Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 98.4%

Time: 16.8s

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: 98.4% 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: 98.4% 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 98.8%

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

Alternative 2: 83.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(ky \cdot ky\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}{Om} \cdot \frac{\ell \cdot 4}{Om}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (pow (sin ky) 2.0) 4e-33)
   (sqrt (+ 0.5 (/ 0.5 (+ 1.0 (* 2.0 (* (* ky ky) (/ (* l l) (* Om Om))))))))
   (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)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (pow(sin(ky), 2.0) <= 4e-33) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	} else {
		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)))))));
	}
	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 ((sin(ky) ** 2.0d0) <= 4d-33) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + (2.0d0 * ((ky * ky) * ((l * l) / (om * om))))))))
    else
        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)))))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Math.pow(Math.sin(ky), 2.0) <= 4e-33) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	} else {
		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)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if math.pow(math.sin(ky), 2.0) <= 4e-33:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))))
	else:
		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)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if ((sin(ky) ^ 2.0) <= 4e-33)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * ky) * Float64(Float64(l * l) / Float64(Om * Om))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(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)))))) / Om) * Float64(Float64(l * 4.0) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((sin(ky) ^ 2.0) <= 4e-33)
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	else
		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)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision], 4e-33], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(2.0 * N[(N[(ky * ky), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(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] / Om), $MachinePrecision] * N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 4.0000000000000002e-33

   1. Initial program 97.3%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 88.4%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 66.6%

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

   10. Simplified 66.6%

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

   if 4.0000000000000002e-33 < (pow.f64 (sin.f64 ky) #s(literal 2 binary64))

   1. Initial program 100.0%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 88.9%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   6. Applied egg-rr 98.6%

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

Alternative 3: 83.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(ky \cdot ky\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + \frac{\frac{\ell \cdot 4}{Om} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{2 \cdot \frac{Om}{\ell}}\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (pow (sin ky) 2.0) 4e-33)
   (sqrt (+ 0.5 (/ 0.5 (+ 1.0 (* 2.0 (* (* ky ky) (/ (* l l) (* Om Om))))))))
   (sqrt
    (+
     0.5
     (*
      0.5
      (pow
       (+
        1.0
        (/ (* (/ (* l 4.0) Om) (- 1.0 (cos (* 2.0 ky)))) (* 2.0 (/ Om l))))
       -0.5))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (pow(sin(ky), 2.0) <= 4e-33) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	} else {
		tmp = sqrt((0.5 + (0.5 * pow((1.0 + ((((l * 4.0) / Om) * (1.0 - cos((2.0 * ky)))) / (2.0 * (Om / l)))), -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 ((sin(ky) ** 2.0d0) <= 4d-33) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + (2.0d0 * ((ky * ky) * ((l * l) / (om * om))))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 * ((1.0d0 + ((((l * 4.0d0) / om) * (1.0d0 - cos((2.0d0 * ky)))) / (2.0d0 * (om / l)))) ** (-0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Math.pow(Math.sin(ky), 2.0) <= 4e-33) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * Math.pow((1.0 + ((((l * 4.0) / Om) * (1.0 - Math.cos((2.0 * ky)))) / (2.0 * (Om / l)))), -0.5))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if math.pow(math.sin(ky), 2.0) <= 4e-33:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 * math.pow((1.0 + ((((l * 4.0) / Om) * (1.0 - math.cos((2.0 * ky)))) / (2.0 * (Om / l)))), -0.5))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if ((sin(ky) ^ 2.0) <= 4e-33)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * ky) * Float64(Float64(l * l) / Float64(Om * Om))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * (Float64(1.0 + Float64(Float64(Float64(Float64(l * 4.0) / Om) * Float64(1.0 - cos(Float64(2.0 * ky)))) / Float64(2.0 * Float64(Om / l)))) ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((sin(ky) ^ 2.0) <= 4e-33)
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	else
		tmp = sqrt((0.5 + (0.5 * ((1.0 + ((((l * 4.0) / Om) * (1.0 - cos((2.0 * ky)))) / (2.0 * (Om / l)))) ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision], 4e-33], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(2.0 * N[(N[(ky * ky), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[(1.0 + N[(N[(N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision] * N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 4.0000000000000002e-33

   1. Initial program 97.3%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 88.4%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 66.6%

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

   10. Simplified 66.6%

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

   if 4.0000000000000002e-33 < (pow.f64 (sin.f64 ky) #s(literal 2 binary64))

   1. Initial program 100.0%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 88.9%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 88.6%

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

   7. Simplified 88.6%

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. sin-mult N/A

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

      5. associate-*r/ N/A

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

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

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

   9. Applied egg-rr 99.0%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

   11. Applied egg-rr 99.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(ky \cdot ky\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + \frac{\frac{\ell \cdot 4}{Om} \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}{2 \cdot \frac{Om}{\ell}}\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(ky \cdot ky\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{\frac{\ell \cdot 4}{Om}}{\frac{Om}{\ell}}}{2}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (pow (sin ky) 2.0) 4e-33)
   (sqrt (+ 0.5 (/ 0.5 (+ 1.0 (* 2.0 (* (* ky ky) (/ (* l l) (* Om Om))))))))
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (/
         (* (- 1.0 (cos (* 2.0 ky))) (/ (/ (* l 4.0) Om) (/ Om l)))
         2.0))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (pow(sin(ky), 2.0) <= 4e-33) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((1.0 - cos((2.0 * ky))) * (((l * 4.0) / Om) / (Om / l))) / 2.0))))));
	}
	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 ((sin(ky) ** 2.0d0) <= 4d-33) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + (2.0d0 * ((ky * ky) * ((l * l) / (om * om))))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((1.0d0 - cos((2.0d0 * ky))) * (((l * 4.0d0) / om) / (om / l))) / 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Math.pow(Math.sin(ky), 2.0) <= 4e-33) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((1.0 - Math.cos((2.0 * ky))) * (((l * 4.0) / Om) / (Om / l))) / 2.0))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if math.pow(math.sin(ky), 2.0) <= 4e-33:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((1.0 - math.cos((2.0 * ky))) * (((l * 4.0) / Om) / (Om / l))) / 2.0))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if ((sin(ky) ^ 2.0) <= 4e-33)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * ky) * Float64(Float64(l * l) / Float64(Om * Om))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * Float64(Float64(Float64(l * 4.0) / Om) / Float64(Om / l))) / 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((sin(ky) ^ 2.0) <= 4e-33)
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	else
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((1.0 - cos((2.0 * ky))) * (((l * 4.0) / Om) / (Om / l))) / 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision], 4e-33], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(2.0 * N[(N[(ky * ky), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 4.0000000000000002e-33

   1. Initial program 97.3%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 88.4%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 66.6%

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

   10. Simplified 66.6%

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

   if 4.0000000000000002e-33 < (pow.f64 (sin.f64 ky) #s(literal 2 binary64))

   1. Initial program 100.0%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 88.9%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 88.6%

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

   7. Simplified 88.6%

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. associate-*l/ N/A

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

      4. sin-mult N/A

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

      5. associate-*r/ N/A

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

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

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

   9. Applied egg-rr 99.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(ky \cdot ky\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot \frac{\frac{\ell \cdot 4}{Om}}{\frac{Om}{\ell}}}{2}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-165}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(0.5 + kx \cdot kx\right) + \cos \left(2 \cdot ky\right) \cdot -0.5\right)\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{Om \cdot 0.25}{\ell \cdot ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 3.1e-165)
   1.0
   (if (<= l 2.75e+125)
     (sqrt
      (+
       0.5
       (/
        0.5
        (+
         1.0
         (/
          (* 2.0 (* (* l l) (+ (+ 0.5 (* kx kx)) (* (cos (* 2.0 ky)) -0.5))))
          (* Om Om))))))
     (sqrt (+ 0.5 (/ (* Om 0.25) (* l ky)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 3.1e-165) {
		tmp = 1.0;
	} else if (l <= 2.75e+125) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l * l) * ((0.5 + (kx * kx)) + (cos((2.0 * ky)) * -0.5)))) / (Om * Om))))));
	} else {
		tmp = sqrt((0.5 + ((Om * 0.25) / (l * ky))));
	}
	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 <= 3.1d-165) then
        tmp = 1.0d0
    else if (l <= 2.75d+125) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + ((2.0d0 * ((l * l) * ((0.5d0 + (kx * kx)) + (cos((2.0d0 * ky)) * (-0.5d0))))) / (om * om))))))
    else
        tmp = sqrt((0.5d0 + ((om * 0.25d0) / (l * ky))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 3.1e-165) {
		tmp = 1.0;
	} else if (l <= 2.75e+125) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l * l) * ((0.5 + (kx * kx)) + (Math.cos((2.0 * ky)) * -0.5)))) / (Om * Om))))));
	} else {
		tmp = Math.sqrt((0.5 + ((Om * 0.25) / (l * ky))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 3.1e-165:
		tmp = 1.0
	elif l <= 2.75e+125:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l * l) * ((0.5 + (kx * kx)) + (math.cos((2.0 * ky)) * -0.5)))) / (Om * Om))))))
	else:
		tmp = math.sqrt((0.5 + ((Om * 0.25) / (l * ky))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 3.1e-165)
		tmp = 1.0;
	elseif (l <= 2.75e+125)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(l * l) * Float64(Float64(0.5 + Float64(kx * kx)) + Float64(cos(Float64(2.0 * ky)) * -0.5)))) / Float64(Om * Om))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(Om * 0.25) / Float64(l * ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 3.1e-165)
		tmp = 1.0;
	elseif (l <= 2.75e+125)
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((l * l) * ((0.5 + (kx * kx)) + (cos((2.0 * ky)) * -0.5)))) / (Om * Om))))));
	else
		tmp = sqrt((0.5 + ((Om * 0.25) / (l * ky))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 3.1e-165], 1.0, If[LessEqual[l, 2.75e+125], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] * N[(N[(0.5 + N[(kx * kx), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(Om * 0.25), $MachinePrecision] / N[(l * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.09999999999999996e-165

   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 87.5%

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

   5. Taylor expanded in l around 0

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

   7. Simplified 70.5%

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

   if 3.09999999999999996e-165 < l < 2.74999999999999998e125

   1. Initial program 96.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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 94.2%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

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

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 83.6%

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

   9. Simplified 83.6%

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

   10. Taylor expanded in l around 0

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

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

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

       2. associate-*r/ N/A

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

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

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

   12. Simplified 76.3%

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

   if 2.74999999999999998e125 < l

   1. Initial program 98.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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 86.5%

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

   5. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

      10. sin-lowering-sin.f64 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 84.6%

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

   10. Simplified 84.6%

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

   11. Taylor expanded in ky around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 85.4%

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

   13. Simplified 85.4%

       \[\leadsto \sqrt{0.5 + \frac{0.25 \cdot Om}{\color{blue}{\ell \cdot ky}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.1 \cdot 10^{-165}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2.75 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(0.5 + kx \cdot kx\right) + \cos \left(2 \cdot ky\right) \cdot -0.5\right)\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{Om \cdot 0.25}{\ell \cdot ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.9 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left(\ell \cdot \ell\right) \cdot \left(\left(ky \cdot ky\right) \cdot 4\right)}{Om \cdot Om}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 2.9e-175)
   (sqrt 0.5)
   (if (<= Om 1.08e+49)
     (sqrt
      (+
       0.5
       (/ 0.5 (sqrt (+ 1.0 (/ (* (* l l) (* (* ky ky) 4.0)) (* Om Om)))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.9e-175) {
		tmp = sqrt(0.5);
	} else if (Om <= 1.08e+49) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l * l) * ((ky * ky) * 4.0)) / (Om * Om)))))));
	} else {
		tmp = 1.0;
	}
	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 (om <= 2.9d-175) then
        tmp = sqrt(0.5d0)
    else if (om <= 1.08d+49) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((l * l) * ((ky * ky) * 4.0d0)) / (om * om)))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.9e-175) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 1.08e+49) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((l * l) * ((ky * ky) * 4.0)) / (Om * Om)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2.9e-175:
		tmp = math.sqrt(0.5)
	elif Om <= 1.08e+49:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((l * l) * ((ky * ky) * 4.0)) / (Om * Om)))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2.9e-175)
		tmp = sqrt(0.5);
	elseif (Om <= 1.08e+49)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(l * l) * Float64(Float64(ky * ky) * 4.0)) / Float64(Om * Om)))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 2.9e-175)
		tmp = sqrt(0.5);
	elseif (Om <= 1.08e+49)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l * l) * ((ky * ky) * 4.0)) / (Om * Om)))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2.9e-175], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 1.08e+49], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(l * l), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 2.89999999999999999e-175

   1. Initial program 98.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. 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 66.8%

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

   5. Simplified 66.8%

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

   if 2.89999999999999999e-175 < Om < 1.08000000000000001e49

   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 100.0%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 89.4%

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

   7. Simplified 89.4%

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

   8. Taylor expanded in ky around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 80.4%

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

   10. Simplified 80.4%

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

   if 1.08000000000000001e49 < Om

   1. Initial program 100.0%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 90.2%

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

   5. Taylor expanded in l around 0

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

   7. Simplified 85.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2.9 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.08 \cdot 10^{+49}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left(\ell \cdot \ell\right) \cdot \left(\left(ky \cdot ky\right) \cdot 4\right)}{Om \cdot Om}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.9 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(ky \cdot ky\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 2.9e-175)
   (sqrt 0.5)
   (if (<= Om 8.2e+47)
     (sqrt (+ 0.5 (/ 0.5 (+ 1.0 (* 2.0 (* (* ky ky) (/ (* l l) (* Om Om))))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.9e-175) {
		tmp = sqrt(0.5);
	} else if (Om <= 8.2e+47) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	} else {
		tmp = 1.0;
	}
	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 (om <= 2.9d-175) then
        tmp = sqrt(0.5d0)
    else if (om <= 8.2d+47) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + (2.0d0 * ((ky * ky) * ((l * l) / (om * om))))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.9e-175) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 8.2e+47) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2.9e-175:
		tmp = math.sqrt(0.5)
	elif Om <= 8.2e+47:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2.9e-175)
		tmp = sqrt(0.5);
	elseif (Om <= 8.2e+47)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * ky) * Float64(Float64(l * l) / Float64(Om * Om))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 2.9e-175)
		tmp = sqrt(0.5);
	elseif (Om <= 8.2e+47)
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((ky * ky) * ((l * l) / (Om * Om))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2.9e-175], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 8.2e+47], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(2.0 * N[(N[(ky * ky), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 2.89999999999999999e-175

   1. Initial program 98.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. 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 66.8%

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

   5. Simplified 66.8%

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

   if 2.89999999999999999e-175 < Om < 8.2000000000000002e47

   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 100.0%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 89.4%

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

   7. Simplified 89.4%

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

   8. Taylor expanded in ky around 0

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 80.0%

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

   10. Simplified 80.0%

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

   if 8.2000000000000002e47 < Om

   1. Initial program 100.0%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 90.2%

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

   5. Taylor expanded in l around 0

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

   7. Simplified 85.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.0% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 2e-10) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 2e-10) {
		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-10) 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-10) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 2e-10:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 2e-10)
		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-10)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 2e-10], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.00000000000000007e-10

   1. Initial program 99.4%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 88.3%

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

   5. Taylor expanded in l around 0

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

   7. Simplified 70.7%

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

   if 2.00000000000000007e-10 < l

   1. Initial program 97.3%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. 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 84.7%

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

   5. Simplified 84.7%

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

Alternative 9: 63.6% 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 98.8%

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

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

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt1-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

3. Simplified 88.7%

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

5. Taylor expanded in l around 0

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

7. Simplified 59.3%

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

Reproduce

herbie shell --seed 2024155 
(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))))))))))