Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.6% → 99.6%

Time: 13.6s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.6% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{2 \cdot l\_m}{Om\_m}\\ \mathbf{if}\;t\_0 \leq 10^{+140}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {t\_0}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om\_m}{l\_m \cdot ky\_m}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l_m) Om_m)))
   (if (<= t_0 1e+140)
     (sqrt
      (*
       (/ 1.0 2.0)
       (+
        1.0
        (/
         1.0
         (sqrt
          (+
           1.0
           (*
            (pow t_0 2.0)
            (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))))))
     (sqrt (fma 0.25 (/ Om_m (* l_m ky_m)) 0.5)))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (2.0 * l_m) / Om_m;
	double tmp;
	if (t_0 <= 1e+140) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(t_0, 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0)))))))));
	} else {
		tmp = sqrt(fma(0.25, (Om_m / (l_m * ky_m)), 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(2.0 * l_m) / Om_m)
	tmp = 0.0
	if (t_0 <= 1e+140)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((t_0 ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))));
	else
		tmp = sqrt(fma(0.25, Float64(Om_m / Float64(l_m * ky_m)), 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]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+140], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.25 * N[(Om$95$m / N[(l$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.00000000000000006e140

   1. Initial program 99.1%

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

   if 1.00000000000000006e140 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 91.4%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in ky around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 96.8

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

   8. Applied rewrites 96.8%

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

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 0.002:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{l\_m}{Om\_m} \cdot 4, \frac{l\_m}{Om\_m} \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(kx\_m + kx\_m\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(ky\_m + ky\_m\right)\right)\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\mathsf{hypot}\left(\sin kx\_m, \sin ky\_m\right) \cdot \left(2 \cdot l\_m\right)}{Om\_m}}\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      0.002)
   (sqrt
    (*
     (/ 1.0 2.0)
     (+
      1.0
      (/
       1.0
       (sqrt
        (fma
         (* (/ l_m Om_m) 4.0)
         (*
          (/ l_m Om_m)
          (+
           (+ 0.5 (* -0.5 (cos (+ kx_m kx_m))))
           (+ 0.5 (* -0.5 (cos (+ ky_m ky_m))))))
         1.0))))))
   (sqrt
    (*
     (/ 1.0 2.0)
     (+ 1.0 (/ 1.0 (/ (* (hypot (sin kx_m) (sin ky_m)) (* 2.0 l_m)) Om_m)))))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 0.002) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt(fma(((l_m / Om_m) * 4.0), ((l_m / Om_m) * ((0.5 + (-0.5 * cos((kx_m + kx_m)))) + (0.5 + (-0.5 * cos((ky_m + ky_m)))))), 1.0))))));
	} else {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / ((hypot(sin(kx_m), sin(ky_m)) * (2.0 * l_m)) / Om_m)))));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 0.002)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(fma(Float64(Float64(l_m / Om_m) * 4.0), Float64(Float64(l_m / Om_m) * Float64(Float64(0.5 + Float64(-0.5 * cos(Float64(kx_m + kx_m)))) + Float64(0.5 + Float64(-0.5 * cos(Float64(ky_m + ky_m)))))), 1.0))))));
	else
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / Float64(Float64(hypot(sin(kx_m), sin(ky_m)) * Float64(2.0 * l_m)) / Om_m)))));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.002], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(N[(0.5 + N[(-0.5 * N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(-0.5 * N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[(N[(N[Sqrt[N[Sin[kx$95$m], $MachinePrecision] ^ 2 + N[Sin[ky$95$m], $MachinePrecision] ^ 2], $MachinePrecision] * N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $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)))) < 2e-3

   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 99.5%

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

   if 2e-3 < (*.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 95.5%

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

   3. Taylor expanded in l around inf

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

      1. associate-*l/ N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.002:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot 4, \frac{\ell}{Om} \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(2 \cdot \ell\right)}{Om}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{l\_m}{Om\_m} \cdot 4, \frac{l\_m}{Om\_m} \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(kx\_m + kx\_m\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(ky\_m + ky\_m\right)\right)\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om\_m}{l\_m \cdot ky\_m}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      4e+15)
   (sqrt
    (*
     (/ 1.0 2.0)
     (+
      1.0
      (/
       1.0
       (sqrt
        (fma
         (* (/ l_m Om_m) 4.0)
         (*
          (/ l_m Om_m)
          (+
           (+ 0.5 (* -0.5 (cos (+ kx_m kx_m))))
           (+ 0.5 (* -0.5 (cos (+ ky_m ky_m))))))
         1.0))))))
   (sqrt (fma 0.25 (/ Om_m (* l_m ky_m)) 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 4e+15) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt(fma(((l_m / Om_m) * 4.0), ((l_m / Om_m) * ((0.5 + (-0.5 * cos((kx_m + kx_m)))) + (0.5 + (-0.5 * cos((ky_m + ky_m)))))), 1.0))))));
	} else {
		tmp = sqrt(fma(0.25, (Om_m / (l_m * ky_m)), 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 4e+15)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(fma(Float64(Float64(l_m / Om_m) * 4.0), Float64(Float64(l_m / Om_m) * Float64(Float64(0.5 + Float64(-0.5 * cos(Float64(kx_m + kx_m)))) + Float64(0.5 + Float64(-0.5 * cos(Float64(ky_m + ky_m)))))), 1.0))))));
	else
		tmp = sqrt(fma(0.25, Float64(Om_m / Float64(l_m * ky_m)), 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]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+15], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(N[(0.5 + N[(-0.5 * N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(-0.5 * N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.25 * N[(Om$95$m / N[(l$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision] + 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)))) < 4e15

   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 99.0%

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

   if 4e15 < (*.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 95.3%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in ky around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.8

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

   8. Applied rewrites 83.8%

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

Alternative 4: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{l\_m \cdot 4}{Om\_m}, \frac{l\_m}{Om\_m} \cdot \mathsf{fma}\left(-0.5, \cos \left(ky\_m \cdot -2\right), 0.5\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om\_m}{l\_m \cdot ky\_m}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      4e+15)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (fma
        (/ (* l_m 4.0) Om_m)
        (* (/ l_m Om_m) (fma -0.5 (cos (* ky_m -2.0)) 0.5))
        1.0)))))
   (sqrt (fma 0.25 (/ Om_m (* l_m ky_m)) 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 4e+15) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(((l_m * 4.0) / Om_m), ((l_m / Om_m) * fma(-0.5, cos((ky_m * -2.0)), 0.5)), 1.0)))));
	} else {
		tmp = sqrt(fma(0.25, (Om_m / (l_m * ky_m)), 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 4e+15)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(Float64(l_m * 4.0) / Om_m), Float64(Float64(l_m / Om_m) * fma(-0.5, cos(Float64(ky_m * -2.0)), 0.5)), 1.0)))));
	else
		tmp = sqrt(fma(0.25, Float64(Om_m / Float64(l_m * ky_m)), 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]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+15], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(-0.5 * N[Cos[N[(ky$95$m * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.25 * N[(Om$95$m / N[(l$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision] + 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)))) < 4e15

   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 87.3%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 86.2

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

   7. Applied rewrites 86.2%

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

   9. Applied rewrites 97.9%

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

   if 4e15 < (*.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 95.3%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.3

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in ky around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 83.8

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

   8. Applied rewrites 83.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 4}{Om}, \frac{\ell}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\ell \cdot ky}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{l\_m}{Om\_m} \cdot \left(\frac{l\_m}{Om\_m} \cdot \mathsf{fma}\left(-0.5, \cos \left(ky\_m \cdot -2\right), 0.5\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om\_m}{l\_m \cdot ky\_m}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      0.002)
   (fma
    -0.5
    (* (/ l_m Om_m) (* (/ l_m Om_m) (fma -0.5 (cos (* ky_m -2.0)) 0.5)))
    1.0)
   (sqrt (fma 0.25 (/ Om_m (* l_m ky_m)) 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 0.002) {
		tmp = fma(-0.5, ((l_m / Om_m) * ((l_m / Om_m) * fma(-0.5, cos((ky_m * -2.0)), 0.5))), 1.0);
	} else {
		tmp = sqrt(fma(0.25, (Om_m / (l_m * ky_m)), 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 0.002)
		tmp = fma(-0.5, Float64(Float64(l_m / Om_m) * Float64(Float64(l_m / Om_m) * fma(-0.5, cos(Float64(ky_m * -2.0)), 0.5))), 1.0);
	else
		tmp = sqrt(fma(0.25, Float64(Om_m / Float64(l_m * ky_m)), 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]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.002], N[(-0.5 * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(-0.5 * N[Cos[N[(ky$95$m * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[Sqrt[N[(0.25 * N[(Om$95$m / N[(l$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision] + 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)))) < 2e-3

   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 88.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.6

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

   7. Applied rewrites 88.6%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 88.6

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

   10. Applied rewrites 88.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*l* N/A

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

       7. times-frac N/A

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

       8. lift-/.f64 N/A

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

       9. associate-*l/ N/A

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

       10. lift-/.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lift-*.f64 N/A

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

       16. lower-*.f64 99.3

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

   12. Applied rewrites 99.3%

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

   if 2e-3 < (*.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 95.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in ky around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

Alternative 6: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 0.002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om\_m}{l\_m \cdot ky\_m}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      0.002)
   1.0
   (sqrt (fma 0.25 (/ Om_m (* l_m ky_m)) 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 0.002) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma(0.25, (Om_m / (l_m * ky_m)), 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 0.002)
		tmp = 1.0;
	else
		tmp = sqrt(fma(0.25, Float64(Om_m / Float64(l_m * ky_m)), 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]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.002], 1.0, N[Sqrt[N[(0.25 * N[(Om$95$m / N[(l$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision] + 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)))) < 2e-3

   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 ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 98.9%

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

   if 2e-3 < (*.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 95.5%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in ky around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 81.8

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

   8. Applied rewrites 81.8%

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

Alternative 7: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(om)
l_m = abs(l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (((((2.0d0 * l_m) / om_m) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))) <= 3.8d0) then
        tmp = 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);
l_m = Math.abs(l);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if ((Math.pow(((2.0 * l_m) / Om_m), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))) <= 3.8) {
		tmp = 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_m = math.fabs(l)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	tmp = 0
	if (math.pow(((2.0 * l_m) / Om_m), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))) <= 3.8:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 3.8)
		tmp = 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_m = abs(l);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0;
	if (((((2.0 * l_m) / Om_m) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 3.8)
		tmp = 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]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.8], 1.0, 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 ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 98.9%

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

   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 95.5%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 97.2%

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

Alternative 8: 62.4% accurate, 581.0× speedup

Math

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

FPCore

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

C

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

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(om)
l_m = abs(l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l_m, om_m, kx_m, ky_m)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Taylor expanded in ky around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in kx around 0

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

8. Applied rewrites 64.4%

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

Reproduce

herbie shell --seed 2024220 
(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))))))))))