Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.5%

Time: 14.0s

Alternatives: 11

Speedup: 0.9×

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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{Om\_m}{\sin ky\_m \cdot \ell}\\ t_1 := \left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2}\\ \mathbf{if}\;t\_1 \leq 10^{+14}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{1 + t\_1}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({t\_0}^{2}, 0.0625, -0.25\right)}{\mathsf{fma}\left(0.25, t\_0, -0.5\right)}}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ Om_m (* (sin ky_m) l)))
        (t_1
         (*
          (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
          (pow (/ (* l 2.0) Om_m) 2.0))))
   (if (<= t_1 1e+14)
     (sqrt (* (+ (/ 1.0 (sqrt (+ 1.0 t_1))) 1.0) (/ 1.0 2.0)))
     (sqrt (/ (fma (pow t_0 2.0) 0.0625 -0.25) (fma 0.25 t_0 -0.5))))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double t_0 = Om_m / (sin(ky_m) * l);
	double t_1 = (pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0);
	double tmp;
	if (t_1 <= 1e+14) {
		tmp = sqrt((((1.0 / sqrt((1.0 + t_1))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = sqrt((fma(pow(t_0, 2.0), 0.0625, -0.25) / fma(0.25, t_0, -0.5)));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	t_0 = Float64(Om_m / Float64(sin(ky_m) * l))
	t_1 = Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0))
	tmp = 0.0
	if (t_1 <= 1e+14)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(Float64(1.0 + t_1))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(Float64(fma((t_0 ^ 2.0), 0.0625, -0.25) / fma(0.25, t_0, -0.5)));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(Om$95$m / N[(N[Sin[ky$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+14], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(1.0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * 0.0625 + -0.25), $MachinePrecision] / N[(0.25 * t$95$0 + -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1e14

   1. Initial program 100.0%

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

   if 1e14 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 92.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 80.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 80.8%

       \[\leadsto \sqrt{\frac{\mathsf{fma}\left({\left(\frac{Om}{\sin ky \cdot \ell}\right)}^{2}, 0.0625, -0.25\right)}{\mathsf{fma}\left(0.25, \color{blue}{\frac{Om}{\sin ky \cdot \ell}}, -0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 10^{+14}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{1 + \left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(\frac{Om}{\sin ky \cdot \ell}\right)}^{2}, 0.0625, -0.25\right)}{\mathsf{fma}\left(0.25, \frac{Om}{\sin ky \cdot \ell}, -0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 10000:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om\_m} \cdot 4, \left(1 - \left(\cos \left(ky\_m \cdot 2\right) + \cos \left(kx\_m + kx\_m\right)\right) \cdot 0.5\right) \cdot \frac{\ell}{Om\_m}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\mathsf{hypot}\left(\sin kx\_m, \sin ky\_m\right) \cdot \left(\frac{2}{Om\_m} \cdot \ell\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
       (pow (/ (* l 2.0) Om_m) 2.0))
      10000.0)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (* (/ l Om_m) 4.0)
         (*
          (- 1.0 (* (+ (cos (* ky_m 2.0)) (cos (+ kx_m kx_m))) 0.5))
          (/ l Om_m))
         1.0)))
      1.0)
     0.5))
   (sqrt
    (*
     (+ (/ 1.0 (* (hypot (sin kx_m) (sin ky_m)) (* (/ 2.0 Om_m) l))) 1.0)
     (/ 1.0 2.0)))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 10000.0) {
		tmp = sqrt((((1.0 / sqrt(fma(((l / Om_m) * 4.0), ((1.0 - ((cos((ky_m * 2.0)) + cos((kx_m + kx_m))) * 0.5)) * (l / Om_m)), 1.0))) + 1.0) * 0.5));
	} else {
		tmp = sqrt((((1.0 / (hypot(sin(kx_m), sin(ky_m)) * ((2.0 / Om_m) * l))) + 1.0) * (1.0 / 2.0)));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 10000.0)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(Float64(l / Om_m) * 4.0), Float64(Float64(1.0 - Float64(Float64(cos(Float64(ky_m * 2.0)) + cos(Float64(kx_m + kx_m))) * 0.5)) * Float64(l / Om_m)), 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(Float64(Float64(Float64(1.0 / Float64(hypot(sin(kx_m), sin(ky_m)) * Float64(Float64(2.0 / Om_m) * l))) + 1.0) * Float64(1.0 / 2.0)));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 10000.0], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(l / Om$95$m), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(N[Cos[N[(ky$95$m * 2.0), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(l / Om$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(1.0 / N[(N[Sqrt[N[Sin[kx$95$m], $MachinePrecision] ^ 2 + N[Sin[ky$95$m], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(2.0 / Om$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1e4

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in kx around inf

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

      1. lower--.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-cos.f64 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-cos.f64 N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   11. Applied rewrites 100.0%

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

   if 1e4 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 92.4%

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

   3. Taylor expanded in Om around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 98.9%

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\left(\ell \cdot \frac{2}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 10000:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot 4, \left(1 - \left(\cos \left(ky \cdot 2\right) + \cos \left(kx + kx\right)\right) \cdot 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 10000:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om\_m} \cdot 4, \left(1 - \left(\cos \left(ky\_m \cdot 2\right) + \cos \left(kx\_m + kx\_m\right)\right) \cdot 0.5\right) \cdot \frac{\ell}{Om\_m}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om\_m}{\sin ky\_m \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
       (pow (/ (* l 2.0) Om_m) 2.0))
      10000.0)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (* (/ l Om_m) 4.0)
         (*
          (- 1.0 (* (+ (cos (* ky_m 2.0)) (cos (+ kx_m kx_m))) 0.5))
          (/ l Om_m))
         1.0)))
      1.0)
     0.5))
   (sqrt (fma (/ Om_m (* (sin ky_m) l)) 0.25 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 10000.0) {
		tmp = sqrt((((1.0 / sqrt(fma(((l / Om_m) * 4.0), ((1.0 - ((cos((ky_m * 2.0)) + cos((kx_m + kx_m))) * 0.5)) * (l / Om_m)), 1.0))) + 1.0) * 0.5));
	} else {
		tmp = sqrt(fma((Om_m / (sin(ky_m) * l)), 0.25, 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 10000.0)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(Float64(l / Om_m) * 4.0), Float64(Float64(1.0 - Float64(Float64(cos(Float64(ky_m * 2.0)) + cos(Float64(kx_m + kx_m))) * 0.5)) * Float64(l / Om_m)), 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(fma(Float64(Om_m / Float64(sin(ky_m) * l)), 0.25, 0.5));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 10000.0], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(l / Om$95$m), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(1.0 - N[(N[(N[Cos[N[(ky$95$m * 2.0), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(l / Om$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(Om$95$m / N[(N[Sin[ky$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1e4

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in kx around inf

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

      1. lower--.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-cos.f64 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-cos.f64 N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   11. Applied rewrites 100.0%

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

   if 1e4 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 92.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 80.0%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 10000:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot 4, \left(1 - \left(\cos \left(ky \cdot 2\right) + \cos \left(kx + kx\right)\right) \cdot 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 10000:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om\_m} \cdot 4, \left(0.5 - \cos \left(ky\_m \cdot 2\right) \cdot 0.5\right) \cdot \frac{\ell}{Om\_m}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om\_m}{\sin ky\_m \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
       (pow (/ (* l 2.0) Om_m) 2.0))
      10000.0)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (* (/ l Om_m) 4.0)
         (* (- 0.5 (* (cos (* ky_m 2.0)) 0.5)) (/ l Om_m))
         1.0)))
      1.0)
     0.5))
   (sqrt (fma (/ Om_m (* (sin ky_m) l)) 0.25 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 10000.0) {
		tmp = sqrt((((1.0 / sqrt(fma(((l / Om_m) * 4.0), ((0.5 - (cos((ky_m * 2.0)) * 0.5)) * (l / Om_m)), 1.0))) + 1.0) * 0.5));
	} else {
		tmp = sqrt(fma((Om_m / (sin(ky_m) * l)), 0.25, 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 10000.0)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(Float64(l / Om_m) * 4.0), Float64(Float64(0.5 - Float64(cos(Float64(ky_m * 2.0)) * 0.5)) * Float64(l / Om_m)), 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(fma(Float64(Om_m / Float64(sin(ky_m) * l)), 0.25, 0.5));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 10000.0], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(l / Om$95$m), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(ky$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(l / Om$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(Om$95$m / N[(N[Sin[ky$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1e4

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-cos.f64 N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.7

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

   9. Applied rewrites 98.7%

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

   if 1e4 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 92.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 80.0%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 10000:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot 4, \left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om\_m} \cdot 4, \mathsf{fma}\left(\cos \left(kx\_m \cdot 2\right), -0.5, 0.5\right) \cdot \frac{\ell}{Om\_m}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om\_m}{\sin ky\_m \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
       (pow (/ (* l 2.0) Om_m) 2.0))
      2.0)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (* (/ l Om_m) 4.0)
         (* (fma (cos (* kx_m 2.0)) -0.5 0.5) (/ l Om_m))
         1.0)))
      1.0)
     0.5))
   (sqrt (fma (/ Om_m (* (sin ky_m) l)) 0.25 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 2.0) {
		tmp = sqrt((((1.0 / sqrt(fma(((l / Om_m) * 4.0), (fma(cos((kx_m * 2.0)), -0.5, 0.5) * (l / Om_m)), 1.0))) + 1.0) * 0.5));
	} else {
		tmp = sqrt(fma((Om_m / (sin(ky_m) * l)), 0.25, 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 2.0)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(Float64(l / Om_m) * 4.0), Float64(fma(cos(Float64(kx_m * 2.0)), -0.5, 0.5) * Float64(l / Om_m)), 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(fma(Float64(Om_m / Float64(sin(ky_m) * l)), 0.25, 0.5));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(l / Om$95$m), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(N[Cos[N[(kx$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(l / Om$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(Om$95$m / N[(N[Sin[ky$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in ky around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lower-cos.f64 N/A

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

      9. distribute-lft-neg-in N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.4

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

   9. Applied rewrites 98.4%

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

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

   1. Initial program 92.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 79.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot 4, \mathsf{fma}\left(\cos \left(kx \cdot 2\right), -0.5, 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om\_m}{\sin ky\_m \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
       (pow (/ (* l 2.0) Om_m) 2.0))
      2.0)
   (sqrt 1.0)
   (sqrt (fma (/ Om_m (* (sin ky_m) l)) 0.25 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 2.0) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(fma((Om_m / (sin(ky_m) * l)), 0.25, 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 2.0)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(fma(Float64(Om_m / Float64(sin(ky_m) * l)), 0.25, 0.5));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(Om$95$m / N[(N[Sin[ky$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in Om around inf

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

   5. Applied rewrites 97.9%

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

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

   1. Initial program 92.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 79.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{Om\_m}{ky\_m \cdot \ell}\\ \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.25, t\_0, -0.5\right)} \cdot \mathsf{fma}\left(t\_0 \cdot t\_0, 0.0625, -0.25\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ Om_m (* ky_m l))))
   (if (<=
        (*
         (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
         (pow (/ (* l 2.0) Om_m) 2.0))
        2.0)
     (sqrt 1.0)
     (sqrt (* (/ 1.0 (fma 0.25 t_0 -0.5)) (fma (* t_0 t_0) 0.0625 -0.25))))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double t_0 = Om_m / (ky_m * l);
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 2.0) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(((1.0 / fma(0.25, t_0, -0.5)) * fma((t_0 * t_0), 0.0625, -0.25)));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	t_0 = Float64(Om_m / Float64(ky_m * l))
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 2.0)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(Float64(Float64(1.0 / fma(0.25, t_0, -0.5)) * fma(Float64(t_0 * t_0), 0.0625, -0.25)));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(Om$95$m / N[(ky$95$m * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(1.0 / N[(0.25 * t$95$0 + -0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.0625 + -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in Om around inf

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

   5. Applied rewrites 97.9%

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

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

   1. Initial program 92.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 79.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 79.6%

       \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, 0.25, 0.5\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 79.5%

       \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{ky \cdot \ell} \cdot \frac{Om}{ky \cdot \ell}, 0.0625, -0.25\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(0.25, \frac{Om}{ky \cdot \ell}, -0.5\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(0.25, \frac{Om}{ky \cdot \ell}, -0.5\right)} \cdot \mathsf{fma}\left(\frac{Om}{ky \cdot \ell} \cdot \frac{Om}{ky \cdot \ell}, 0.0625, -0.25\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\frac{Om\_m}{\ell}, 0.25, 0.5 \cdot ky\_m\right)}{ky\_m}}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
       (pow (/ (* l 2.0) Om_m) 2.0))
      2.0)
   (sqrt 1.0)
   (sqrt (/ (fma (/ Om_m l) 0.25 (* 0.5 ky_m)) ky_m))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 2.0) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt((fma((Om_m / l), 0.25, (0.5 * ky_m)) / ky_m));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 2.0)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(Float64(fma(Float64(Om_m / l), 0.25, Float64(0.5 * ky_m)) / ky_m));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(N[(Om$95$m / l), $MachinePrecision] * 0.25 + N[(0.5 * ky$95$m), $MachinePrecision]), $MachinePrecision] / ky$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in Om around inf

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

   5. Applied rewrites 97.9%

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

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

   1. Initial program 92.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 79.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 79.6%

       \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{Om}{\ell}, 0.25, ky \cdot 0.5\right)}{ky}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{\ell}, 0.25, 0.5 \cdot ky\right)}{ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om\_m}{ky\_m \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
       (pow (/ (* l 2.0) Om_m) 2.0))
      2.0)
   (sqrt 1.0)
   (sqrt (fma (/ Om_m (* ky_m l)) 0.25 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 2.0) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(fma((Om_m / (ky_m * l)), 0.25, 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 2.0)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(fma(Float64(Om_m / Float64(ky_m * l)), 0.25, 0.5));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(Om$95$m / N[(ky$95$m * l), $MachinePrecision]), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in Om around inf

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

   5. Applied rewrites 97.9%

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

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

   1. Initial program 92.4%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 79.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 79.6%

       \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, 0.25, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om\_m}\right)}^{2} \leq 3.8:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
       (pow (/ (* l 2.0) Om_m) 2.0))
      3.8)
   (sqrt 1.0)
   (sqrt 0.5)))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om_m), 2.0)) <= 3.8) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om_m, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if ((((sin(ky_m) ** 2.0d0) + (sin(kx_m) ** 2.0d0)) * (((l * 2.0d0) / om_m) ** 2.0d0)) <= 3.8d0) then
        tmp = sqrt(1.0d0)
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

ky_m = Math.abs(ky);
kx_m = Math.abs(kx);
Om_m = Math.abs(Om);
assert l < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (((Math.pow(Math.sin(ky_m), 2.0) + Math.pow(Math.sin(kx_m), 2.0)) * Math.pow(((l * 2.0) / Om_m), 2.0)) <= 3.8) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
Om_m = math.fabs(Om)
[l, Om_m, kx_m, ky_m] = sort([l, Om_m, kx_m, ky_m])
def code(l, Om_m, kx_m, ky_m):
	tmp = 0
	if ((math.pow(math.sin(ky_m), 2.0) + math.pow(math.sin(kx_m), 2.0)) * math.pow(((l * 2.0) / Om_m), 2.0)) <= 3.8:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l, Om_m, kx_m, ky_m = sort([l, Om_m, kx_m, ky_m])
function code(l, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om_m) ^ 2.0)) <= 3.8)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
kx_m = abs(kx);
Om_m = abs(Om);
l, Om_m, kx_m, ky_m = num2cell(sort([l, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l, Om_m, kx_m, ky_m)
	tmp = 0.0;
	if ((((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (((l * 2.0) / Om_m) ^ 2.0)) <= 3.8)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.8], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 3.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in Om around inf

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

   5. Applied rewrites 97.9%

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

   if 3.7999999999999998 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 92.4%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 97.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 3.8:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.1% accurate, 52.8× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ [l, Om_m, kx_m, ky_m] = \mathsf{sort}([l, Om_m, kx_m, ky_m])\\ \\ \sqrt{0.5} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx_m ky_m) :precision binary64 (sqrt 0.5))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l, double Om_m, double kx_m, double ky_m) {
	return sqrt(0.5);
}

Fortran

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

Java

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

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
Om_m = math.fabs(Om)
[l, Om_m, kx_m, ky_m] = sort([l, Om_m, kx_m, ky_m])
def code(l, Om_m, kx_m, ky_m):
	return math.sqrt(0.5)

Julia

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

MATLAB

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

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx$95$m_, ky$95$m_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 96.5%

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

3. Taylor expanded in Om around 0

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

5. Applied rewrites 56.1%

   \[\leadsto \sqrt{\color{blue}{0.5}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))))))))))