Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.9%

Time: 13.7s

Alternatives: 7

Speedup: 5.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ kx_m = \left|kx\right| \\ Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := 4 \cdot \frac{l\_m}{Om\_m}\\ \mathbf{if}\;\frac{l\_m \cdot 2}{Om\_m} \leq 500000000000:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(t\_0, \left(\left(\left(0.5 - \cos \left(kx\_m + kx\_m\right) \cdot 0.5\right) - \cos \left(ky\_m + ky\_m\right) \cdot 0.5\right) + 0.5\right) \cdot \frac{l\_m}{Om\_m}, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left(\frac{l\_m}{Om\_m} \cdot ky\_m\right) \cdot ky\_m, t\_0, 1\right)}} + 0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
kx_m = (fabs.f64 kx)
Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ l_m Om_m))))
   (if (<= (/ (* l_m 2.0) Om_m) 500000000000.0)
     (sqrt
      (*
       (+
        (/
         1.0
         (sqrt
          (fma
           t_0
           (*
            (+
             (-
              (- 0.5 (* (cos (+ kx_m kx_m)) 0.5))
              (* (cos (+ ky_m ky_m)) 0.5))
             0.5)
            (/ l_m Om_m))
           1.0)))
        1.0)
       (/ 1.0 2.0)))
     (sqrt
      (+ (/ 0.5 (sqrt (fma (* (* (/ l_m Om_m) ky_m) ky_m) t_0 1.0))) 0.5)))))

C

ky_m = fabs(ky);
kx_m = fabs(kx);
Om_m = fabs(Om);
l_m = fabs(l);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = 4.0 * (l_m / Om_m);
	double tmp;
	if (((l_m * 2.0) / Om_m) <= 500000000000.0) {
		tmp = sqrt((((1.0 / sqrt(fma(t_0, ((((0.5 - (cos((kx_m + kx_m)) * 0.5)) - (cos((ky_m + ky_m)) * 0.5)) + 0.5) * (l_m / Om_m)), 1.0))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = sqrt(((0.5 / sqrt(fma((((l_m / Om_m) * ky_m) * ky_m), t_0, 1.0))) + 0.5));
	}
	return tmp;
}

Julia

ky_m = abs(ky)
kx_m = abs(kx)
Om_m = abs(Om)
l_m = abs(l)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(4.0 * Float64(l_m / Om_m))
	tmp = 0.0
	if (Float64(Float64(l_m * 2.0) / Om_m) <= 500000000000.0)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(t_0, Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(kx_m + kx_m)) * 0.5)) - Float64(cos(Float64(ky_m + ky_m)) * 0.5)) + 0.5) * Float64(l_m / Om_m)), 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(Float64(Float64(0.5 / sqrt(fma(Float64(Float64(Float64(l_m / Om_m) * ky_m) * ky_m), t_0, 1.0))) + 0.5));
	end
	return tmp
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 500000000000.0], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(t$95$0 * N[(N[(N[(N[(0.5 - N[(N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[(N[(N[(l$95$m / Om$95$m), $MachinePrecision] * ky$95$m), $MachinePrecision] * ky$95$m), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 5e11

   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%

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

   if 5e11 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 96.6%

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in ky around 0

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

      1. associate-/l* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 96.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

   12. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\ell \cdot 2}{Om} \leq 500000000000:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \left(\left(\left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right) - \cos \left(ky + ky\right) \cdot 0.5\right) + 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot ky\right) \cdot ky, 4 \cdot \frac{\ell}{Om}, 1\right)}} + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Reproduce

herbie shell --seed 2024229 
(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))))))))))