Toniolo and Linder, Equation (3a)

Percentage Accurate: 39.5% → 73.5%

Time: 42.3s

Precision: binary64

Cost: 43600

• Unsound rule application detected in e-graph. Results may not be sound.

• 60.6% of points produce a very large (infinite) output. You may want to add a precondition.

\[ \begin{array}{c}[kx, ky] = \mathsf{sort}([kx, ky])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := {\sin ky}^{3}\\ t_1 := 0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{ky \cdot \ell}\\ t_2 := \frac{kx \cdot \left(kx \cdot Om\right)}{t_0 \cdot \ell}\\ t_3 := \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}}\\ t_4 := -0.027777777777777776 \cdot \frac{Om}{\ell} + \frac{Om}{\ell} \cdot 0.008333333333333333\\ t_5 := t_4 \cdot {ky}^{3}\\ \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{t_0 \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{elif}\;kx \leq -2.1 \cdot 10^{-202}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;kx \leq -4.8 \cdot 10^{-238}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, t_2, t_1 - t_5\right)\right)}^{0.5}\\ \mathbf{elif}\;kx \leq 6.5 \cdot 10^{-225}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, t_2, \left(t_1 - {ky}^{5} \cdot \left(\frac{Om}{\ell} \cdot 0.001388888888888889 + \left(\frac{Om}{\ell} \cdot -0.0001984126984126984 + t_4 \cdot 0.16666666666666666\right)\right)\right) - t_5\right)\right)}^{0.5}\\ \end{array} \]

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))))))))))

↓

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (pow (sin ky) 3.0))
        (t_1 (+ (* 0.16666666666666666 (/ (* Om ky) l)) (/ Om (* ky l))))
        (t_2 (/ (* kx (* kx Om)) (* t_0 l)))
        (t_3
         (sqrt
          (+
           0.5
           (*
            0.5
            (/
             1.0
             (hypot 1.0 (* (* 2.0 (/ l Om)) (hypot (sin kx) (sin ky)))))))))
        (t_4
         (+
          (* -0.027777777777777776 (/ Om l))
          (* (/ Om l) 0.008333333333333333)))
        (t_5 (* t_4 (pow ky 3.0))))
   (if (<= kx -1.02e+79)
     (pow (+ 0.5 (* (* -0.5 (/ Om (* t_0 (/ l (* kx kx))))) -0.25)) 0.5)
     (if (<= kx -2.1e-202)
       t_3
       (if (<= kx -4.8e-238)
         (pow (+ 0.5 (* -0.25 (fma -0.5 t_2 (- t_1 t_5)))) 0.5)
         (if (<= kx 6.5e-225)
           t_3
           (pow
            (+
             0.5
             (*
              -0.25
              (fma
               -0.5
               t_2
               (-
                (-
                 t_1
                 (*
                  (pow ky 5.0)
                  (+
                   (* (/ Om l) 0.001388888888888889)
                   (+
                    (* (/ Om l) -0.0001984126984126984)
                    (* t_4 0.16666666666666666)))))
                t_5))))
            0.5)))))))

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)))))))));
}

↓

double code(double l, double Om, double kx, double ky) {
	double t_0 = pow(sin(ky), 3.0);
	double t_1 = (0.16666666666666666 * ((Om * ky) / l)) + (Om / (ky * l));
	double t_2 = (kx * (kx * Om)) / (t_0 * l);
	double t_3 = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 * (l / Om)) * hypot(sin(kx), sin(ky))))))));
	double t_4 = (-0.027777777777777776 * (Om / l)) + ((Om / l) * 0.008333333333333333);
	double t_5 = t_4 * pow(ky, 3.0);
	double tmp;
	if (kx <= -1.02e+79) {
		tmp = pow((0.5 + ((-0.5 * (Om / (t_0 * (l / (kx * kx))))) * -0.25)), 0.5);
	} else if (kx <= -2.1e-202) {
		tmp = t_3;
	} else if (kx <= -4.8e-238) {
		tmp = pow((0.5 + (-0.25 * fma(-0.5, t_2, (t_1 - t_5)))), 0.5);
	} else if (kx <= 6.5e-225) {
		tmp = t_3;
	} else {
		tmp = pow((0.5 + (-0.25 * fma(-0.5, t_2, ((t_1 - (pow(ky, 5.0) * (((Om / l) * 0.001388888888888889) + (((Om / l) * -0.0001984126984126984) + (t_4 * 0.16666666666666666))))) - t_5)))), 0.5);
	}
	return tmp;
}

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

↓

function code(l, Om, kx, ky)
	t_0 = sin(ky) ^ 3.0
	t_1 = Float64(Float64(0.16666666666666666 * Float64(Float64(Om * ky) / l)) + Float64(Om / Float64(ky * l)))
	t_2 = Float64(Float64(kx * Float64(kx * Om)) / Float64(t_0 * l))
	t_3 = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(Float64(2.0 * Float64(l / Om)) * hypot(sin(kx), sin(ky))))))))
	t_4 = Float64(Float64(-0.027777777777777776 * Float64(Om / l)) + Float64(Float64(Om / l) * 0.008333333333333333))
	t_5 = Float64(t_4 * (ky ^ 3.0))
	tmp = 0.0
	if (kx <= -1.02e+79)
		tmp = Float64(0.5 + Float64(Float64(-0.5 * Float64(Om / Float64(t_0 * Float64(l / Float64(kx * kx))))) * -0.25)) ^ 0.5;
	elseif (kx <= -2.1e-202)
		tmp = t_3;
	elseif (kx <= -4.8e-238)
		tmp = Float64(0.5 + Float64(-0.25 * fma(-0.5, t_2, Float64(t_1 - t_5)))) ^ 0.5;
	elseif (kx <= 6.5e-225)
		tmp = t_3;
	else
		tmp = Float64(0.5 + Float64(-0.25 * fma(-0.5, t_2, Float64(Float64(t_1 - Float64((ky ^ 5.0) * Float64(Float64(Float64(Om / l) * 0.001388888888888889) + Float64(Float64(Float64(Om / l) * -0.0001984126984126984) + Float64(t_4 * 0.16666666666666666))))) - t_5)))) ^ 0.5;
	end
	return tmp
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]

↓

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Power[N[Sin[ky], $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.16666666666666666 * N[(N[(Om * ky), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] + N[(Om / N[(ky * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(kx * N[(kx * Om), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(-0.027777777777777776 * N[(Om / l), $MachinePrecision]), $MachinePrecision] + N[(N[(Om / l), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Power[ky, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[kx, -1.02e+79], N[Power[N[(0.5 + N[(N[(-0.5 * N[(Om / N[(t$95$0 * N[(l / N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[kx, -2.1e-202], t$95$3, If[LessEqual[kx, -4.8e-238], N[Power[N[(0.5 + N[(-0.25 * N[(-0.5 * t$95$2 + N[(t$95$1 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], If[LessEqual[kx, 6.5e-225], t$95$3, N[Power[N[(0.5 + N[(-0.25 * N[(-0.5 * t$95$2 + N[(N[(t$95$1 - N[(N[Power[ky, 5.0], $MachinePrecision] * N[(N[(N[(Om / l), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + N[(N[(N[(Om / l), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] + N[(t$95$4 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if kx < -1.02000000000000006e79

   1. Initial program 3.1%

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

   2. Simplified 3.1%

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

      [Start] 3.1%	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
      distribute-rgt-in [=>] 3.1%	\[ \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]
      metadata-eval [=>] 3.1%	\[ \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      metadata-eval [=>] 3.1%	\[ \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      associate-/l* [=>] 3.1%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      metadata-eval [=>] 3.1%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Taylor expanded in l around -inf 6.9%

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

   4. Simplified 6.9%

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

      [Start] 6.9%	\[ \sqrt{0.5 + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \frac{Om}{\ell}\right)\right) \cdot 0.5} \]
      *-commutative [=>] 6.9%	\[ \sqrt{0.5 + \color{blue}{\left(\left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \frac{Om}{\ell}\right) \cdot -0.5\right)} \cdot 0.5} \]

   5. Taylor expanded in kx around 0 34.2%

      \[\leadsto \sqrt{0.5 + \left(\color{blue}{\left(-0.5 \cdot \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}} + \frac{Om}{\ell \cdot \sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]

   6. Simplified 36.4%

      \[\leadsto \sqrt{0.5 + \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]
      Step-by-step derivation

      [Start] 34.2%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}} + \frac{Om}{\ell \cdot \sin ky}\right) \cdot -0.5\right) \cdot 0.5} \]
      fma-def [=>] 34.2%	\[ \sqrt{0.5 + \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}}, \frac{Om}{\ell \cdot \sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]
      unpow2 [=>] 34.2%	\[ \sqrt{0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \color{blue}{\left(kx \cdot kx\right)}}{\ell \cdot {\sin ky}^{3}}, \frac{Om}{\ell \cdot \sin ky}\right) \cdot -0.5\right) \cdot 0.5} \]
      associate-/r* [=>] 36.4%	\[ \sqrt{0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \color{blue}{\frac{\frac{Om}{\ell}}{\sin ky}}\right) \cdot -0.5\right) \cdot 0.5} \]

   7. Taylor expanded in kx around inf 38.7%

      \[\leadsto \sqrt{0.5 + \left(\color{blue}{\left(-0.5 \cdot \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}}\right)} \cdot -0.5\right) \cdot 0.5} \]

   8. Simplified 36.5%

      \[\leadsto \sqrt{0.5 + \left(\color{blue}{\left(-0.5 \cdot \frac{\frac{Om}{\frac{\ell}{kx \cdot kx}}}{{\sin ky}^{3}}\right)} \cdot -0.5\right) \cdot 0.5} \]
      Step-by-step derivation

      [Start] 38.7%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5} \]
      unpow2 [=>] 38.7%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{Om \cdot \color{blue}{\left(kx \cdot kx\right)}}{\ell \cdot {\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5} \]
      associate-*l* [<=] 36.5%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{\color{blue}{\left(Om \cdot kx\right) \cdot kx}}{\ell \cdot {\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5} \]
      associate-/r* [=>] 36.5%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \color{blue}{\frac{\frac{\left(Om \cdot kx\right) \cdot kx}{\ell}}{{\sin ky}^{3}}}\right) \cdot -0.5\right) \cdot 0.5} \]
      associate-*l* [=>] 38.7%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{\frac{\color{blue}{Om \cdot \left(kx \cdot kx\right)}}{\ell}}{{\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5} \]
      unpow2 [<=] 38.7%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{\frac{Om \cdot \color{blue}{{kx}^{2}}}{\ell}}{{\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5} \]
      associate-/l* [=>] 36.5%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{\color{blue}{\frac{Om}{\frac{\ell}{{kx}^{2}}}}}{{\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5} \]
      unpow2 [=>] 36.5%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{\frac{Om}{\frac{\ell}{\color{blue}{kx \cdot kx}}}}{{\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5} \]

   9. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{{\left(0.5 + \left(-0.5 \cdot \frac{Om}{{\sin ky}^{3} \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 36.5%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{\frac{Om}{\frac{\ell}{kx \cdot kx}}}{{\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5} \]
      pow1/2 [=>] 93.4%	\[ \color{blue}{{\left(0.5 + \left(\left(-0.5 \cdot \frac{\frac{Om}{\frac{\ell}{kx \cdot kx}}}{{\sin ky}^{3}}\right) \cdot -0.5\right) \cdot 0.5\right)}^{0.5}} \]
      associate-*l* [=>] 93.4%	\[ {\left(0.5 + \color{blue}{\left(-0.5 \cdot \frac{\frac{Om}{\frac{\ell}{kx \cdot kx}}}{{\sin ky}^{3}}\right) \cdot \left(-0.5 \cdot 0.5\right)}\right)}^{0.5} \]
      associate-/l/ [=>] 95.6%	\[ {\left(0.5 + \left(-0.5 \cdot \color{blue}{\frac{Om}{{\sin ky}^{3} \cdot \frac{\ell}{kx \cdot kx}}}\right) \cdot \left(-0.5 \cdot 0.5\right)\right)}^{0.5} \]
      metadata-eval [=>] 95.6%	\[ {\left(0.5 + \left(-0.5 \cdot \frac{Om}{{\sin ky}^{3} \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot \color{blue}{-0.25}\right)}^{0.5} \]

   if -1.02000000000000006e79 < kx < -2.09999999999999985e-202 or -4.7999999999999997e-238 < kx < 6.5000000000000005e-225

   1. Initial program 60.6%

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

   2. Simplified 60.6%

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

      [Start] 60.6%	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
      distribute-rgt-in [=>] 60.6%	\[ \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]
      metadata-eval [=>] 60.6%	\[ \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      metadata-eval [=>] 60.6%	\[ \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      associate-/l* [=>] 60.6%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      metadata-eval [=>] 60.6%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Applied egg-rr 66.6%

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

      [Start] 60.6%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot 0.5} \]
      add-sqr-sqrt [=>] 60.6%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \cdot \sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}} \cdot 0.5} \]
      hypot-1-def [=>] 60.6%	\[ \sqrt{0.5 + \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)}} \cdot 0.5} \]
      sqrt-prod [=>] 60.6%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot 0.5} \]
      unpow2 [=>] 60.6%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{2}{\frac{Om}{\ell}} \cdot \frac{2}{\frac{Om}{\ell}}}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \cdot 0.5} \]
      sqrt-prod [=>] 38.2%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{2}{\frac{Om}{\ell}}} \cdot \sqrt{\frac{2}{\frac{Om}{\ell}}}\right)} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \cdot 0.5} \]
      add-sqr-sqrt [<=] 65.0%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{2}{\frac{Om}{\ell}}} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \cdot 0.5} \]
      div-inv [=>] 65.0%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(2 \cdot \frac{1}{\frac{Om}{\ell}}\right)} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \cdot 0.5} \]
      clear-num [<=] 65.0%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \color{blue}{\frac{\ell}{Om}}\right) \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)} \cdot 0.5} \]
      unpow2 [=>] 65.0%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sqrt{\color{blue}{\sin kx \cdot \sin kx} + {\sin ky}^{2}}\right)} \cdot 0.5} \]
      unpow2 [=>] 65.0%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sqrt{\sin kx \cdot \sin kx + \color{blue}{\sin ky \cdot \sin ky}}\right)} \cdot 0.5} \]
      hypot-def [=>] 66.6%	\[ \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\right)} \cdot 0.5} \]

   if -2.09999999999999985e-202 < kx < -4.7999999999999997e-238

   1. Initial program 41.9%

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

   2. Simplified 41.9%

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

      [Start] 41.9%	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
      distribute-rgt-in [=>] 41.9%	\[ \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]
      metadata-eval [=>] 41.9%	\[ \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      metadata-eval [=>] 41.9%	\[ \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      associate-/l* [=>] 41.9%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      metadata-eval [=>] 41.9%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Taylor expanded in l around -inf 40.6%

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

   4. Simplified 40.6%

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

      [Start] 40.6%	\[ \sqrt{0.5 + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \frac{Om}{\ell}\right)\right) \cdot 0.5} \]
      *-commutative [=>] 40.6%	\[ \sqrt{0.5 + \color{blue}{\left(\left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \frac{Om}{\ell}\right) \cdot -0.5\right)} \cdot 0.5} \]

   5. Taylor expanded in kx around 0 41.3%

      \[\leadsto \sqrt{0.5 + \left(\color{blue}{\left(-0.5 \cdot \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}} + \frac{Om}{\ell \cdot \sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]

   6. Simplified 41.3%

      \[\leadsto \sqrt{0.5 + \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]
      Step-by-step derivation

      [Start] 41.3%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}} + \frac{Om}{\ell \cdot \sin ky}\right) \cdot -0.5\right) \cdot 0.5} \]
      fma-def [=>] 41.3%	\[ \sqrt{0.5 + \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}}, \frac{Om}{\ell \cdot \sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]
      unpow2 [=>] 41.3%	\[ \sqrt{0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \color{blue}{\left(kx \cdot kx\right)}}{\ell \cdot {\sin ky}^{3}}, \frac{Om}{\ell \cdot \sin ky}\right) \cdot -0.5\right) \cdot 0.5} \]
      associate-/r* [=>] 41.3%	\[ \sqrt{0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \color{blue}{\frac{\frac{Om}{\ell}}{\sin ky}}\right) \cdot -0.5\right) \cdot 0.5} \]

   7. Applied egg-rr 41.3%

      \[\leadsto \color{blue}{{\left(0.5 + \mathsf{fma}\left(-0.5, \frac{\left(Om \cdot kx\right) \cdot kx}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot -0.25\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 41.3%	\[ \sqrt{0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot -0.5\right) \cdot 0.5} \]
      pow1/2 [=>] 41.3%	\[ \color{blue}{{\left(0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot -0.5\right) \cdot 0.5\right)}^{0.5}} \]
      associate-*l* [=>] 41.3%	\[ {\left(0.5 + \color{blue}{\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot \left(-0.5 \cdot 0.5\right)}\right)}^{0.5} \]
      associate-*r* [=>] 41.3%	\[ {\left(0.5 + \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(Om \cdot kx\right) \cdot kx}}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot \left(-0.5 \cdot 0.5\right)\right)}^{0.5} \]
      metadata-eval [=>] 41.3%	\[ {\left(0.5 + \mathsf{fma}\left(-0.5, \frac{\left(Om \cdot kx\right) \cdot kx}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot \color{blue}{-0.25}\right)}^{0.5} \]

   8. Taylor expanded in ky around 0 100.0%

      \[\leadsto {\left(0.5 + \mathsf{fma}\left(-0.5, \frac{\left(Om \cdot kx\right) \cdot kx}{\ell \cdot {\sin ky}^{3}}, \color{blue}{-1 \cdot \left(\left(-0.027777777777777776 \cdot \frac{Om}{\ell} + 0.008333333333333333 \cdot \frac{Om}{\ell}\right) \cdot {ky}^{3}\right) + \left(0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{\ell \cdot ky}\right)}\right) \cdot -0.25\right)}^{0.5} \]

   if 6.5000000000000005e-225 < kx

   1. Initial program 34.6%

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

   2. Simplified 34.6%

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

      [Start] 34.6%	\[ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
      distribute-rgt-in [=>] 34.6%	\[ \sqrt{\color{blue}{1 \cdot \frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}}} \]
      metadata-eval [=>] 34.6%	\[ \sqrt{1 \cdot \color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      metadata-eval [=>] 34.6%	\[ \sqrt{\color{blue}{0.5} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      associate-/l* [=>] 34.6%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\color{blue}{\left(\frac{2}{\frac{Om}{\ell}}\right)}}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}} \]
      metadata-eval [=>] 34.6%	\[ \sqrt{0.5 + \frac{1}{\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \color{blue}{0.5}} \]

   3. Taylor expanded in l around -inf 20.8%

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

   4. Simplified 20.8%

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

      [Start] 20.8%	\[ \sqrt{0.5 + \left(-0.5 \cdot \left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \frac{Om}{\ell}\right)\right) \cdot 0.5} \]
      *-commutative [=>] 20.8%	\[ \sqrt{0.5 + \color{blue}{\left(\left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \frac{Om}{\ell}\right) \cdot -0.5\right)} \cdot 0.5} \]

   5. Taylor expanded in kx around 0 27.5%

      \[\leadsto \sqrt{0.5 + \left(\color{blue}{\left(-0.5 \cdot \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}} + \frac{Om}{\ell \cdot \sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]

   6. Simplified 27.6%

      \[\leadsto \sqrt{0.5 + \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]
      Step-by-step derivation

      [Start] 27.5%	\[ \sqrt{0.5 + \left(\left(-0.5 \cdot \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}} + \frac{Om}{\ell \cdot \sin ky}\right) \cdot -0.5\right) \cdot 0.5} \]
      fma-def [=>] 27.5%	\[ \sqrt{0.5 + \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{Om \cdot {kx}^{2}}{\ell \cdot {\sin ky}^{3}}, \frac{Om}{\ell \cdot \sin ky}\right)} \cdot -0.5\right) \cdot 0.5} \]
      unpow2 [=>] 27.5%	\[ \sqrt{0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \color{blue}{\left(kx \cdot kx\right)}}{\ell \cdot {\sin ky}^{3}}, \frac{Om}{\ell \cdot \sin ky}\right) \cdot -0.5\right) \cdot 0.5} \]
      associate-/r* [=>] 27.6%	\[ \sqrt{0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \color{blue}{\frac{\frac{Om}{\ell}}{\sin ky}}\right) \cdot -0.5\right) \cdot 0.5} \]

   7. Applied egg-rr 40.0%

      \[\leadsto \color{blue}{{\left(0.5 + \mathsf{fma}\left(-0.5, \frac{\left(Om \cdot kx\right) \cdot kx}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot -0.25\right)}^{0.5}} \]
      Step-by-step derivation

      [Start] 27.6%	\[ \sqrt{0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot -0.5\right) \cdot 0.5} \]
      pow1/2 [=>] 43.2%	\[ \color{blue}{{\left(0.5 + \left(\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot -0.5\right) \cdot 0.5\right)}^{0.5}} \]
      associate-*l* [=>] 43.2%	\[ {\left(0.5 + \color{blue}{\mathsf{fma}\left(-0.5, \frac{Om \cdot \left(kx \cdot kx\right)}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot \left(-0.5 \cdot 0.5\right)}\right)}^{0.5} \]
      associate-*r* [=>] 40.0%	\[ {\left(0.5 + \mathsf{fma}\left(-0.5, \frac{\color{blue}{\left(Om \cdot kx\right) \cdot kx}}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot \left(-0.5 \cdot 0.5\right)\right)}^{0.5} \]
      metadata-eval [=>] 40.0%	\[ {\left(0.5 + \mathsf{fma}\left(-0.5, \frac{\left(Om \cdot kx\right) \cdot kx}{\ell \cdot {\sin ky}^{3}}, \frac{\frac{Om}{\ell}}{\sin ky}\right) \cdot \color{blue}{-0.25}\right)}^{0.5} \]

   8. Taylor expanded in ky around 0 51.8%

      \[\leadsto {\left(0.5 + \mathsf{fma}\left(-0.5, \frac{\left(Om \cdot kx\right) \cdot kx}{\ell \cdot {\sin ky}^{3}}, \color{blue}{-1 \cdot \left(\left(-0.027777777777777776 \cdot \frac{Om}{\ell} + 0.008333333333333333 \cdot \frac{Om}{\ell}\right) \cdot {ky}^{3}\right) + \left(-1 \cdot \left({ky}^{5} \cdot \left(0.001388888888888889 \cdot \frac{Om}{\ell} + \left(-0.0001984126984126984 \cdot \frac{Om}{\ell} + 0.16666666666666666 \cdot \left(-0.027777777777777776 \cdot \frac{Om}{\ell} + 0.008333333333333333 \cdot \frac{Om}{\ell}\right)\right)\right)\right) + \left(0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{\ell \cdot ky}\right)\right)}\right) \cdot -0.25\right)}^{0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{{\sin ky}^{3} \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{elif}\;kx \leq -2.1 \cdot 10^{-202}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}}\\ \mathbf{elif}\;kx \leq -4.8 \cdot 10^{-238}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, \frac{kx \cdot \left(kx \cdot Om\right)}{{\sin ky}^{3} \cdot \ell}, \left(0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{ky \cdot \ell}\right) - \left(-0.027777777777777776 \cdot \frac{Om}{\ell} + \frac{Om}{\ell} \cdot 0.008333333333333333\right) \cdot {ky}^{3}\right)\right)}^{0.5}\\ \mathbf{elif}\;kx \leq 6.5 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, \frac{kx \cdot \left(kx \cdot Om\right)}{{\sin ky}^{3} \cdot \ell}, \left(\left(0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{ky \cdot \ell}\right) - {ky}^{5} \cdot \left(\frac{Om}{\ell} \cdot 0.001388888888888889 + \left(\frac{Om}{\ell} \cdot -0.0001984126984126984 + \left(-0.027777777777777776 \cdot \frac{Om}{\ell} + \frac{Om}{\ell} \cdot 0.008333333333333333\right) \cdot 0.16666666666666666\right)\right)\right) - \left(-0.027777777777777776 \cdot \frac{Om}{\ell} + \frac{Om}{\ell} \cdot 0.008333333333333333\right) \cdot {ky}^{3}\right)\right)}^{0.5}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.5%
Cost	43600

\[\begin{array}{l} t_0 := {\sin ky}^{3}\\ t_1 := 0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{ky \cdot \ell}\\ t_2 := \frac{kx \cdot \left(kx \cdot Om\right)}{t_0 \cdot \ell}\\ t_3 := \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}}\\ t_4 := -0.027777777777777776 \cdot \frac{Om}{\ell} + \frac{Om}{\ell} \cdot 0.008333333333333333\\ t_5 := t_4 \cdot {ky}^{3}\\ \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{t_0 \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{elif}\;kx \leq -2.1 \cdot 10^{-202}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;kx \leq -4.8 \cdot 10^{-238}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, t_2, t_1 - t_5\right)\right)}^{0.5}\\ \mathbf{elif}\;kx \leq 6.5 \cdot 10^{-225}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, t_2, \left(t_1 - {ky}^{5} \cdot \left(\frac{Om}{\ell} \cdot 0.001388888888888889 + \left(\frac{Om}{\ell} \cdot -0.0001984126984126984 + t_4 \cdot 0.16666666666666666\right)\right)\right) - t_5\right)\right)}^{0.5}\\ \end{array} \]

Alternative 2
Accuracy	73.3%
Cost	35345

\[\begin{array}{l} t_0 := {\sin ky}^{3}\\ \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{t_0 \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{elif}\;kx \leq -5.5 \cdot 10^{-203} \lor \neg \left(kx \leq -4.8 \cdot 10^{-238}\right) \land kx \leq 6.5 \cdot 10^{-225}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, \frac{kx \cdot \left(kx \cdot Om\right)}{t_0 \cdot \ell}, \left(0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{ky \cdot \ell}\right) - \left(-0.027777777777777776 \cdot \frac{Om}{\ell} + \frac{Om}{\ell} \cdot 0.008333333333333333\right) \cdot {ky}^{3}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 3
Accuracy	33.7%
Cost	33616

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(kx \cdot Om\right) \cdot 0.16666666666666666 + \frac{Om}{kx}\right)\right)}\\ \mathbf{if}\;\sin kx \leq -5 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\sin kx \leq -4 \cdot 10^{-200}:\\ \;\;\;\;1\\ \mathbf{elif}\;\sin kx \leq -2 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\sin kx \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \left(0.041666666666666664 \cdot \frac{kx \cdot Om}{\ell} + 0.25 \cdot \frac{Om}{kx \cdot \ell}\right)}\\ \end{array} \]

Alternative 4
Accuracy	71.1%
Cost	33489

\[\begin{array}{l} t_0 := {\sin ky}^{3}\\ \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{t_0 \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{elif}\;kx \leq -2.6 \cdot 10^{-198} \lor \neg \left(kx \leq -3.8 \cdot 10^{-231}\right) \land kx \leq 6.5 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, \frac{kx \cdot \left(kx \cdot Om\right)}{t_0 \cdot \ell}, 0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{ky \cdot \ell}\right)\right)}^{0.5}\\ \end{array} \]

Alternative 5
Accuracy	68.8%
Cost	27985

\[\begin{array}{l} t_0 := {\sin ky}^{3}\\ \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{t_0 \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{elif}\;kx \leq -2.6 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin kx\right)}}\\ \mathbf{elif}\;kx \leq -1.2 \cdot 10^{-234} \lor \neg \left(kx \leq 1.4 \cdot 10^{-35}\right):\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \mathsf{fma}\left(-0.5, \frac{kx \cdot \left(kx \cdot Om\right)}{t_0 \cdot \ell}, 0.16666666666666666 \cdot \frac{Om \cdot ky}{\ell} + \frac{Om}{ky \cdot \ell}\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \frac{kx \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 6
Accuracy	69.5%
Cost	26700

\[\begin{array}{l} t_0 := Om \cdot -0.027777777777777776 + Om \cdot 0.008333333333333333\\ \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{{\sin ky}^{3} \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{elif}\;kx \leq -2.45 \cdot 10^{-208}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin kx\right)}}\\ \mathbf{elif}\;kx \leq -2.4 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{Om}{\sin kx}\right)\right)\right)}\\ \mathbf{elif}\;kx \leq 1.35 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \frac{kx \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(\left(kx \cdot Om\right) \cdot 0.16666666666666666 + \left(\frac{Om}{kx} - t_0 \cdot {kx}^{3}\right)\right) - {kx}^{5} \cdot \left(Om \cdot -0.0001984126984126984 + \left(0.16666666666666666 \cdot t_0 + Om \cdot 0.001388888888888889\right)\right)\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	70.1%
Cost	22600

\[\begin{array}{l} t_0 := Om \cdot -0.027777777777777776 + Om \cdot 0.008333333333333333\\ \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{{\sin ky}^{3} \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{elif}\;kx \leq 1.25 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(\left(kx \cdot Om\right) \cdot 0.16666666666666666 + \left(\frac{Om}{kx} - t_0 \cdot {kx}^{3}\right)\right) - {kx}^{5} \cdot \left(Om \cdot -0.0001984126984126984 + \left(0.16666666666666666 \cdot t_0 + Om \cdot 0.001388888888888889\right)\right)\right)\right)}\\ \end{array} \]

Alternative 8
Accuracy	70.9%
Cost	20553

\[\begin{array}{l} \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79} \lor \neg \left(kx \leq 1.35 \cdot 10^{+76}\right):\\ \;\;\;\;{\left(0.5 + \left(-0.5 \cdot \frac{Om}{{\sin ky}^{3} \cdot \frac{\ell}{kx \cdot kx}}\right) \cdot -0.25\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin kx\right)}}\\ \end{array} \]

Alternative 9
Accuracy	59.2%
Cost	20496

\[\begin{array}{l} t_0 := {\left(0.5 + -0.25 \cdot \left(-0.5 \cdot \left(\frac{Om}{\ell} \cdot \frac{kx \cdot kx}{{ky}^{3}}\right)\right)\right)}^{0.5}\\ t_1 := \sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(kx \cdot Om\right) \cdot 0.16666666666666666 + \left(\frac{Om}{kx} - \left(Om \cdot -0.027777777777777776 + Om \cdot 0.008333333333333333\right) \cdot {kx}^{3}\right)\right)\right)}\\ \mathbf{if}\;kx \leq -2 \cdot 10^{+241}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;kx \leq -4.5 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;kx \leq 1.35 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	58.6%
Cost	15120

\[\begin{array}{l} t_0 := {\left(0.5 + -0.25 \cdot \left(-0.5 \cdot \left(\frac{Om}{\ell} \cdot \frac{kx \cdot kx}{{ky}^{3}}\right)\right)\right)}^{0.5}\\ t_1 := \sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(kx \cdot Om\right) \cdot 0.16666666666666666 + \left(\frac{Om}{kx} - \left(Om \cdot -0.027777777777777776 + Om \cdot 0.008333333333333333\right) \cdot {kx}^{3}\right)\right)\right)}\\ \mathbf{if}\;kx \leq -2 \cdot 10^{+241}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;kx \leq -5.8 \cdot 10^{+213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;kx \leq 3.3 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \frac{kx \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	57.6%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;kx \leq -1.02 \cdot 10^{+79}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \left(-0.5 \cdot \left(\frac{Om}{\ell} \cdot \frac{kx \cdot kx}{{ky}^{3}}\right)\right)\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \frac{kx \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 12
Accuracy	46.7%
Cost	13960

\[\begin{array}{l} \mathbf{if}\;kx \leq -1.4 \cdot 10^{+211}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(kx \cdot Om\right) \cdot 0.16666666666666666 + \frac{Om}{kx}\right)\right)}\\ \mathbf{elif}\;kx \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \frac{Om}{\sin ky \cdot \ell}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \frac{kx \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 13
Accuracy	35.9%
Cost	13704

\[\begin{array}{l} \mathbf{if}\;kx \leq -2.7 \cdot 10^{+210}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(kx \cdot Om\right) \cdot 0.16666666666666666 + \frac{Om}{kx}\right)\right)}\\ \mathbf{elif}\;kx \leq -1.15 \cdot 10^{+85}:\\ \;\;\;\;{\left(0.5 + -0.25 \cdot \frac{Om}{\sin ky \cdot \ell}\right)}^{0.5}\\ \mathbf{elif}\;kx \leq -3.45 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	33.7%
Cost	7636

\[\begin{array}{l} \mathbf{if}\;kx \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \left(\frac{0.5}{\ell} \cdot \left(\left(kx \cdot Om\right) \cdot 0.16666666666666666 + \frac{Om}{kx}\right)\right)}\\ \mathbf{elif}\;kx \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;kx \leq -3.2 \cdot 10^{-105}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;kx \leq -1.42 \cdot 10^{-201}:\\ \;\;\;\;1\\ \mathbf{elif}\;kx \leq -2.4 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{0.5 + \frac{Om}{kx} \cdot \frac{0.25}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Accuracy	32.2%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-54}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 16
Accuracy	22.6%
Cost	6464

\[\sqrt{0.5} \]

Reproduce

herbie shell --seed 2023255 
(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))))))))))