Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.5%

Time: 15.9s

Alternatives: 5

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;ky\_m \leq 6.6 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \frac{\sin ky\_m}{\frac{Om}{\ell}} \cdot \frac{4 \cdot \left(ky\_m \cdot \ell\right)}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \left(2 \cdot \frac{1 - \cos \left(ky\_m \cdot 2\right)}{Om}\right) \cdot \frac{\ell}{\frac{Om}{\ell}}}}}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= ky_m 6.6e-11)
   (sqrt
    (-
     0.5
     (/
      -0.5
      (sqrt (+ 1.0 (* (/ (sin ky_m) (/ Om l)) (/ (* 4.0 (* ky_m l)) Om)))))))
   (sqrt
    (-
     0.5
     (/
      -0.5
      (sqrt
       (+
        1.0
        (* (* 2.0 (/ (- 1.0 (cos (* ky_m 2.0))) Om)) (/ l (/ Om l))))))))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (ky_m <= 6.6e-11) {
		tmp = sqrt((0.5 - (-0.5 / sqrt((1.0 + ((sin(ky_m) / (Om / l)) * ((4.0 * (ky_m * l)) / Om)))))));
	} else {
		tmp = sqrt((0.5 - (-0.5 / sqrt((1.0 + ((2.0 * ((1.0 - cos((ky_m * 2.0))) / Om)) * (l / (Om / l))))))));
	}
	return tmp;
}

Fortran

kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (ky_m <= 6.6d-11) then
        tmp = sqrt((0.5d0 - ((-0.5d0) / sqrt((1.0d0 + ((sin(ky_m) / (om / l)) * ((4.0d0 * (ky_m * l)) / om)))))))
    else
        tmp = sqrt((0.5d0 - ((-0.5d0) / sqrt((1.0d0 + ((2.0d0 * ((1.0d0 - cos((ky_m * 2.0d0))) / om)) * (l / (om / l))))))))
    end if
    code = tmp
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (ky_m <= 6.6e-11) {
		tmp = Math.sqrt((0.5 - (-0.5 / Math.sqrt((1.0 + ((Math.sin(ky_m) / (Om / l)) * ((4.0 * (ky_m * l)) / Om)))))));
	} else {
		tmp = Math.sqrt((0.5 - (-0.5 / Math.sqrt((1.0 + ((2.0 * ((1.0 - Math.cos((ky_m * 2.0))) / Om)) * (l / (Om / l))))))));
	}
	return tmp;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if ky_m <= 6.6e-11:
		tmp = math.sqrt((0.5 - (-0.5 / math.sqrt((1.0 + ((math.sin(ky_m) / (Om / l)) * ((4.0 * (ky_m * l)) / Om)))))))
	else:
		tmp = math.sqrt((0.5 - (-0.5 / math.sqrt((1.0 + ((2.0 * ((1.0 - math.cos((ky_m * 2.0))) / Om)) * (l / (Om / l))))))))
	return tmp

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (ky_m <= 6.6e-11)
		tmp = sqrt(Float64(0.5 - Float64(-0.5 / sqrt(Float64(1.0 + Float64(Float64(sin(ky_m) / Float64(Om / l)) * Float64(Float64(4.0 * Float64(ky_m * l)) / Om)))))));
	else
		tmp = sqrt(Float64(0.5 - Float64(-0.5 / sqrt(Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(1.0 - cos(Float64(ky_m * 2.0))) / Om)) * Float64(l / Float64(Om / l))))))));
	end
	return tmp
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (ky_m <= 6.6e-11)
		tmp = sqrt((0.5 - (-0.5 / sqrt((1.0 + ((sin(ky_m) / (Om / l)) * ((4.0 * (ky_m * l)) / Om)))))));
	else
		tmp = sqrt((0.5 - (-0.5 / sqrt((1.0 + ((2.0 * ((1.0 - cos((ky_m * 2.0))) / Om)) * (l / (Om / l))))))));
	end
	tmp_2 = tmp;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[ky$95$m, 6.6e-11], N[Sqrt[N[(0.5 - N[(-0.5 / N[Sqrt[N[(1.0 + N[(N[(N[Sin[ky$95$m], $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(N[(4.0 * N[(ky$95$m * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 - N[(-0.5 / N[Sqrt[N[(1.0 + N[(N[(2.0 * N[(N[(1.0 - N[Cos[N[(ky$95$m * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 6.6000000000000005e-11

   1. Initial program 96.4%

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

   4. Applied egg-rr 86.3%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 70.6%

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

   7. Simplified 70.6%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

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

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

      6. sqr-sin-a N/A

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

      7. associate-/r/ N/A

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

      8. times-frac N/A

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

   9. Applied egg-rr 89.9%

      \[\leadsto \sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \color{blue}{\frac{\sin ky}{\frac{\frac{Om}{\ell}}{4}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}}}}} \]

   10. Taylor expanded in ky around 0

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. *-lowering-*.f64 79.6%

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

   12. Simplified 79.6%

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

   if 6.6000000000000005e-11 < ky

   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 egg-rr 100.0%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. associate-/r/ N/A

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

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

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

   9. Applied egg-rr 99.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 6.6 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{4 \cdot \left(ky \cdot \ell\right)}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \left(2 \cdot \frac{1 - \cos \left(ky \cdot 2\right)}{Om}\right) \cdot \frac{\ell}{\frac{Om}{\ell}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = sqrt((0.5d0 - ((-0.5d0) / sqrt((1.0d0 + ((sin(ky_m) / ((om / l) / 4.0d0)) * (sin(ky_m) / (om / l))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

4. Applied egg-rr 89.5%

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

5. Taylor expanded in kx around 0

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

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

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

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

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

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

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

   4. *-lowering-*.f64 77.4%

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

7. Simplified 77.4%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

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

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

   6. sqr-sin-a N/A

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

   7. associate-/r/ N/A

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

   8. times-frac N/A

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

9. Applied egg-rr 92.2%

   \[\leadsto \sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \color{blue}{\frac{\sin ky}{\frac{\frac{Om}{\ell}}{4}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}}}}} \]
10. Add Preprocessing

Alternative 3: 71.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \frac{\sin ky\_m}{\frac{Om}{\ell}} \cdot \frac{4 \cdot \left(ky\_m \cdot \ell\right)}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= l 4.5e-171)
   1.0
   (if (<= l 1.2e+107)
     (sqrt
      (-
       0.5
       (/
        -0.5
        (sqrt (+ 1.0 (* (/ (sin ky_m) (/ Om l)) (/ (* 4.0 (* ky_m l)) Om)))))))
     (sqrt 0.5))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 4.5e-171) {
		tmp = 1.0;
	} else if (l <= 1.2e+107) {
		tmp = sqrt((0.5 - (-0.5 / sqrt((1.0 + ((sin(ky_m) / (Om / l)) * ((4.0 * (ky_m * l)) / Om)))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (l <= 4.5d-171) then
        tmp = 1.0d0
    else if (l <= 1.2d+107) then
        tmp = sqrt((0.5d0 - ((-0.5d0) / sqrt((1.0d0 + ((sin(ky_m) / (om / l)) * ((4.0d0 * (ky_m * l)) / om)))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 4.5e-171) {
		tmp = 1.0;
	} else if (l <= 1.2e+107) {
		tmp = Math.sqrt((0.5 - (-0.5 / Math.sqrt((1.0 + ((Math.sin(ky_m) / (Om / l)) * ((4.0 * (ky_m * l)) / Om)))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if l <= 4.5e-171:
		tmp = 1.0
	elif l <= 1.2e+107:
		tmp = math.sqrt((0.5 - (-0.5 / math.sqrt((1.0 + ((math.sin(ky_m) / (Om / l)) * ((4.0 * (ky_m * l)) / Om)))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (l <= 4.5e-171)
		tmp = 1.0;
	elseif (l <= 1.2e+107)
		tmp = sqrt(Float64(0.5 - Float64(-0.5 / sqrt(Float64(1.0 + Float64(Float64(sin(ky_m) / Float64(Om / l)) * Float64(Float64(4.0 * Float64(ky_m * l)) / Om)))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (l <= 4.5e-171)
		tmp = 1.0;
	elseif (l <= 1.2e+107)
		tmp = sqrt((0.5 - (-0.5 / sqrt((1.0 + ((sin(ky_m) / (Om / l)) * ((4.0 * (ky_m * l)) / Om)))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[l, 4.5e-171], 1.0, If[LessEqual[l, 1.2e+107], N[Sqrt[N[(0.5 - N[(-0.5 / N[Sqrt[N[(1.0 + N[(N[(N[Sin[ky$95$m], $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(N[(4.0 * N[(ky$95$m * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.5000000000000004e-171

   1. Initial program 98.8%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 85.1%

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

   5. Taylor expanded in l around 0

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

   7. Simplified 72.1%

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

   if 4.5000000000000004e-171 < l < 1.2e107

   1. Initial program 95.9%

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

   4. Applied egg-rr 88.6%

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

   5. Taylor expanded in kx around 0

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

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

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

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

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

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

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

      4. *-lowering-*.f64 80.0%

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

   7. Simplified 80.0%

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

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

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

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

      6. sqr-sin-a N/A

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

      7. associate-/r/ N/A

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

      8. times-frac N/A

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

   9. Applied egg-rr 90.0%

      \[\leadsto \sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \color{blue}{\frac{\sin ky}{\frac{\frac{Om}{\ell}}{4}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}}}}} \]

   10. Taylor expanded in ky around 0

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. *-lowering-*.f64 74.1%

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

   12. Simplified 74.1%

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

   if 1.2e107 < l

   1. Initial program 92.5%

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

   3. Taylor expanded in l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right) \]
   4. Step-by-step derivation

   5. Simplified 83.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.5 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+107}:\\ \;\;\;\;\sqrt{0.5 - \frac{-0.5}{\sqrt{1 + \frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{4 \cdot \left(ky \cdot \ell\right)}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.5 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= Om 1.5e-70) (sqrt 0.5) 1.0))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (Om <= 1.5e-70) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (om <= 1.5d-70) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (Om <= 1.5e-70) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if Om <= 1.5e-70:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (Om <= 1.5e-70)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (Om <= 1.5e-70)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[Om, 1.5e-70], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.5000000000000001e-70

   1. Initial program 97.3%

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

   3. Taylor expanded in l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right) \]
   4. Step-by-step derivation

   5. Simplified 60.6%

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

   if 1.5000000000000001e-70 < Om

   1. Initial program 97.1%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 86.0%

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

   5. Taylor expanded in l around 0

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

   7. Simplified 82.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.3% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

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

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt1-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

3. Simplified 82.6%

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

5. Taylor expanded in l around 0

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

7. Simplified 64.6%

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

Reproduce

herbie shell --seed 2024164 
(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))))))))))