = sqrt((0.5 + (0.5 * ((1.0 + ((l * ((0.5 - (0.5 * cos((2.0 * kx)))) + (0.5 - (0.5 * cos((2.0 * ky)))))) / (Om / ((l * 4.0) / Om)))) ^ -0.5))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 1e+96], N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[(1.0 + N[(N[(l * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.00000000000000005e96

   1. Initial program 99.1%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 98.3%

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

      1. metadata-eval N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

   6. Applied egg-rr 95.0%

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

   if 1.00000000000000005e96 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around inf

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

      1. sqrt-lowering-sqrt.f64 97.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 97.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}{\frac{Om}{\frac{\ell \cdot 4}{Om}}}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}{\frac{Om}{\frac{\ell \cdot 4}{Om}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 1e+96)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (/
         (*
          l
          (+
           (- 0.5 (* 0.5 (cos (* 2.0 kx))))
           (- 0.5 (* 0.5 (cos (* 2.0 ky))))))
         (/ Om (/ (* l 4.0) Om))))))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((l * ((0.5 - (0.5 * cos((2.0 * kx)))) + (0.5 - (0.5 * cos((2.0 * ky)))))) / (Om / ((l * 4.0) / Om))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l) / om) <= 1d+96) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((l * ((0.5d0 - (0.5d0 * cos((2.0d0 * kx)))) + (0.5d0 - (0.5d0 * cos((2.0d0 * ky)))))) / (om / ((l * 4.0d0) / om))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((l * ((0.5 - (0.5 * Math.cos((2.0 * kx)))) + (0.5 - (0.5 * Math.cos((2.0 * ky)))))) / (Om / ((l * 4.0) / Om))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((2.0 * l) / Om) <= 1e+96:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((l * ((0.5 - (0.5 * math.cos((2.0 * kx)))) + (0.5 - (0.5 * math.cos((2.0 * ky)))))) / (Om / ((l * 4.0) / Om))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 1e+96)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(l * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky)))))) / Float64(Om / Float64(Float64(l * 4.0) / Om))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((2.0 * l) / Om) <= 1e+96)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((l * ((0.5 - (0.5 * cos((2.0 * kx)))) + (0.5 - (0.5 * cos((2.0 * ky)))))) / (Om / ((l * 4.0) / Om))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 1e+96], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(l * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.00000000000000005e96

   1. Initial program 99.1%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 98.3%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

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

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

   6. Applied egg-rr 94.9%

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

   if 1.00000000000000005e96 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around inf

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

      1. sqrt-lowering-sqrt.f64 97.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 97.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}{\frac{Om}{\frac{\ell \cdot 4}{Om}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + \frac{\left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right) \cdot \frac{\ell}{Om}}{\frac{Om}{\ell \cdot 4}}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 1e+96)
   (sqrt
    (+
     0.5
     (*
      0.5
      (pow
       (+
        1.0
        (/ (* (+ 0.5 (* (cos (* 2.0 ky)) -0.5)) (/ l Om)) (/ Om (* l 4.0))))
       -0.5))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = sqrt((0.5 + (0.5 * pow((1.0 + (((0.5 + (cos((2.0 * ky)) * -0.5)) * (l / Om)) / (Om / (l * 4.0)))), -0.5))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l) / om) <= 1d+96) then
        tmp = sqrt((0.5d0 + (0.5d0 * ((1.0d0 + (((0.5d0 + (cos((2.0d0 * ky)) * (-0.5d0))) * (l / om)) / (om / (l * 4.0d0)))) ** (-0.5d0)))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = Math.sqrt((0.5 + (0.5 * Math.pow((1.0 + (((0.5 + (Math.cos((2.0 * ky)) * -0.5)) * (l / Om)) / (Om / (l * 4.0)))), -0.5))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((2.0 * l) / Om) <= 1e+96:
		tmp = math.sqrt((0.5 + (0.5 * math.pow((1.0 + (((0.5 + (math.cos((2.0 * ky)) * -0.5)) * (l / Om)) / (Om / (l * 4.0)))), -0.5))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 1e+96)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * (Float64(1.0 + Float64(Float64(Float64(0.5 + Float64(cos(Float64(2.0 * ky)) * -0.5)) * Float64(l / Om)) / Float64(Om / Float64(l * 4.0)))) ^ -0.5))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((2.0 * l) / Om) <= 1e+96)
		tmp = sqrt((0.5 + (0.5 * ((1.0 + (((0.5 + (cos((2.0 * ky)) * -0.5)) * (l / Om)) / (Om / (l * 4.0)))) ^ -0.5))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 1e+96], N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[(1.0 + N[(N[(N[(0.5 + N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(l * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.00000000000000005e96

   1. Initial program 99.1%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 98.3%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

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

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 80.9%

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

   9. Simplified 80.9%

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

   10. Taylor expanded in kx around 0

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

       1. cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f64 87.2%

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

   12. Simplified 87.2%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

   14. Applied egg-rr 88.3%

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

   if 1.00000000000000005e96 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around inf

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

      1. sqrt-lowering-sqrt.f64 97.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 97.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + \frac{\left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right) \cdot \frac{\ell}{Om}}{\frac{Om}{\ell \cdot 4}}\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right)}{\frac{Om}{\frac{\ell \cdot 4}{Om}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 1e+96)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (/
         (* l (+ 0.5 (* (cos (* 2.0 ky)) -0.5)))
         (/ Om (/ (* l 4.0) Om))))))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((l * (0.5 + (cos((2.0 * ky)) * -0.5))) / (Om / ((l * 4.0) / Om))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l) / om) <= 1d+96) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((l * (0.5d0 + (cos((2.0d0 * ky)) * (-0.5d0)))) / (om / ((l * 4.0d0) / om))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((l * (0.5 + (Math.cos((2.0 * ky)) * -0.5))) / (Om / ((l * 4.0) / Om))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((2.0 * l) / Om) <= 1e+96:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((l * (0.5 + (math.cos((2.0 * ky)) * -0.5))) / (Om / ((l * 4.0) / Om))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 1e+96)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(l * Float64(0.5 + Float64(cos(Float64(2.0 * ky)) * -0.5))) / Float64(Om / Float64(Float64(l * 4.0) / Om))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((2.0 * l) / Om) <= 1e+96)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((l * (0.5 + (cos((2.0 * ky)) * -0.5))) / (Om / ((l * 4.0) / Om))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 1e+96], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(l * N[(0.5 + N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om / N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.00000000000000005e96

   1. Initial program 99.1%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 98.3%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

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

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 80.9%

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

   9. Simplified 80.9%

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

   10. Taylor expanded in kx around 0

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

       1. cancel-sign-sub-inv N/A

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

       2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

       7. *-lowering-*.f64 87.2%

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

   12. Simplified 87.2%

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

   if 1.00000000000000005e96 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around inf

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

      1. sqrt-lowering-sqrt.f64 97.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 97.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right)}{\frac{Om}{\frac{\ell \cdot 4}{Om}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2}{Om} \cdot \left(\ell \cdot \left(\left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right) \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 1e+96)
   (sqrt
    (+
     0.5
     (/
      0.5
      (+
       1.0
       (* (/ 2.0 Om) (* l (* (+ 0.5 (* (cos (* 2.0 ky)) -0.5)) (/ l Om))))))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 / Om) * (l * ((0.5 + (cos((2.0 * ky)) * -0.5)) * (l / Om))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l) / om) <= 1d+96) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + ((2.0d0 / om) * (l * ((0.5d0 + (cos((2.0d0 * ky)) * (-0.5d0))) * (l / om))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+96) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 / Om) * (l * ((0.5 + (Math.cos((2.0 * ky)) * -0.5)) * (l / Om))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((2.0 * l) / Om) <= 1e+96:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 / Om) * (l * ((0.5 + (math.cos((2.0 * ky)) * -0.5)) * (l / Om))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 1e+96)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(2.0 / Om) * Float64(l * Float64(Float64(0.5 + Float64(cos(Float64(2.0 * ky)) * -0.5)) * Float64(l / Om))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((2.0 * l) / Om) <= 1e+96)
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 / Om) * (l * ((0.5 + (cos((2.0 * ky)) * -0.5)) * (l / Om))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 1e+96], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(2.0 / Om), $MachinePrecision] * N[(l * N[(N[(0.5 + N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.00000000000000005e96

   1. Initial program 99.1%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 98.3%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

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

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. cancel-sign-sub-inv N/A

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

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

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

      11. metadata-eval N/A

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

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

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

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

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

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

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

      15. unpow2 N/A

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

   9. Simplified 75.3%

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

   10. Taylor expanded in l around 0

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

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

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

       2. associate-*r/ N/A

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

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

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f64 74.6%

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

   12. Simplified 74.6%

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

       1. times-frac N/A

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

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

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

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

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

       4. metadata-eval N/A

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

       5. cancel-sign-sub-inv N/A

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

       6. associate-*l* N/A

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

       7. cancel-sign-sub-inv N/A

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

       8. metadata-eval N/A

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

       9. associate-/l* N/A

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

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

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

   14. Applied egg-rr 87.1%

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

   if 1.00000000000000005e96 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 100.0%

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

   3. Taylor expanded in l around inf

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

      1. sqrt-lowering-sqrt.f64 97.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 97.8%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2}{Om} \cdot \left(\ell \cdot \left(\left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right) \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 2e+79) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 2e+79) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 2d+79) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 2e+79) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 2e+79:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 2e+79)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 2e+79)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 2e+79], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.99999999999999993e79

   1. Initial program 100.0%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 99.1%

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

   5. Taylor expanded in l around 0

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

   7. Simplified 72.3%

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

   if 1.99999999999999993e79 < l

   1. Initial program 96.0%

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

   3. Taylor expanded in l around inf

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

      1. sqrt-lowering-sqrt.f64 81.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 81.6%

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

Alternative 9: 62.5% accurate, 722.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 1.0)

C

double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

Python

def code(l, Om, kx, ky):
	return 1.0

Julia

function code(l, Om, kx, ky)
	return 1.0
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = 1.0;
end

Wolfram

code[l_, Om_, kx_, ky_] := 1.0

Derivation

1. Initial program 99.2%

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

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

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt1-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

3. Simplified 98.6%

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

5. Taylor expanded in l around 0

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

7. Simplified 64.5%

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

Reproduce

herbie shell --seed 2024138 
(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))))))))))