Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.6%

Time: 11.9s

Alternatives: 3

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.3× speedup

Math

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

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 (* 2.0 (* (/ l Om) (sin ky)))))))))

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, (2.0 * ((l / Om) * sin(ky))))))));
}

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, (2.0 * ((l / Om) * Math.sin(ky))))))));
}

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, (2.0 * ((l / Om) * math.sin(ky))))))))

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(2.0 * Float64(Float64(l / Om) * sin(ky))))))))
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, (2.0 * ((l / Om) * sin(ky))))))));
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[(2.0 * N[(N[(l / Om), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.7%

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

   1. distribute-rgt-in 97.7%

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

   2. metadata-eval 97.7%

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

   3. metadata-eval 97.7%

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

   4. associate-/l* 97.7%

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

   5. metadata-eval 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in kx around 0 77.6%

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

   1. associate-/l* 76.3%

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

   2. associate-/r/ 77.1%

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

   3. unpow2 77.1%

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

   4. unpow2 77.1%

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

   5. times-frac 87.8%

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

6. Simplified 87.8%

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

   1. add-sqr-sqrt 87.8%

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

   2. hypot-1-def 87.8%

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

   3. sqrt-prod 87.8%

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

   4. metadata-eval 87.8%

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

   5. sqrt-prod 87.8%

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

   6. sqrt-prod 51.6%

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

   7. add-sqr-sqrt 89.6%

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

   8. unpow2 89.6%

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

   9. sqrt-prod 46.1%

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

   10. add-sqr-sqrt 94.1%

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

8. Applied egg-rr 94.1%

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

9. Final simplification 94.1%

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

Alternative 2: 70.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+58}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

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 (<= l 2.1e+58) 1.0 (sqrt 0.5)))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 2.1e+58) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	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 (l <= 2.1d+58) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    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 (l <= 2.1e+58) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if l <= 2.1e+58:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 2.1e+58)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	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 (l <= 2.1e+58)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	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[l, 2.1e+58], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.10000000000000012e58

   1. Initial program 98.4%

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

      1. distribute-rgt-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. metadata-eval 98.4%

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

      4. associate-/l* 98.4%

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

      5. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in kx around 0 78.1%

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

      1. associate-/l* 78.5%

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

      2. associate-/r/ 78.1%

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

      3. unpow2 78.1%

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

      4. unpow2 78.1%

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

      5. times-frac 88.5%

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

   6. Simplified 88.5%

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

      1. add-sqr-sqrt 88.5%

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

      2. hypot-1-def 88.5%

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

      3. sqrt-prod 88.5%

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

      4. metadata-eval 88.5%

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

      5. sqrt-prod 88.5%

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

      6. sqrt-prod 55.4%

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

      7. add-sqr-sqrt 89.9%

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

      8. unpow2 89.9%

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

      9. sqrt-prod 45.8%

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

      10. add-sqr-sqrt 93.4%

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

   8. Applied egg-rr 93.4%

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

   9. Taylor expanded in l around 0 66.9%

      \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]

   if 2.10000000000000012e58 < l

   1. Initial program 95.4%

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

      1. distribute-rgt-in 95.4%

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

      2. metadata-eval 95.4%

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

      3. metadata-eval 95.4%

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

      4. associate-/l* 95.4%

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

      5. metadata-eval 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in Om around 0 72.7%

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

      1. associate-*r* 72.7%

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

      2. *-commutative 72.7%

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

      3. associate-*r* 72.7%

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

      4. unpow2 72.7%

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

      5. unpow2 72.7%

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

      6. hypot-def 75.8%

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

   6. Simplified 75.8%

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

   7. Taylor expanded in l around inf 79.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.1 \cdot 10^{+58}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3: 56.0% 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 97.7%

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

   1. distribute-rgt-in 97.7%

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

   2. metadata-eval 97.7%

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

   3. metadata-eval 97.7%

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

   4. associate-/l* 97.7%

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

   5. metadata-eval 97.7%

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

3. Simplified 97.7%

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

4. Taylor expanded in Om around 0 49.7%

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

   1. associate-*r* 49.7%

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

   2. *-commutative 49.7%

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

   3. associate-*r* 49.7%

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

   4. unpow2 49.7%

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

   5. unpow2 49.7%

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

   6. hypot-def 50.9%

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

6. Simplified 50.9%

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

7. Taylor expanded in l around inf 58.8%

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

8. Final simplification 58.8%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023238 
(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))))))))))