Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 98.1%

Time: 14.8s

Alternatives: 8

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: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{\ell}{Om} \cdot 2\\ \mathbf{if}\;{\sin ky\_m}^{2} \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(t\_0, \left(kx\_m \cdot kx\_m + ky\_m \cdot ky\_m\right) \cdot t\_0, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \left(1 - \left(\cos \left(kx\_m \cdot 2\right) + \cos \left(ky\_m + ky\_m\right)\right) \cdot 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \end{array} \end{array} \]

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 (* (/ l Om) 2.0)))
   (if (<= (pow (sin ky_m) 2.0) 2e-38)
     (sqrt
      (*
       (+
        (/ 1.0 (sqrt (fma t_0 (* (+ (* kx_m kx_m) (* ky_m ky_m)) t_0) 1.0)))
        1.0)
       0.5))
     (sqrt
      (*
       (+
        (/
         1.0
         (sqrt
          (fma
           (* 4.0 (/ l Om))
           (*
            (- 1.0 (* (+ (cos (* kx_m 2.0)) (cos (+ ky_m ky_m))) 0.5))
            (/ l Om))
           1.0)))
        1.0)
       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 = (l / Om) * 2.0;
	double tmp;
	if (pow(sin(ky_m), 2.0) <= 2e-38) {
		tmp = sqrt((((1.0 / sqrt(fma(t_0, (((kx_m * kx_m) + (ky_m * ky_m)) * t_0), 1.0))) + 1.0) * 0.5));
	} else {
		tmp = sqrt((((1.0 / sqrt(fma((4.0 * (l / Om)), ((1.0 - ((cos((kx_m * 2.0)) + cos((ky_m + ky_m))) * 0.5)) * (l / Om)), 1.0))) + 1.0) * 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(l / Om) * 2.0)
	tmp = 0.0
	if ((sin(ky_m) ^ 2.0) <= 2e-38)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(t_0, Float64(Float64(Float64(kx_m * kx_m) + Float64(ky_m * ky_m)) * t_0), 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(4.0 * Float64(l / Om)), Float64(Float64(1.0 - Float64(Float64(cos(Float64(kx_m * 2.0)) + cos(Float64(ky_m + ky_m))) * 0.5)) * Float64(l / Om)), 1.0))) + 1.0) * 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[(N[(l / Om), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision], 2e-38], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(t$95$0 * N[(N[(N[(kx$95$m * kx$95$m), $MachinePrecision] + N[(ky$95$m * ky$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(4.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(N[Cos[N[(kx$95$m * 2.0), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 1.9999999999999999e-38

   1. Initial program 96.9%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.7

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

   10. Applied rewrites 98.7%

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

       1. lift-/.f64 N/A

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

       2. metadata-eval 98.7

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

   12. Applied rewrites 98.7%

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

   if 1.9999999999999999e-38 < (pow.f64 (sin.f64 ky) #s(literal 2 binary64))

   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%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in ky around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. metadata-eval N/A

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

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

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

      14. lower-cos.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   9. Applied rewrites 100.0%

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

   11. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 2 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot 2, \left(kx \cdot kx + ky \cdot ky\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right), 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \left(1 - \left(\cos \left(kx \cdot 2\right) + \cos \left(ky + ky\right)\right) \cdot 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{\ell}{Om} \cdot 2\\ \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky\_m}^{2} + {\sin kx\_m}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.0001:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky\_m \cdot \ell}, -0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(kx\_m + kx\_m\right), 0.5\right) \cdot t\_0, t\_0, 1\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

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 (* (/ l Om) 2.0)))
   (if (<=
        (/
         1.0
         (sqrt
          (+
           (*
            (+ (pow (sin ky_m) 2.0) (pow (sin kx_m) 2.0))
            (pow (/ (* l 2.0) Om) 2.0))
           1.0)))
        0.0001)
     (sqrt (fma (/ Om (* (sin ky_m) l)) -0.25 0.5))
     (sqrt
      (fma
       (sqrt (/ 1.0 (fma (* (fma -0.5 (cos (+ kx_m kx_m)) 0.5) t_0) t_0 1.0)))
       0.5
       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 = (l / Om) * 2.0;
	double tmp;
	if ((1.0 / sqrt((((pow(sin(ky_m), 2.0) + pow(sin(kx_m), 2.0)) * pow(((l * 2.0) / Om), 2.0)) + 1.0))) <= 0.0001) {
		tmp = sqrt(fma((Om / (sin(ky_m) * l)), -0.25, 0.5));
	} else {
		tmp = sqrt(fma(sqrt((1.0 / fma((fma(-0.5, cos((kx_m + kx_m)), 0.5) * t_0), t_0, 1.0))), 0.5, 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(l / Om) * 2.0)
	tmp = 0.0
	if (Float64(1.0 / sqrt(Float64(Float64(Float64((sin(ky_m) ^ 2.0) + (sin(kx_m) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) + 1.0))) <= 0.0001)
		tmp = sqrt(fma(Float64(Om / Float64(sin(ky_m) * l)), -0.25, 0.5));
	else
		tmp = sqrt(fma(sqrt(Float64(1.0 / fma(Float64(fma(-0.5, cos(Float64(kx_m + kx_m)), 0.5) * t_0), t_0, 1.0))), 0.5, 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[(N[(l / Om), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[N[(1.0 / N[Sqrt[N[(N[(N[(N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0001], N[Sqrt[N[(N[(Om / N[(N[Sin[ky$95$m], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(N[(N[(-0.5 * N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.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))))))) < 1.00000000000000005e-4

   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 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-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 90.0%

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

   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 99.4%

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

   if 1.00000000000000005e-4 < (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.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 97.3%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 96.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(kx + kx\right), 0.5\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right), \frac{\ell}{Om} \cdot 2, 1\right)}}, 0.5, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.0001:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, -0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(kx + kx\right), 0.5\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right), \frac{\ell}{Om} \cdot 2, 1\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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))))))))))