Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 98.1%

Time: 17.4s

Alternatives: 6

Speedup: 1.0×

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: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 10^{+60}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left(1 + -0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right) \cdot \frac{l\_m}{Om\_m}}{\frac{Om\_m}{l\_m \cdot 4}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 1e+60)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (/
         (*
          (+ 1.0 (* -0.5 (+ (cos (* 2.0 kx)) (cos (* 2.0 ky)))))
          (/ l_m Om_m))
         (/ Om_m (* l_m 4.0))))))))
   (sqrt 0.5)))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 1e+60) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((1.0 + (-0.5 * (cos((2.0 * kx)) + cos((2.0 * ky))))) * (l_m / Om_m)) / (Om_m / (l_m * 4.0))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l_m) / om_m) <= 1d+60) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((1.0d0 + ((-0.5d0) * (cos((2.0d0 * kx)) + cos((2.0d0 * ky))))) * (l_m / om_m)) / (om_m / (l_m * 4.0d0))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 1e+60) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((1.0 + (-0.5 * (Math.cos((2.0 * kx)) + Math.cos((2.0 * ky))))) * (l_m / Om_m)) / (Om_m / (l_m * 4.0))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 1e+60:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((1.0 + (-0.5 * (math.cos((2.0 * kx)) + math.cos((2.0 * ky))))) * (l_m / Om_m)) / (Om_m / (l_m * 4.0))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 1e+60)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(-0.5 * Float64(cos(Float64(2.0 * kx)) + cos(Float64(2.0 * ky))))) * Float64(l_m / Om_m)) / Float64(Om_m / Float64(l_m * 4.0))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 1e+60)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((1.0 + (-0.5 * (cos((2.0 * kx)) + cos((2.0 * ky))))) * (l_m / Om_m)) / (Om_m / (l_m * 4.0))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 1e+60], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(1.0 + N[(-0.5 * N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m / N[(l$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 9.9999999999999995e59

   1. Initial program 98.6%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 97.7%

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

   6. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}{\frac{Om}{\frac{\ell \cdot 4}{Om}}}}}\right)}^{0.25}\right)}^{2}} \]

   8. Applied egg-rr 92.4%

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

      1. +-commutative N/A

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

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

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

   10. Applied egg-rr 94.2%

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

   if 9.9999999999999995e59 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 97.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \]
   4. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 98.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 98.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+60}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left(1 + -0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right) \cdot \frac{\ell}{Om}}{\frac{Om}{\ell \cdot 4}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l_m) Om_m) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

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

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    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_m) / om_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

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

Python

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

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, 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_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, 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$95$m), $MachinePrecision] / Om$95$m), $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]

Derivation

1. Initial program 98.4%

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

Alternative 3: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \sqrt{0.5 + \frac{0.5}{\sqrt{1 + l\_m \cdot \left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{\frac{l\_m \cdot 4}{Om\_m}}{Om\_m}\right)}}} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (/
    0.5
    (sqrt
     (+
      1.0
      (*
       l_m
       (*
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))
        (/ (/ (* l_m 4.0) Om_m) Om_m)))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	return sqrt((0.5 + (0.5 / sqrt((1.0 + (l_m * ((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * (((l_m * 4.0) / Om_m) / Om_m))))))));
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (l_m * (((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)) * (((l_m * 4.0d0) / om_m) / om_m))))))))
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (l_m * ((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)) * (((l_m * 4.0) / Om_m) / Om_m))))))));
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	return math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (l_m * ((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)) * (((l_m * 4.0) / Om_m) / Om_m))))))))

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(l_m * Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * Float64(Float64(Float64(l_m * 4.0) / Om_m) / Om_m))))))))
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp = code(l_m, Om_m, kx, ky)
	tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (l_m * (((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (((l_m * 4.0) / Om_m) / Om_m))))))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(l$95$m * N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

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

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt1-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

3. Simplified 98.1%

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

Alternative 4: 95.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 10^{+60}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{l\_m \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)}{\frac{Om\_m}{\frac{l\_m \cdot 4}{Om\_m}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 1e+60)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (/
         (* l_m (+ 0.5 (* -0.5 (cos (* 2.0 ky)))))
         (/ Om_m (/ (* l_m 4.0) Om_m))))))))
   (sqrt 0.5)))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 1e+60) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((l_m * (0.5 + (-0.5 * cos((2.0 * ky))))) / (Om_m / ((l_m * 4.0) / Om_m))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l_m) / om_m) <= 1d+60) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((l_m * (0.5d0 + ((-0.5d0) * cos((2.0d0 * ky))))) / (om_m / ((l_m * 4.0d0) / om_m))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 1e+60) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((l_m * (0.5 + (-0.5 * Math.cos((2.0 * ky))))) / (Om_m / ((l_m * 4.0) / Om_m))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 1e+60:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((l_m * (0.5 + (-0.5 * math.cos((2.0 * ky))))) / (Om_m / ((l_m * 4.0) / Om_m))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 1e+60)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(l_m * Float64(0.5 + Float64(-0.5 * cos(Float64(2.0 * ky))))) / Float64(Om_m / Float64(Float64(l_m * 4.0) / Om_m))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 1e+60)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((l_m * (0.5 + (-0.5 * cos((2.0 * ky))))) / (Om_m / ((l_m * 4.0) / Om_m))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 1e+60], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(l$95$m * N[(0.5 + N[(-0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m / N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 9.9999999999999995e59

   1. Initial program 98.6%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 97.7%

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

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

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

   6. Applied egg-rr 92.4%

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

   7. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 74.6%

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

   9. Simplified 74.6%

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

   10. Taylor expanded in kx around 0

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

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

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

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

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

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

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

       4. metadata-eval N/A

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

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

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

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

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

       7. *-lowering-*.f64 82.8%

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

   12. Simplified 82.8%

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

   if 9.9999999999999995e59 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 97.8%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \]
   4. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 98.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 98.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+60}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)}{\frac{Om}{\frac{\ell \cdot 4}{Om}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 4 \cdot 10^{+20}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 4e+20) 1.0 (sqrt 0.5)))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 4e+20) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((2.0d0 * l_m) / om_m) <= 4d+20) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 4e+20) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 4e+20:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 4e+20)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 4e+20)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 4e+20], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 4e20

   1. Initial program 98.5%

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

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

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

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

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

      6. metadata-eval N/A

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

      7. associate-*l/ N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

   3. Simplified 98.0%

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

   5. Taylor expanded in l around 0

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

   7. Simplified 71.4%

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

   if 4e20 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 98.2%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}}} \]
   4. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 95.8%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   5. Simplified 95.8%

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

Alternative 6: 62.4% accurate, 722.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky) :precision binary64 1.0)

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	return 1.0;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	return 1.0;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	return 1.0

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	return 1.0
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp = code(l_m, Om_m, kx, ky)
	tmp = 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := 1.0

Derivation

1. Initial program 98.4%

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

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

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt1-in N/A

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

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

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

   6. metadata-eval N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

3. Simplified 98.1%

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

5. Taylor expanded in l around 0

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

7. Simplified 60.9%

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

Reproduce

herbie shell --seed 2024139 
(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))))))))))