Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.5%

Time: 12.2s

Alternatives: 5

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: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \sin ky_m\right) \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (cbrt (pow (+ 0.5 (/ 0.5 (hypot 1.0 (* (* 2.0 (sin ky_m)) (/ l Om))))) 1.5)))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return cbrt(pow((0.5 + (0.5 / hypot(1.0, ((2.0 * sin(ky_m)) * (l / Om))))), 1.5));
}

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	return Math.cbrt(Math.pow((0.5 + (0.5 / Math.hypot(1.0, ((2.0 * Math.sin(ky_m)) * (l / Om))))), 1.5));
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	return cbrt((Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * sin(ky_m)) * Float64(l / Om))))) ^ 1.5))
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := N[Power[N[Power[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[Sin[ky$95$m], $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

   1. expm1-log1p-u 97.6%

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

   2. expm1-udef 97.6%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 90.6%

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

   1. add-cbrt-cube 90.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}}} \]

   2. pow1/3 90.6%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

10. Applied egg-rr 90.6%

    \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
11. Step-by-step derivation

    1. unpow1/3 90.6%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

    2. associate-*r* 90.6%

       \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\left(\sin ky \cdot 2\right) \cdot \frac{\ell}{Om}}\right)}\right)}^{1.5}} \]

    3. *-commutative 90.6%

       \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\left(2 \cdot \sin ky\right)} \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}} \]

12. Simplified 90.6%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \sin ky\right) \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}}} \]

13. Final simplification 90.6%

    \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \sin ky\right) \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}} \]

Alternative 2: 99.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

   1. expm1-log1p-u 97.6%

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

   2. expm1-udef 97.6%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 90.6%

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

   1. expm1-log1p-u 90.1%

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

   2. expm1-udef 90.1%

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

   3. associate-*l/ 90.1%

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

   4. metadata-eval 90.1%

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

   5. associate-*r/ 90.1%

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

   6. div-inv 90.1%

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

   7. clear-num 90.1%

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

   8. associate-*r* 90.1%

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

10. Applied egg-rr 90.1%

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

    1. expm1-def 90.1%

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

    2. expm1-log1p 90.6%

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

    3. associate-*r* 90.6%

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

    4. *-commutative 90.6%

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

12. Simplified 90.6%

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

13. Final simplification 90.6%

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

Alternative 3: 74.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{-91}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky_m \cdot \ell}{Om}\right)}}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= l 2.35e-91)
   1.0
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* ky_m l) Om))))))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 2.35e-91) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky_m * l) / Om))))));
	}
	return tmp;
}

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 2.35e-91) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((ky_m * l) / Om))))));
	}
	return tmp;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if l <= 2.35e-91:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((ky_m * l) / Om))))))
	return tmp

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (l <= 2.35e-91)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(ky_m * l) / Om))))));
	end
	return tmp
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (l <= 2.35e-91)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky_m * l) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[l, 2.35e-91], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(ky$95$m * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.35000000000000003e-91

   1. Initial program 98.8%

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

   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. Step-by-step derivation

      1. expm1-log1p-u 98.8%

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

      2. expm1-udef 98.8%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 91.7%

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

      1. add-cbrt-cube 91.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}}} \]

      2. pow1/3 91.7%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 91.7%

       \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. unpow1/3 91.7%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

       2. associate-*r* 91.7%

          \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\left(\sin ky \cdot 2\right) \cdot \frac{\ell}{Om}}\right)}\right)}^{1.5}} \]

       3. *-commutative 91.7%

          \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\left(2 \cdot \sin ky\right)} \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}} \]

   12. Simplified 91.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \sin ky\right) \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}}} \]

   13. Taylor expanded in ky around 0 69.9%

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

   if 2.35000000000000003e-91 < l

   1. Initial program 95.6%

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

   3. Simplified 95.6%

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

      1. expm1-log1p-u 95.6%

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

      2. expm1-udef 95.6%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 88.6%

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

      1. expm1-log1p-u 88.0%

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

      2. expm1-udef 88.0%

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

      3. associate-*l/ 88.0%

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

      4. metadata-eval 88.0%

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

      5. associate-*r/ 88.0%

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

      6. div-inv 88.0%

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

      7. clear-num 88.0%

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

      8. associate-*r* 88.0%

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

   10. Applied egg-rr 88.0%

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

       1. expm1-def 88.0%

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

       2. expm1-log1p 88.6%

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

       3. associate-*r* 88.6%

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

       4. *-commutative 88.6%

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

   12. Simplified 88.6%

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

   13. Taylor expanded in ky around 0 79.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{-91}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 4: 70.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= l 4.7e-85)
   1.0
   (if (<= l 9e-54) (sqrt 0.5) (if (<= l 1.6e+50) 1.0 (sqrt 0.5)))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 4.7e-85) {
		tmp = 1.0;
	} else if (l <= 9e-54) {
		tmp = sqrt(0.5);
	} else if (l <= 1.6e+50) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (l <= 4.7d-85) then
        tmp = 1.0d0
    else if (l <= 9d-54) then
        tmp = sqrt(0.5d0)
    else if (l <= 1.6d+50) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 4.7e-85) {
		tmp = 1.0;
	} else if (l <= 9e-54) {
		tmp = Math.sqrt(0.5);
	} else if (l <= 1.6e+50) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if l <= 4.7e-85:
		tmp = 1.0
	elif l <= 9e-54:
		tmp = math.sqrt(0.5)
	elif l <= 1.6e+50:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (l <= 4.7e-85)
		tmp = 1.0;
	elseif (l <= 9e-54)
		tmp = sqrt(0.5);
	elseif (l <= 1.6e+50)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (l <= 4.7e-85)
		tmp = 1.0;
	elseif (l <= 9e-54)
		tmp = sqrt(0.5);
	elseif (l <= 1.6e+50)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[l, 4.7e-85], 1.0, If[LessEqual[l, 9e-54], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, 1.6e+50], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 4.70000000000000009e-85 or 8.9999999999999997e-54 < l < 1.59999999999999991e50

   1. Initial program 98.4%

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

   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. Step-by-step derivation

      1. expm1-log1p-u 98.4%

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

      2. expm1-udef 98.4%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. associate-/l* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 92.9%

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

      1. add-cbrt-cube 92.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}}} \]

      2. pow1/3 92.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 92.9%

       \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. unpow1/3 92.9%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

       2. associate-*r* 92.9%

          \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\left(\sin ky \cdot 2\right) \cdot \frac{\ell}{Om}}\right)}\right)}^{1.5}} \]

       3. *-commutative 92.9%

          \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\left(2 \cdot \sin ky\right)} \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}} \]

   12. Simplified 92.9%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \sin ky\right) \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}}} \]

   13. Taylor expanded in ky around 0 71.7%

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

   if 4.70000000000000009e-85 < l < 8.9999999999999997e-54 or 1.59999999999999991e50 < l

   1. Initial program 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in Om around 0 71.4%

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

      1. associate-*r* 71.4%

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

      2. *-commutative 71.4%

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

      3. unpow2 71.4%

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

      4. unpow2 71.4%

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

      5. hypot-def 76.2%

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

      6. associate-*r/ 76.2%

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

      7. associate-/l* 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in Om around 0 80.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-85}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 62.4% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

3. Simplified 97.6%

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

   1. expm1-log1p-u 97.6%

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

   2. expm1-udef 97.6%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*r/ 100.0%

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

   5. associate-/l* 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 90.6%

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

   1. add-cbrt-cube 90.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}}} \]

   2. pow1/3 90.6%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sin ky \cdot \frac{2}{\frac{Om}{\ell}}\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

10. Applied egg-rr 90.6%

    \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
11. Step-by-step derivation

    1. unpow1/3 90.6%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

    2. associate-*r* 90.6%

       \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\left(\sin ky \cdot 2\right) \cdot \frac{\ell}{Om}}\right)}\right)}^{1.5}} \]

    3. *-commutative 90.6%

       \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\left(2 \cdot \sin ky\right)} \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}} \]

12. Simplified 90.6%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \sin ky\right) \cdot \frac{\ell}{Om}\right)}\right)}^{1.5}}} \]

13. Taylor expanded in ky around 0 63.5%

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

14. Final simplification 63.5%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023333 
(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))))))))))