Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.6% → 99.7%

Time: 11.8s

Alternatives: 7

Speedup: 2.3×

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: 99.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2 \cdot \ell}{Om}\right)}} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (* (sin ky) (/ (* 2.0 l) Om))))))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, (sin(ky) * ((2.0 * l) / Om)))))));
}

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, (Math.sin(ky) * ((2.0 * l) / Om)))))));
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, (math.sin(ky) * ((2.0 * l) / Om)))))))

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(sin(ky) * Float64(Float64(2.0 * l) / Om)))))))
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, (sin(ky) * ((2.0 * l) / Om)))))));
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[ky], $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in 99.2%

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

   2. metadata-eval 99.2%

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

   3. metadata-eval 99.2%

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

   4. associate-/l* 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in kx around 0 81.2%

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

   1. associate-/l* 81.3%

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

   2. associate-/r/ 82.2%

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

   3. unpow2 82.2%

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

   4. unpow2 82.2%

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

   5. times-frac 90.8%

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

6. Simplified 90.8%

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

   1. expm1-log1p-u 90.8%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

   2. expm1-udef 90.8%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)} - 1}} \cdot 0.5} \]

8. Applied egg-rr 95.6%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
9. Step-by-step derivation

   1. expm1-def 95.6%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

   2. expm1-log1p 95.6%

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

   3. *-commutative 95.6%

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

   4. *-commutative 95.6%

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

   5. associate-*l* 95.6%

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

   6. associate-*l/ 95.6%

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

   7. *-commutative 95.6%

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

10. Simplified 95.6%

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

11. Final simplification 95.6%

    \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2 \cdot \ell}{Om}\right)}} \]

Alternative 2: 71.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{2 \cdot \frac{ky \cdot \ell}{Om} + 0.25 \cdot \frac{Om}{ky \cdot \ell}}}\\ \mathbf{elif}\;Om \leq 10^{+122}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \frac{\left(ky \cdot \ell\right) \cdot \left(ky \cdot \ell\right)}{Om \cdot Om}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 5e-166)
   (sqrt
    (+
     0.5
     (* 0.5 (/ 1.0 (+ (* 2.0 (/ (* ky l) Om)) (* 0.25 (/ Om (* ky l))))))))
   (if (<= Om 1e+122)
     (sqrt
      (+
       0.5
       (*
        0.5
        (/ 1.0 (sqrt (+ 1.0 (* 4.0 (/ (* (* ky l) (* ky l)) (* Om Om)))))))))
     1.0)))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5e-166) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))));
	} else if (Om <= 1e+122) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (4.0 * (((ky * l) * (ky * l)) / (Om * Om)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 5d-166) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / ((2.0d0 * ((ky * l) / om)) + (0.25d0 * (om / (ky * l))))))))
    else if (om <= 1d+122) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + (4.0d0 * (((ky * l) * (ky * l)) / (om * om)))))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5e-166) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))));
	} else if (Om <= 1e+122) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.sqrt((1.0 + (4.0 * (((ky * l) * (ky * l)) / (Om * Om)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 5e-166:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))))
	elif Om <= 1e+122:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + (4.0 * (((ky * l) * (ky * l)) / (Om * Om)))))))))
	else:
		tmp = 1.0
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 5e-166)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(Float64(2.0 * Float64(Float64(ky * l) / Om)) + Float64(0.25 * Float64(Om / Float64(ky * l))))))));
	elseif (Om <= 1e+122)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64(4.0 * Float64(Float64(Float64(ky * l) * Float64(ky * l)) / Float64(Om * Om)))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 5e-166)
		tmp = sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))));
	elseif (Om <= 1e+122)
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (4.0 * (((ky * l) * (ky * l)) / (Om * Om)))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 5e-166], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(Om / N[(ky * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1e+122], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(4.0 * N[(N[(N[(ky * l), $MachinePrecision] * N[(ky * l), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 5e-166

   1. Initial program 98.7%

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

      1. distribute-rgt-in 98.7%

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

      2. metadata-eval 98.7%

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

      3. metadata-eval 98.7%

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

      4. associate-/l* 98.7%

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

      5. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in kx around 0 76.5%

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

      1. associate-/l* 77.5%

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

      2. associate-/r/ 78.2%

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

      3. unpow2 78.2%

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

      4. unpow2 78.2%

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

      5. times-frac 88.3%

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

   6. Simplified 88.3%

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

   7. Taylor expanded in ky around 0 59.5%

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

      1. unpow2 59.5%

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

      2. unpow2 59.5%

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

      3. unswap-sqr 70.6%

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

      4. unpow2 70.6%

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

   9. Simplified 70.6%

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

   10. Taylor expanded in l around inf 57.8%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \frac{\ell \cdot ky}{Om} + 0.25 \cdot \frac{Om}{\ell \cdot ky}}} \cdot 0.5} \]

   if 5e-166 < Om < 1.00000000000000001e122

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 91.7%

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

      1. associate-/l* 89.8%

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

      2. associate-/r/ 91.7%

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

      3. unpow2 91.7%

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

      4. unpow2 91.7%

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

      5. times-frac 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in ky around 0 76.2%

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

      1. unpow2 76.2%

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

      2. unpow2 76.2%

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

      3. unswap-sqr 80.4%

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

      4. unpow2 80.4%

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

   9. Simplified 80.4%

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

   if 1.00000000000000001e122 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 85.3%

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

      1. associate-/l* 84.4%

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

      2. associate-/r/ 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

      5. times-frac 100.0%

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

   6. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

      2. expm1-udef 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)} - 1}} \cdot 0.5} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   9. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. associate-*l/ 100.0%

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

      7. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in ky around 0 94.1%

       \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{2 \cdot \frac{ky \cdot \ell}{Om} + 0.25 \cdot \frac{Om}{ky \cdot \ell}}}\\ \mathbf{elif}\;Om \leq 10^{+122}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \frac{\left(ky \cdot \ell\right) \cdot \left(ky \cdot \ell\right)}{Om \cdot Om}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 69.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.7 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{2 \cdot \frac{ky \cdot \ell}{Om} + 0.25 \cdot \frac{Om}{ky \cdot \ell}}}\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{fma}\left(2, \frac{\left(ky \cdot \ell\right) \cdot \left(ky \cdot \ell\right)}{Om \cdot Om}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.7e-202)
   (sqrt
    (+
     0.5
     (* 0.5 (/ 1.0 (+ (* 2.0 (/ (* ky l) Om)) (* 0.25 (/ Om (* ky l))))))))
   (if (<= Om 1.15e-56)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (fma 2.0 (/ (* (* ky l) (* ky l)) (* Om Om)) 1.0)))))
     1.0)))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.7e-202) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))));
	} else if (Om <= 1.15e-56) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / fma(2.0, (((ky * l) * (ky * l)) / (Om * Om)), 1.0)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.7e-202)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(Float64(2.0 * Float64(Float64(ky * l) / Om)) + Float64(0.25 * Float64(Om / Float64(ky * l))))))));
	elseif (Om <= 1.15e-56)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / fma(2.0, Float64(Float64(Float64(ky * l) * Float64(ky * l)) / Float64(Om * Om)), 1.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.7e-202], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(Om / N[(ky * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.15e-56], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(2.0 * N[(N[(N[(ky * l), $MachinePrecision] * N[(ky * l), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 1.70000000000000006e-202

   1. Initial program 98.7%

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

      1. distribute-rgt-in 98.7%

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

      2. metadata-eval 98.7%

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

      3. metadata-eval 98.7%

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

      4. associate-/l* 98.7%

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

      5. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in kx around 0 76.6%

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

      1. associate-/l* 77.6%

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

      2. associate-/r/ 78.2%

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

      3. unpow2 78.2%

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

      4. unpow2 78.2%

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

      5. times-frac 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in ky around 0 59.1%

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

      1. unpow2 59.1%

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

      2. unpow2 59.1%

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

      3. unswap-sqr 70.4%

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

      4. unpow2 70.4%

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

   9. Simplified 70.4%

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

   10. Taylor expanded in l around inf 57.3%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \frac{\ell \cdot ky}{Om} + 0.25 \cdot \frac{Om}{\ell \cdot ky}}} \cdot 0.5} \]

   if 1.70000000000000006e-202 < Om < 1.15000000000000001e-56

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 82.4%

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

      1. associate-/l* 82.4%

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

      2. associate-/r/ 82.4%

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

      3. unpow2 82.4%

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

      4. unpow2 82.4%

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

      5. times-frac 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in ky around 0 78.4%

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

      1. +-commutative 78.4%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \frac{{\ell}^{2} \cdot {ky}^{2}}{{Om}^{2}} + 1}} \cdot 0.5} \]

      2. fma-def 78.4%

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

      3. unpow2 78.4%

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

      4. unpow2 78.4%

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

      5. unswap-sqr 80.0%

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

      6. unpow2 80.0%

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

   9. Simplified 80.0%

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

   if 1.15000000000000001e-56 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 90.1%

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

      1. associate-/l* 88.3%

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

      2. associate-/r/ 90.1%

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

      3. unpow2 90.1%

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

      4. unpow2 90.1%

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

      5. times-frac 97.9%

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

   6. Simplified 97.9%

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

      1. expm1-log1p-u 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

      2. expm1-udef 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)} - 1}} \cdot 0.5} \]

   8. Applied egg-rr 97.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   9. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

      2. expm1-log1p 97.9%

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

      3. *-commutative 97.9%

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

      4. *-commutative 97.9%

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

      5. associate-*l* 97.9%

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

      6. associate-*l/ 97.9%

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

      7. *-commutative 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in ky around 0 89.3%

       \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.7 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{2 \cdot \frac{ky \cdot \ell}{Om} + 0.25 \cdot \frac{Om}{ky \cdot \ell}}}\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{fma}\left(2, \frac{\left(ky \cdot \ell\right) \cdot \left(ky \cdot \ell\right)}{Om \cdot Om}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 68.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 8.2 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 8.2e-139)
   (sqrt 0.5)
   (if (<= Om 1.1e-56)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* l l) (/ (* Om Om) (* ky ky)))))))))
     1.0)))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8.2e-139) {
		tmp = sqrt(0.5);
	} else if (Om <= 1.1e-56) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 8.2d-139) then
        tmp = sqrt(0.5d0)
    else if (om <= 1.1d-56) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((l * l) / ((om * om) / (ky * ky)))))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8.2e-139) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 1.1e-56) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 8.2e-139:
		tmp = math.sqrt(0.5)
	elif Om <= 1.1e-56:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))))
	else:
		tmp = 1.0
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 8.2e-139)
		tmp = sqrt(0.5);
	elseif (Om <= 1.1e-56)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(l * l) / Float64(Float64(Om * Om) / Float64(ky * ky)))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 8.2e-139)
		tmp = sqrt(0.5);
	elseif (Om <= 1.1e-56)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 8.2e-139], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 1.1e-56], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 8.20000000000000028e-139

   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. distribute-rgt-in 98.8%

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

      2. metadata-eval 98.8%

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

      3. metadata-eval 98.8%

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

      4. associate-/l* 98.8%

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

      5. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in Om around 0 50.0%

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

      1. associate-*r* 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-*r* 50.0%

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

      4. unpow2 50.0%

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

      5. unpow2 50.0%

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

      6. hypot-def 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in l around inf 59.4%

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

   if 8.20000000000000028e-139 < Om < 1.10000000000000002e-56

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 77.3%

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

      1. associate-/l* 77.3%

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

      2. associate-/r/ 77.3%

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

      3. unpow2 77.3%

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

      4. unpow2 77.3%

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

      5. times-frac 77.3%

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

   6. Simplified 77.3%

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

   7. Taylor expanded in ky around 0 70.7%

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

      1. associate-/l* 64.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{ky}^{2}}}}} \cdot 0.5} \]

      2. unpow2 64.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{{Om}^{2}}{{ky}^{2}}}} \cdot 0.5} \]

      3. unpow2 64.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{\color{blue}{Om \cdot Om}}{{ky}^{2}}}} \cdot 0.5} \]

      4. unpow2 64.0%

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

   9. Simplified 64.0%

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

   if 1.10000000000000002e-56 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 90.1%

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

      1. associate-/l* 88.3%

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

      2. associate-/r/ 90.1%

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

      3. unpow2 90.1%

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

      4. unpow2 90.1%

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

      5. times-frac 97.9%

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

   6. Simplified 97.9%

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

      1. expm1-log1p-u 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

      2. expm1-udef 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)} - 1}} \cdot 0.5} \]

   8. Applied egg-rr 97.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   9. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

      2. expm1-log1p 97.9%

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

      3. *-commutative 97.9%

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

      4. *-commutative 97.9%

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

      5. associate-*l* 97.9%

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

      6. associate-*l/ 97.9%

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

      7. *-commutative 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in ky around 0 89.3%

       \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 8.2 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 68.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.26 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{2 \cdot \frac{ky \cdot \ell}{Om} + 0.25 \cdot \frac{Om}{ky \cdot \ell}}}\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.26e-138)
   (sqrt
    (+
     0.5
     (* 0.5 (/ 1.0 (+ (* 2.0 (/ (* ky l) Om)) (* 0.25 (/ Om (* ky l))))))))
   (if (<= Om 1.15e-56)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* l l) (/ (* Om Om) (* ky ky)))))))))
     1.0)))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.26e-138) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))));
	} else if (Om <= 1.15e-56) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 1.26d-138) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / ((2.0d0 * ((ky * l) / om)) + (0.25d0 * (om / (ky * l))))))))
    else if (om <= 1.15d-56) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((l * l) / ((om * om) / (ky * ky)))))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.26e-138) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))));
	} else if (Om <= 1.15e-56) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.26e-138:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))))
	elif Om <= 1.15e-56:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))))
	else:
		tmp = 1.0
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.26e-138)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(Float64(2.0 * Float64(Float64(ky * l) / Om)) + Float64(0.25 * Float64(Om / Float64(ky * l))))))));
	elseif (Om <= 1.15e-56)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(l * l) / Float64(Float64(Om * Om) / Float64(ky * ky)))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1.26e-138)
		tmp = sqrt((0.5 + (0.5 * (1.0 / ((2.0 * ((ky * l) / Om)) + (0.25 * (Om / (ky * l))))))));
	elseif (Om <= 1.15e-56)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.26e-138], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] + N[(0.25 * N[(Om / N[(ky * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[Om, 1.15e-56], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 1.26e-138

   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. distribute-rgt-in 98.8%

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

      2. metadata-eval 98.8%

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

      3. metadata-eval 98.8%

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

      4. associate-/l* 98.8%

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

      5. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in kx around 0 77.4%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 78.9%

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

      3. unpow2 78.9%

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

      4. unpow2 78.9%

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

      5. times-frac 88.8%

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

   6. Simplified 88.8%

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

   7. Taylor expanded in ky around 0 61.0%

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

      1. unpow2 61.0%

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

      2. unpow2 61.0%

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

      3. unswap-sqr 71.6%

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

      4. unpow2 71.6%

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

   9. Simplified 71.6%

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

   10. Taylor expanded in l around inf 59.4%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{2 \cdot \frac{\ell \cdot ky}{Om} + 0.25 \cdot \frac{Om}{\ell \cdot ky}}} \cdot 0.5} \]

   if 1.26e-138 < Om < 1.15000000000000001e-56

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 77.3%

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

      1. associate-/l* 77.3%

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

      2. associate-/r/ 77.3%

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

      3. unpow2 77.3%

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

      4. unpow2 77.3%

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

      5. times-frac 77.3%

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

   6. Simplified 77.3%

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

   7. Taylor expanded in ky around 0 70.7%

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

      1. associate-/l* 64.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{ky}^{2}}}}} \cdot 0.5} \]

      2. unpow2 64.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{{Om}^{2}}{{ky}^{2}}}} \cdot 0.5} \]

      3. unpow2 64.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{\color{blue}{Om \cdot Om}}{{ky}^{2}}}} \cdot 0.5} \]

      4. unpow2 64.0%

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

   9. Simplified 64.0%

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

   if 1.15000000000000001e-56 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 90.1%

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

      1. associate-/l* 88.3%

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

      2. associate-/r/ 90.1%

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

      3. unpow2 90.1%

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

      4. unpow2 90.1%

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

      5. times-frac 97.9%

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

   6. Simplified 97.9%

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

      1. expm1-log1p-u 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

      2. expm1-udef 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)} - 1}} \cdot 0.5} \]

   8. Applied egg-rr 97.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   9. Step-by-step derivation

      1. expm1-def 97.9%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

      2. expm1-log1p 97.9%

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

      3. *-commutative 97.9%

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

      4. *-commutative 97.9%

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

      5. associate-*l* 97.9%

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

      6. associate-*l/ 97.9%

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

      7. *-commutative 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in ky around 0 89.3%

       \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.26 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{2 \cdot \frac{ky \cdot \ell}{Om} + 0.25 \cdot \frac{Om}{ky \cdot \ell}}}\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 67.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{-82}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 5e-108)
   (sqrt 0.5)
   (if (<= Om 1.6e-82) 1.0 (if (<= Om 1.05e-56) (sqrt 0.5) 1.0))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5e-108) {
		tmp = sqrt(0.5);
	} else if (Om <= 1.6e-82) {
		tmp = 1.0;
	} else if (Om <= 1.05e-56) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 5d-108) then
        tmp = sqrt(0.5d0)
    else if (om <= 1.6d-82) then
        tmp = 1.0d0
    else if (om <= 1.05d-56) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5e-108) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 1.6e-82) {
		tmp = 1.0;
	} else if (Om <= 1.05e-56) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 5e-108:
		tmp = math.sqrt(0.5)
	elif Om <= 1.6e-82:
		tmp = 1.0
	elif Om <= 1.05e-56:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 5e-108)
		tmp = sqrt(0.5);
	elseif (Om <= 1.6e-82)
		tmp = 1.0;
	elseif (Om <= 1.05e-56)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 5e-108)
		tmp = sqrt(0.5);
	elseif (Om <= 1.6e-82)
		tmp = 1.0;
	elseif (Om <= 1.05e-56)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 5e-108], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 1.6e-82], 1.0, If[LessEqual[Om, 1.05e-56], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if Om < 5e-108 or 1.6000000000000001e-82 < Om < 1.05000000000000003e-56

   1. Initial program 98.9%

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

      1. distribute-rgt-in 98.9%

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

      2. metadata-eval 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-/l* 98.9%

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

      5. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in Om around 0 51.7%

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

      1. associate-*r* 51.7%

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

      2. *-commutative 51.7%

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

      3. associate-*r* 51.7%

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

      4. unpow2 51.7%

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

      5. unpow2 51.7%

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

      6. hypot-def 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in l around inf 60.8%

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

   if 5e-108 < Om < 1.6000000000000001e-82 or 1.05000000000000003e-56 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 90.5%

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

      1. associate-/l* 88.8%

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

      2. associate-/r/ 90.5%

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

      3. unpow2 90.5%

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

      4. unpow2 90.5%

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

      5. times-frac 98.0%

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

   6. Simplified 98.0%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)\right)}} \cdot 0.5} \]

      2. expm1-udef 98.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot {\sin ky}^{2}\right)}\right)} - 1}} \cdot 0.5} \]

   8. Applied egg-rr 98.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   9. Step-by-step derivation

      1. expm1-def 98.0%

         \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \sin ky\right)\right)\right)\right)}} \cdot 0.5} \]

      2. expm1-log1p 98.0%

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

      3. *-commutative 98.0%

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

      4. *-commutative 98.0%

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

      5. associate-*l* 98.0%

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

      6. associate-*l/ 98.0%

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

      7. *-commutative 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in ky around 0 89.7%

       \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{-82}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 1.05 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 56.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \sqrt{0.5} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky) :precision binary64 (sqrt 0.5))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

Fortran

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

Java

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

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

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

MATLAB

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

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in 99.2%

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

   2. metadata-eval 99.2%

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

   3. metadata-eval 99.2%

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

   4. associate-/l* 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in Om around 0 40.4%

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

   1. associate-*r* 40.4%

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

   2. *-commutative 40.4%

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

   3. associate-*r* 40.4%

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

   4. unpow2 40.4%

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

   5. unpow2 40.4%

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

   6. hypot-def 41.2%

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

6. Simplified 41.2%

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

7. Taylor expanded in l around inf 51.1%

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

8. Final simplification 51.1%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023217 
(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))))))))))