Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.1%

Time: 10.2s

Alternatives: 7

Speedup: 0.8×

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}\\ t_1 := {\sin kx\_m}^{2}\\ \mathbf{if}\;t\_0 \cdot \left(t\_1 + {\sin ky\_m}^{2}\right) \leq 10^{+19}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + {\left(\sqrt{1 + t\_0 \cdot \left(t\_1 + \mathsf{fma}\left(\cos \left(ky\_m \cdot 2\right), -0.5, 0.5\right)\right)}\right)}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky\_m \cdot \frac{\ell \cdot 2}{Om}} + 0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (let* ((t_0 (pow (/ (* 2.0 l) Om) 2.0)) (t_1 (pow (sin kx_m) 2.0)))
   (if (<= (* t_0 (+ t_1 (pow (sin ky_m) 2.0))) 1e+19)
     (sqrt
      (*
       0.5
       (+
        1.0
        (pow
         (sqrt (+ 1.0 (* t_0 (+ t_1 (fma (cos (* ky_m 2.0)) -0.5 0.5)))))
         -1.0))))
     (sqrt (+ (/ 0.5 (* (sin ky_m) (/ (* l 2.0) Om))) 0.5)))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double t_0 = pow(((2.0 * l) / Om), 2.0);
	double t_1 = pow(sin(kx_m), 2.0);
	double tmp;
	if ((t_0 * (t_1 + pow(sin(ky_m), 2.0))) <= 1e+19) {
		tmp = sqrt((0.5 * (1.0 + pow(sqrt((1.0 + (t_0 * (t_1 + fma(cos((ky_m * 2.0)), -0.5, 0.5))))), -1.0))));
	} else {
		tmp = sqrt(((0.5 / (sin(ky_m) * ((l * 2.0) / Om))) + 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	t_0 = Float64(Float64(2.0 * l) / Om) ^ 2.0
	t_1 = sin(kx_m) ^ 2.0
	tmp = 0.0
	if (Float64(t_0 * Float64(t_1 + (sin(ky_m) ^ 2.0))) <= 1e+19)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + (sqrt(Float64(1.0 + Float64(t_0 * Float64(t_1 + fma(cos(Float64(ky_m * 2.0)), -0.5, 0.5))))) ^ -1.0))));
	else
		tmp = sqrt(Float64(Float64(0.5 / Float64(sin(ky_m) * Float64(Float64(l * 2.0) / Om))) + 0.5));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[(t$95$1 + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+19], N[Sqrt[N[(0.5 * N[(1.0 + N[Power[N[Sqrt[N[(1.0 + N[(t$95$0 * N[(t$95$1 + N[(N[Cos[N[(ky$95$m * 2.0), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[(N[Sin[ky$95$m], $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1e19

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. count-2 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-cos.f64 N/A

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

      18. count-2 N/A

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

      19. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 99.7

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

   6. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval 99.7

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

   8. Applied rewrites 99.7%

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

   if 1e19 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 98.7%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

   10. Applied rewrites 76.9%

       \[\leadsto \sqrt{\color{blue}{\frac{0.5}{\sin ky \cdot \frac{\ell \cdot 2}{Om}} + 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 10^{+19}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + {\left(\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + \mathsf{fma}\left(\cos \left(ky \cdot 2\right), -0.5, 0.5\right)\right)}\right)}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky \cdot \frac{\ell \cdot 2}{Om}} + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := {\sin kx\_m}^{2}\\ \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left(t\_0 + {\sin ky\_m}^{2}\right) \leq 0.4:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(\frac{t\_0}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)\right)}^{-1}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{2}^{-1} \cdot \left(1 + {\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin ky\_m, \sin kx\_m\right)\right)}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (let* ((t_0 (pow (sin kx_m) 2.0)))
   (if (<= (* (pow (/ (* 2.0 l) Om) 2.0) (+ t_0 (pow (sin ky_m) 2.0))) 0.4)
     (sqrt
      (fma
       (sqrt (pow (fma (* (/ t_0 Om) (* (/ l Om) l)) 4.0 1.0) -1.0))
       0.5
       0.5))
     (sqrt
      (*
       (pow 2.0 -1.0)
       (+
        1.0
        (pow (* (* 2.0 (/ l Om)) (hypot (sin ky_m) (sin kx_m))) -1.0)))))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double t_0 = pow(sin(kx_m), 2.0);
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (t_0 + pow(sin(ky_m), 2.0))) <= 0.4) {
		tmp = sqrt(fma(sqrt(pow(fma(((t_0 / Om) * ((l / Om) * l)), 4.0, 1.0), -1.0)), 0.5, 0.5));
	} else {
		tmp = sqrt((pow(2.0, -1.0) * (1.0 + pow(((2.0 * (l / Om)) * hypot(sin(ky_m), sin(kx_m))), -1.0))));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	t_0 = sin(kx_m) ^ 2.0
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64(t_0 + (sin(ky_m) ^ 2.0))) <= 0.4)
		tmp = sqrt(fma(sqrt((fma(Float64(Float64(t_0 / Om) * Float64(Float64(l / Om) * l)), 4.0, 1.0) ^ -1.0)), 0.5, 0.5));
	else
		tmp = sqrt(Float64((2.0 ^ -1.0) * Float64(1.0 + (Float64(Float64(2.0 * Float64(l / Om)) * hypot(sin(ky_m), sin(kx_m))) ^ -1.0))));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$0 + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4], N[Sqrt[N[(N[Sqrt[N[Power[N[(N[(N[(t$95$0 / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(1.0 + N[Power[N[(N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + N[Sin[kx$95$m], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 98.8%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin kx}^{2}}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)}}, 0.5, 0.5\right)} \]

   if 0.40000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 98.7%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.4:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(\frac{{\sin kx}^{2}}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)\right)}^{-1}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{2}^{-1} \cdot \left(1 + {\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right)\right)}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := {\sin kx\_m}^{2}\\ \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left(t\_0 + {\sin ky\_m}^{2}\right) \leq 0.4:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(\frac{t\_0}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)\right)}^{-1}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky\_m \cdot \frac{\ell \cdot 2}{Om}} + 0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (let* ((t_0 (pow (sin kx_m) 2.0)))
   (if (<= (* (pow (/ (* 2.0 l) Om) 2.0) (+ t_0 (pow (sin ky_m) 2.0))) 0.4)
     (sqrt
      (fma
       (sqrt (pow (fma (* (/ t_0 Om) (* (/ l Om) l)) 4.0 1.0) -1.0))
       0.5
       0.5))
     (sqrt (+ (/ 0.5 (* (sin ky_m) (/ (* l 2.0) Om))) 0.5)))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double t_0 = pow(sin(kx_m), 2.0);
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (t_0 + pow(sin(ky_m), 2.0))) <= 0.4) {
		tmp = sqrt(fma(sqrt(pow(fma(((t_0 / Om) * ((l / Om) * l)), 4.0, 1.0), -1.0)), 0.5, 0.5));
	} else {
		tmp = sqrt(((0.5 / (sin(ky_m) * ((l * 2.0) / Om))) + 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	t_0 = sin(kx_m) ^ 2.0
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64(t_0 + (sin(ky_m) ^ 2.0))) <= 0.4)
		tmp = sqrt(fma(sqrt((fma(Float64(Float64(t_0 / Om) * Float64(Float64(l / Om) * l)), 4.0, 1.0) ^ -1.0)), 0.5, 0.5));
	else
		tmp = sqrt(Float64(Float64(0.5 / Float64(sin(ky_m) * Float64(Float64(l * 2.0) / Om))) + 0.5));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$0 + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4], N[Sqrt[N[(N[Sqrt[N[Power[N[(N[(N[(t$95$0 / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[(N[Sin[ky$95$m], $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 98.8%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin kx}^{2}}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)}}, 0.5, 0.5\right)} \]

   if 0.40000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 98.7%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

   10. Applied rewrites 76.3%

       \[\leadsto \sqrt{\color{blue}{\frac{0.5}{\sin ky \cdot \frac{\ell \cdot 2}{Om}} + 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.4:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(\frac{{\sin kx}^{2}}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)\right)}^{-1}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky \cdot \frac{\ell \cdot 2}{Om}} + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = sqrt(((2.0d0 ** (-1.0d0)) * (1.0d0 + (sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))))) ** (-1.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Final simplification 99.4%

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

Alternative 5: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 0.4:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky\_m \cdot \frac{\ell \cdot 2}{Om}} + 0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l) Om) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      0.4)
   (sqrt 1.0)
   (sqrt (+ (/ 0.5 (* (sin ky_m) (/ (* l 2.0) Om))) 0.5))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 0.4) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(((0.5 / (sin(ky_m) * ((l * 2.0) / Om))) + 0.5));
	}
	return tmp;
}

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))) <= 0.4d0) then
        tmp = sqrt(1.0d0)
    else
        tmp = sqrt(((0.5d0 / (sin(ky_m) * ((l * 2.0d0) / om))) + 0.5d0))
    end if
    code = tmp
end function

Java

ky_m = Math.abs(ky);
kx_m = Math.abs(kx);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))) <= 0.4) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(((0.5 / (Math.sin(ky_m) * ((l * 2.0) / Om))) + 0.5));
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))) <= 0.4:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(((0.5 / (math.sin(ky_m) * ((l * 2.0) / Om))) + 0.5))
	return tmp

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 0.4)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(Float64(Float64(0.5 / Float64(sin(ky_m) * Float64(Float64(l * 2.0) / Om))) + 0.5));
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
kx_m = abs(kx);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (((((2.0 * l) / Om) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 0.4)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(((0.5 / (sin(ky_m) * ((l * 2.0) / Om))) + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[(N[Sin[ky$95$m], $MachinePrecision] * N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 98.3%

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

   if 0.40000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 98.7%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

   10. Applied rewrites 76.3%

       \[\leadsto \sqrt{\color{blue}{\frac{0.5}{\sin ky \cdot \frac{\ell \cdot 2}{Om}} + 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right) \leq 0.4:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l) Om) 2.0)
       (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))
      0.4)
   (sqrt 1.0)
   (sqrt 0.5)))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))) <= 0.4) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

ky_m = abs(ky)
kx_m = abs(kx)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))) <= 0.4d0) then
        tmp = sqrt(1.0d0)
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

ky_m = Math.abs(ky);
kx_m = Math.abs(kx);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if ((Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))) <= 0.4) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
kx_m = math.fabs(kx)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))) <= 0.4:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

ky_m = abs(ky)
kx_m = abs(kx)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 0.4)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
kx_m = abs(kx);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (((((2.0 * l) / Om) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))) <= 0.4)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 98.3%

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

   if 0.40000000000000002 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 98.7%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 97.9%

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

Alternative 7: 57.4% accurate, 52.8× speedup

Math

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

FPCore

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

C

ky_m = fabs(ky);
kx_m = fabs(kx);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return sqrt(0.5);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in l around inf

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

5. Applied rewrites 56.5%

   \[\leadsto \sqrt{\color{blue}{0.5}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(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))))))))))