Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 99.1%

Time: 16.1s

Alternatives: 9

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: 98.5% 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, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 5000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot 4, \frac{l\_m}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right) + \cos \left(ky \cdot -2\right), 1\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(2 \cdot \frac{l\_m}{Om}\right)}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (l_m Om kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      5000000000.0)
   (sqrt
    (*
     0.5
     (+
      1.0
      (/
       1.0
       (sqrt
        (fma
         (* (/ l_m Om) 4.0)
         (* (/ l_m Om) (fma -0.5 (+ (cos (* kx -2.0)) (cos (* ky -2.0))) 1.0))
         1.0))))))
   (sqrt
    (*
     (/ 1.0 2.0)
     (+ 1.0 (/ 1.0 (* (hypot (sin kx) (sin ky)) (* 2.0 (/ l_m Om)))))))))

C

l_m = fabs(l);
double code(double l_m, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 5000000000.0) {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(((l_m / Om) * 4.0), ((l_m / Om) * fma(-0.5, (cos((kx * -2.0)) + cos((ky * -2.0))), 1.0)), 1.0))))));
	} else {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (hypot(sin(kx), sin(ky)) * (2.0 * (l_m / Om)))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(l_m, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 5000000000.0)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(Float64(Float64(l_m / Om) * 4.0), Float64(Float64(l_m / Om) * fma(-0.5, Float64(cos(Float64(kx * -2.0)) + cos(Float64(ky * -2.0))), 1.0)), 1.0))))));
	else
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) * Float64(2.0 * Float64(l_m / Om)))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om_, kx_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $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], 5000000000.0], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(N[(N[(l$95$m / Om), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(-0.5 * N[(N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(2.0 * N[(l$95$m / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)))) < 5e9

   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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.3

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

   7. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 97.3

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

   9. Applied rewrites 97.3%

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

   10. Taylor expanded in kx around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-out N/A

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

       3. metadata-eval N/A

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

       4. distribute-lft-neg-in N/A

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

       5. cos-neg N/A

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

       6. metadata-eval N/A

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

       7. distribute-lft-neg-in N/A

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

       8. cos-neg N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-cos.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

   if 5e9 < (*.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 98.2%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 5000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot 4, \frac{\ell}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right) + \cos \left(ky \cdot -2\right), 1\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 5000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot 4, \frac{l\_m}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right) + \cos \left(ky \cdot -2\right), 1\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{l\_m \cdot \sin ky}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (l_m Om kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      5000000000.0)
   (sqrt
    (*
     0.5
     (+
      1.0
      (/
       1.0
       (sqrt
        (fma
         (* (/ l_m Om) 4.0)
         (* (/ l_m Om) (fma -0.5 (+ (cos (* kx -2.0)) (cos (* ky -2.0))) 1.0))
         1.0))))))
   (sqrt (fma 0.25 (/ Om (* l_m (sin ky))) 0.5))))

C

l_m = fabs(l);
double code(double l_m, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 5000000000.0) {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(((l_m / Om) * 4.0), ((l_m / Om) * fma(-0.5, (cos((kx * -2.0)) + cos((ky * -2.0))), 1.0)), 1.0))))));
	} else {
		tmp = sqrt(fma(0.25, (Om / (l_m * sin(ky))), 0.5));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(l_m, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 5000000000.0)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(Float64(Float64(l_m / Om) * 4.0), Float64(Float64(l_m / Om) * fma(-0.5, Float64(cos(Float64(kx * -2.0)) + cos(Float64(ky * -2.0))), 1.0)), 1.0))))));
	else
		tmp = sqrt(fma(0.25, Float64(Om / Float64(l_m * sin(ky))), 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om_, kx_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $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], 5000000000.0], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(N[(N[(l$95$m / Om), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(-0.5 * N[(N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.25 * N[(Om / N[(l$95$m * N[Sin[ky], $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)))) < 5e9

   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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.3

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

   7. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 97.3

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

   9. Applied rewrites 97.3%

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

   10. Taylor expanded in kx around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-out N/A

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

       3. metadata-eval N/A

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

       4. distribute-lft-neg-in N/A

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

       5. cos-neg N/A

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

       6. metadata-eval N/A

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

       7. distribute-lft-neg-in N/A

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

       8. cos-neg N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-+.f64 N/A

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

       11. lower-cos.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower-cos.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

   if 5e9 < (*.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 98.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 82.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{Om}{\ell \cdot \sin ky}}, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 5000000000:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot 4, \frac{l\_m}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{l\_m \cdot \sin ky}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (l_m Om kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      5000000000.0)
   (sqrt
    (*
     0.5
     (+
      1.0
      (/
       1.0
       (sqrt
        (fma
         (* (/ l_m Om) 4.0)
         (* (/ l_m Om) (fma -0.5 (cos (* ky -2.0)) 0.5))
         1.0))))))
   (sqrt (fma 0.25 (/ Om (* l_m (sin ky))) 0.5))))

C

l_m = fabs(l);
double code(double l_m, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 5000000000.0) {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(((l_m / Om) * 4.0), ((l_m / Om) * fma(-0.5, cos((ky * -2.0)), 0.5)), 1.0))))));
	} else {
		tmp = sqrt(fma(0.25, (Om / (l_m * sin(ky))), 0.5));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(l_m, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 5000000000.0)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(Float64(Float64(l_m / Om) * 4.0), Float64(Float64(l_m / Om) * fma(-0.5, cos(Float64(ky * -2.0)), 0.5)), 1.0))))));
	else
		tmp = sqrt(fma(0.25, Float64(Om / Float64(l_m * sin(ky))), 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om_, kx_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $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], 5000000000.0], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(N[(N[(l$95$m / Om), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.25 * N[(Om / N[(l$95$m * N[Sin[ky], $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)))) < 5e9

   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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.3

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

   7. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 97.3

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

   9. Applied rewrites 97.3%

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

   if 5e9 < (*.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 98.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 82.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{Om}{\ell \cdot \sin ky}}, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.5:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{l\_m}{Om} \cdot 4, \frac{l\_m}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(kx \cdot -2\right), 0.5\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{l\_m \cdot \sin ky}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (l_m Om kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      0.5)
   (sqrt
    (*
     0.5
     (+
      1.0
      (/
       1.0
       (sqrt
        (fma
         (* (/ l_m Om) 4.0)
         (* (/ l_m Om) (fma -0.5 (cos (* kx -2.0)) 0.5))
         1.0))))))
   (sqrt (fma 0.25 (/ Om (* l_m (sin ky))) 0.5))))

C

l_m = fabs(l);
double code(double l_m, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 0.5) {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(((l_m / Om) * 4.0), ((l_m / Om) * fma(-0.5, cos((kx * -2.0)), 0.5)), 1.0))))));
	} else {
		tmp = sqrt(fma(0.25, (Om / (l_m * sin(ky))), 0.5));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(l_m, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.5)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(Float64(Float64(l_m / Om) * 4.0), Float64(Float64(l_m / Om) * fma(-0.5, cos(Float64(kx * -2.0)), 0.5)), 1.0))))));
	else
		tmp = sqrt(fma(0.25, Float64(Om / Float64(l_m * sin(ky))), 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om_, kx_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $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], 0.5], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(N[(N[(l$95$m / Om), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(l$95$m / Om), $MachinePrecision] * N[(-0.5 * N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.25 * N[(Om / N[(l$95$m * N[Sin[ky], $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)))) < 0.5

   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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. cos-neg N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.7

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

   7. Applied rewrites 97.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 97.7

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

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in ky around 0

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

       1. +-commutative N/A

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

       2. metadata-eval N/A

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

       3. distribute-lft-neg-in N/A

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

       4. cos-neg N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 98.9

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

   12. Applied rewrites 98.9%

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

   if 0.5 < (*.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 98.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 82.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{Om}{\ell \cdot \sin ky}}, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{l\_m \cdot \sin ky}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (l_m Om kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      0.5)
   1.0
   (sqrt (fma 0.25 (/ Om (* l_m (sin ky))) 0.5))))

C

l_m = fabs(l);
double code(double l_m, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 0.5) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma(0.25, (Om / (l_m * sin(ky))), 0.5));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(l_m, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.5)
		tmp = 1.0;
	else
		tmp = sqrt(fma(0.25, Float64(Om / Float64(l_m * sin(ky))), 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om_, kx_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $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], 0.5], 1.0, N[Sqrt[N[(0.25 * N[(Om / N[(l$95$m * N[Sin[ky], $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)))) < 0.5

   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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 97.7%

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

   if 0.5 < (*.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 98.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 82.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{Om}{\ell \cdot \sin ky}}, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

Alternative 7: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{l\_m \cdot ky}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (l_m Om kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      0.5)
   1.0
   (sqrt (fma 0.25 (/ Om (* l_m ky)) 0.5))))

C

l_m = fabs(l);
double code(double l_m, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 0.5) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma(0.25, (Om / (l_m * ky)), 0.5));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(l_m, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.5)
		tmp = 1.0;
	else
		tmp = sqrt(fma(0.25, Float64(Om / Float64(l_m * ky)), 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om_, kx_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $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], 0.5], 1.0, N[Sqrt[N[(0.25 * N[(Om / N[(l$95$m * ky), $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)))) < 0.5

   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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 97.7%

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

   if 0.5 < (*.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 98.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in l around inf

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

   8. Applied rewrites 82.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \color{blue}{\frac{Om}{\ell \cdot \sin ky}}, 0.5\right)} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{4}, \frac{Om}{ky \cdot \ell}, \frac{1}{2}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.4%

       \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \frac{Om}{ky \cdot \ell}, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\ell \cdot ky}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 3.75:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (l_m Om kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      3.75)
   1.0
   (sqrt 0.5)))

C

l_m = fabs(l);
double code(double l_m, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l_m) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 3.75) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
real(8) function code(l_m, om, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((((2.0d0 * l_m) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))) <= 3.75d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double l_m, double Om, double kx, double ky) {
	double tmp;
	if ((Math.pow(((2.0 * l_m) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))) <= 3.75) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(l_m, Om, kx, ky):
	tmp = 0
	if (math.pow(((2.0 * l_m) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))) <= 3.75:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
function code(l_m, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l_m) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 3.75)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
function tmp_2 = code(l_m, Om, kx, ky)
	tmp = 0.0;
	if (((((2.0 * l_m) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 3.75)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om_, kx_, ky_] := If[LessEqual[N[(N[Power[N[(N[(2.0 * l$95$m), $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], 3.75], 1.0, 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)))) < 3.75

   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

   4. Applied rewrites 80.7%

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

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 97.7%

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

   if 3.75 < (*.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 98.2%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 98.1%

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

Alternative 9: 62.1% accurate, 581.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 79.4%

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

5. Taylor expanded in l around 0

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

7. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2024225 
(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))))))))))