Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.1% → 99.6%

Time: 14.9s

Alternatives: 9

Speedup: 2.0×

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: 98.1% 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: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{Om}{\sin ky\_m}\\ \sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{t\_0}, \frac{\ell}{t\_0}, 1\right)}}, 0.5\right)} \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
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 (/ Om (sin ky_m))))
   (sqrt (fma 0.5 (sqrt (/ 1.0 (fma (/ (* l 4.0) t_0) (/ l t_0) 1.0))) 0.5))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
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 = Om / sin(ky_m);
	return sqrt(fma(0.5, sqrt((1.0 / fma(((l * 4.0) / t_0), (l / t_0), 1.0))), 0.5));
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	t_0 = Float64(Om / sin(ky_m))
	return sqrt(fma(0.5, sqrt(Float64(1.0 / fma(Float64(Float64(l * 4.0) / t_0), Float64(l / t_0), 1.0))), 0.5))
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $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[(Om / N[Sin[ky$95$m], $MachinePrecision]), $MachinePrecision]}, N[Sqrt[N[(0.5 * N[Sqrt[N[(1.0 / N[(N[(N[(l * 4.0), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(l / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

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 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. accelerator-lowering-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. Simplified 80.4%

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

   1. pow2 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. times-frac N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

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

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

   11. /-lowering-/.f64 N/A

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

   12. sin-lowering-sin.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. sin-lowering-sin.f64 96.2

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

7. Applied egg-rr 96.2%

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

Alternative 2: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;{\sin ky\_m}^{2} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{\frac{Om}{\sin ky\_m}}, \frac{\ell \cdot ky\_m}{Om}, 1\right)}}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \frac{\ell \cdot \mathsf{fma}\left(-0.5, \cos \left(ky\_m \cdot -2\right), 0.5\right)}{Om}, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= (pow (sin ky_m) 2.0) 2e-34)
   (sqrt
    (fma
     0.5
     (sqrt (/ 1.0 (fma (/ (* l 4.0) (/ Om (sin ky_m))) (/ (* l ky_m) Om) 1.0)))
     0.5))
   (sqrt
    (*
     0.5
     (+
      1.0
      (/
       1.0
       (sqrt
        (fma
         (* 4.0 (/ l Om))
         (/ (* l (fma -0.5 (cos (* ky_m -2.0)) 0.5)) Om)
         1.0))))))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (pow(sin(ky_m), 2.0) <= 2e-34) {
		tmp = sqrt(fma(0.5, sqrt((1.0 / fma(((l * 4.0) / (Om / sin(ky_m))), ((l * ky_m) / Om), 1.0))), 0.5));
	} else {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt(fma((4.0 * (l / Om)), ((l * fma(-0.5, cos((ky_m * -2.0)), 0.5)) / Om), 1.0))))));
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if ((sin(ky_m) ^ 2.0) <= 2e-34)
		tmp = sqrt(fma(0.5, sqrt(Float64(1.0 / fma(Float64(Float64(l * 4.0) / Float64(Om / sin(ky_m))), Float64(Float64(l * ky_m) / Om), 1.0))), 0.5));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(Float64(4.0 * Float64(l / Om)), Float64(Float64(l * fma(-0.5, cos(Float64(ky_m * -2.0)), 0.5)) / Om), 1.0))))));
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision], 2e-34], N[Sqrt[N[(0.5 * N[Sqrt[N[(1.0 / N[(N[(N[(l * 4.0), $MachinePrecision] / N[(Om / N[Sin[ky$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * ky$95$m), $MachinePrecision] / Om), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(N[(4.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(-0.5 * N[Cos[N[(ky$95$m * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 1.99999999999999986e-34

   1. Initial program 93.3%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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-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. accelerator-lowering-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. Simplified 67.9%

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

      1. pow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. times-frac N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. /-lowering-/.f64 N/A

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

      12. sin-lowering-sin.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sin-lowering-sin.f64 91.9

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in ky around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 91.9

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

   10. Simplified 91.9%

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

   if 1.99999999999999986e-34 < (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 egg-rr 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. Step-by-step derivation

      1. metadata-eval 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in kx around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. cos-neg N/A

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

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

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

      9. *-lowering-*.f64 100.0

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

   9. Simplified 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{\frac{Om}{\sin ky}}, \frac{\ell \cdot ky}{Om}, 1\right)}}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \frac{\ell \cdot \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right)}{Om}, 1\right)}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{Om}, \frac{\ell}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(kx\_m + kx\_m\right), 0.5\right), 1\right)}}, 0.5\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(-0.5, \cos \left(ky\_m \cdot -2\right), 0.5\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{\frac{Om}{\sin ky\_m}}, \frac{\ell \cdot ky\_m}{Om}, 1\right)}}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= l 2e-177)
   (sqrt
    (fma
     0.5
     (sqrt
      (/
       1.0
       (fma
        (/ (* l 4.0) Om)
        (* (/ l Om) (fma -0.5 (cos (+ kx_m kx_m)) 0.5))
        1.0)))
     0.5))
   (if (<= l 1.6e+128)
     (sqrt
      (+
       0.5
       (*
        0.5
        (sqrt
         (/
          1.0
          (fma
           4.0
           (/ (* (fma -0.5 (cos (* ky_m -2.0)) 0.5) (* l l)) (* Om Om))
           1.0))))))
     (sqrt
      (fma
       0.5
       (sqrt
        (/ 1.0 (fma (/ (* l 4.0) (/ Om (sin ky_m))) (/ (* l ky_m) Om) 1.0)))
       0.5)))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 2e-177) {
		tmp = sqrt(fma(0.5, sqrt((1.0 / fma(((l * 4.0) / Om), ((l / Om) * fma(-0.5, cos((kx_m + kx_m)), 0.5)), 1.0))), 0.5));
	} else if (l <= 1.6e+128) {
		tmp = sqrt((0.5 + (0.5 * sqrt((1.0 / fma(4.0, ((fma(-0.5, cos((ky_m * -2.0)), 0.5) * (l * l)) / (Om * Om)), 1.0))))));
	} else {
		tmp = sqrt(fma(0.5, sqrt((1.0 / fma(((l * 4.0) / (Om / sin(ky_m))), ((l * ky_m) / Om), 1.0))), 0.5));
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (l <= 2e-177)
		tmp = sqrt(fma(0.5, sqrt(Float64(1.0 / fma(Float64(Float64(l * 4.0) / Om), Float64(Float64(l / Om) * fma(-0.5, cos(Float64(kx_m + kx_m)), 0.5)), 1.0))), 0.5));
	elseif (l <= 1.6e+128)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * sqrt(Float64(1.0 / fma(4.0, Float64(Float64(fma(-0.5, cos(Float64(ky_m * -2.0)), 0.5) * Float64(l * l)) / Float64(Om * Om)), 1.0))))));
	else
		tmp = sqrt(fma(0.5, sqrt(Float64(1.0 / fma(Float64(Float64(l * 4.0) / Float64(Om / sin(ky_m))), Float64(Float64(l * ky_m) / Om), 1.0))), 0.5));
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[l, 2e-177], N[Sqrt[N[(0.5 * N[Sqrt[N[(1.0 / N[(N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(-0.5 * N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.6e+128], N[Sqrt[N[(0.5 + N[(0.5 * N[Sqrt[N[(1.0 / N[(4.0 * N[(N[(N[(-0.5 * N[Cos[N[(ky$95$m * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[Sqrt[N[(1.0 / N[(N[(N[(l * 4.0), $MachinePrecision] / N[(Om / N[Sin[ky$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * ky$95$m), $MachinePrecision] / Om), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.9999999999999999e-177

   1. Initial program 98.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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. accelerator-lowering-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. Simplified 79.9%

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

   7. Applied egg-rr 83.6%

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

   if 1.9999999999999999e-177 < l < 1.59999999999999993e128

   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 egg-rr 96.0%

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

   5. Taylor expanded in kx around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 89.2%

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

   if 1.59999999999999993e128 < l

   1. Initial program 88.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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. accelerator-lowering-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. Simplified 66.7%

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

      1. pow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. times-frac N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

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

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

      11. /-lowering-/.f64 N/A

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

      12. sin-lowering-sin.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sin-lowering-sin.f64 94.3

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

   7. Applied egg-rr 94.3%

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

   8. Taylor expanded in ky around 0

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 94.3

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

   10. Simplified 94.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{Om}, \frac{\ell}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(kx + kx\right), 0.5\right), 1\right)}}, 0.5\right)}\\ \mathbf{elif}\;\ell \leq 1.6 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{\frac{Om}{\sin ky}}, \frac{\ell \cdot ky}{Om}, 1\right)}}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{Om}, \frac{\ell}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(kx\_m + kx\_m\right), 0.5\right), 1\right)}}, 0.5\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(-0.5, \cos \left(ky\_m \cdot -2\right), 0.5\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= l 2e-177)
   (sqrt
    (fma
     0.5
     (sqrt
      (/
       1.0
       (fma
        (/ (* l 4.0) Om)
        (* (/ l Om) (fma -0.5 (cos (+ kx_m kx_m)) 0.5))
        1.0)))
     0.5))
   (if (<= l 3.4e+130)
     (sqrt
      (+
       0.5
       (*
        0.5
        (sqrt
         (/
          1.0
          (fma
           4.0
           (/ (* (fma -0.5 (cos (* ky_m -2.0)) 0.5) (* l l)) (* Om Om))
           1.0))))))
     (sqrt 0.5))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 2e-177) {
		tmp = sqrt(fma(0.5, sqrt((1.0 / fma(((l * 4.0) / Om), ((l / Om) * fma(-0.5, cos((kx_m + kx_m)), 0.5)), 1.0))), 0.5));
	} else if (l <= 3.4e+130) {
		tmp = sqrt((0.5 + (0.5 * sqrt((1.0 / fma(4.0, ((fma(-0.5, cos((ky_m * -2.0)), 0.5) * (l * l)) / (Om * Om)), 1.0))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (l <= 2e-177)
		tmp = sqrt(fma(0.5, sqrt(Float64(1.0 / fma(Float64(Float64(l * 4.0) / Om), Float64(Float64(l / Om) * fma(-0.5, cos(Float64(kx_m + kx_m)), 0.5)), 1.0))), 0.5));
	elseif (l <= 3.4e+130)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * sqrt(Float64(1.0 / fma(4.0, Float64(Float64(fma(-0.5, cos(Float64(ky_m * -2.0)), 0.5) * Float64(l * l)) / Float64(Om * Om)), 1.0))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[l, 2e-177], N[Sqrt[N[(0.5 * N[Sqrt[N[(1.0 / N[(N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(-0.5 * N[Cos[N[(kx$95$m + kx$95$m), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.4e+130], N[Sqrt[N[(0.5 + N[(0.5 * N[Sqrt[N[(1.0 / N[(4.0 * N[(N[(N[(-0.5 * N[Cos[N[(ky$95$m * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.9999999999999999e-177

   1. Initial program 98.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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. accelerator-lowering-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. Simplified 79.9%

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

   7. Applied egg-rr 83.6%

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

   if 1.9999999999999999e-177 < l < 3.4000000000000001e130

   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 egg-rr 96.0%

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

   5. Taylor expanded in kx around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 89.2%

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

   if 3.4000000000000001e130 < l

   1. Initial program 88.1%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 87.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, \sqrt{\frac{1}{\mathsf{fma}\left(\frac{\ell \cdot 4}{Om}, \frac{\ell}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(kx + kx\right), 0.5\right), 1\right)}}, 0.5\right)}\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+130}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-177}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.28 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(-0.5, \cos \left(ky\_m \cdot -2\right), 0.5\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= l 2e-177)
   1.0
   (if (<= l 1.28e+129)
     (sqrt
      (+
       0.5
       (*
        0.5
        (sqrt
         (/
          1.0
          (fma
           4.0
           (/ (* (fma -0.5 (cos (* ky_m -2.0)) 0.5) (* l l)) (* Om Om))
           1.0))))))
     (sqrt 0.5))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 2e-177) {
		tmp = 1.0;
	} else if (l <= 1.28e+129) {
		tmp = sqrt((0.5 + (0.5 * sqrt((1.0 / fma(4.0, ((fma(-0.5, cos((ky_m * -2.0)), 0.5) * (l * l)) / (Om * Om)), 1.0))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (l <= 2e-177)
		tmp = 1.0;
	elseif (l <= 1.28e+129)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * sqrt(Float64(1.0 / fma(4.0, Float64(Float64(fma(-0.5, cos(Float64(ky_m * -2.0)), 0.5) * Float64(l * l)) / Float64(Om * Om)), 1.0))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[l, 2e-177], 1.0, If[LessEqual[l, 1.28e+129], N[Sqrt[N[(0.5 + N[(0.5 * N[Sqrt[N[(1.0 / N[(4.0 * N[(N[(N[(-0.5 * N[Cos[N[(ky$95$m * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.9999999999999999e-177

   1. Initial program 98.1%

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

   3. Taylor expanded in l around 0

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

   5. Simplified 70.6%

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

      1. metadata-eval 70.6

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

   7. Applied egg-rr 70.6%

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

   if 1.9999999999999999e-177 < l < 1.27999999999999994e129

   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 egg-rr 96.0%

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

   5. Taylor expanded in kx around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 89.2%

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

   if 1.27999999999999994e129 < l

   1. Initial program 88.1%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 87.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-177}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.28 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(4, \frac{\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right) \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 6.9× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\ell \cdot \left(\ell \cdot 4\right), \frac{ky\_m \cdot ky\_m}{Om \cdot Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= l 2e-30)
   1.0
   (if (<= l 6.5e+134)
     (sqrt
      (+
       0.5
       (/ 0.5 (sqrt (fma (* l (* l 4.0)) (/ (* ky_m ky_m) (* Om Om)) 1.0)))))
     (sqrt 0.5))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 2e-30) {
		tmp = 1.0;
	} else if (l <= 6.5e+134) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma((l * (l * 4.0)), ((ky_m * ky_m) / (Om * Om)), 1.0)))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (l <= 2e-30)
		tmp = 1.0;
	elseif (l <= 6.5e+134)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(l * Float64(l * 4.0)), Float64(Float64(ky_m * ky_m) / Float64(Om * Om)), 1.0)))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[l, 2e-30], 1.0, If[LessEqual[l, 6.5e+134], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(l * N[(l * 4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(ky$95$m * ky$95$m), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2e-30

   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. Taylor expanded in l around 0

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

   5. Simplified 73.4%

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

      1. metadata-eval 73.4

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

   7. Applied egg-rr 73.4%

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

   if 2e-30 < l < 6.5e134

   1. Initial program 97.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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. accelerator-lowering-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. Simplified 84.1%

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

   6. Taylor expanded in ky around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 79.0

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

   8. Simplified 79.0%

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

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

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

   10. Applied egg-rr 79.0%

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

   if 6.5e134 < l

   1. Initial program 89.7%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 88.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\ell \cdot \left(\ell \cdot 4\right), \frac{ky \cdot ky}{Om \cdot Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 7.8× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, \left(ky\_m \cdot ky\_m\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= l 2e-30)
   1.0
   (if (<= l 1.25e+135)
     (sqrt
      (+ 0.5 (/ 0.5 (fma 2.0 (* (* ky_m ky_m) (/ (* l l) (* Om Om))) 1.0))))
     (sqrt 0.5))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (l <= 2e-30) {
		tmp = 1.0;
	} else if (l <= 1.25e+135) {
		tmp = sqrt((0.5 + (0.5 / fma(2.0, ((ky_m * ky_m) * ((l * l) / (Om * Om))), 1.0))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (l <= 2e-30)
		tmp = 1.0;
	elseif (l <= 1.25e+135)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / fma(2.0, Float64(Float64(ky_m * ky_m) * Float64(Float64(l * l) / Float64(Om * Om))), 1.0))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[l, 2e-30], 1.0, If[LessEqual[l, 1.25e+135], N[Sqrt[N[(0.5 + N[(0.5 / N[(2.0 * N[(N[(ky$95$m * ky$95$m), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2e-30

   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. Taylor expanded in l around 0

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

   5. Simplified 73.4%

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

      1. metadata-eval 73.4

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

   7. Applied egg-rr 73.4%

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

   if 2e-30 < l < 1.25000000000000007e135

   1. Initial program 97.1%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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. accelerator-lowering-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. Simplified 84.1%

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

   6. Taylor expanded in ky around 0

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

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

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 79.0

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

   8. Simplified 79.0%

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

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

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

   10. Applied egg-rr 79.0%

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

   11. Taylor expanded in l around 0

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

       1. +-commutative N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. associate-/l* N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. /-lowering-/.f64 N/A

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 79.0

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

   13. Simplified 79.0%

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

   if 1.25000000000000007e135 < l

   1. Initial program 89.7%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 88.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, \left(ky \cdot ky\right) \cdot \frac{\ell \cdot \ell}{Om \cdot Om}, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.7% accurate, 34.1× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.2 \cdot 10^{-85}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<= Om 2.2e-85) (sqrt 0.5) 1.0))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (Om <= 2.2e-85) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

kx_m = abs(kx)
ky_m = abs(ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx_m, ky_m)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (om <= 2.2d-85) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	double tmp;
	if (Om <= 2.2e-85) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	tmp = 0
	if Om <= 2.2e-85:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	tmp = 0.0
	if (Om <= 2.2e-85)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (Om <= 2.2e-85)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[Om, 2.2e-85], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 2.2e-85

   1. Initial program 96.2%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 63.4%

      \[\leadsto \sqrt{\color{blue}{0.5}} \]

   if 2.2e-85 < Om

   1. Initial program 98.6%

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

   3. Taylor expanded in l around 0

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

   5. Simplified 81.4%

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

      1. metadata-eval 81.4

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

   7. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.3% accurate, 581.0× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ 1 \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
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 1.0)

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return 1.0;
}

Fortran

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

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	return 1.0;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	return 1.0

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	return 1.0
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp = code(l, Om, kx_m, ky_m)
	tmp = 1.0;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $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_] := 1.0

Derivation

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 l around 0

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

5. Simplified 62.8%

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

   1. metadata-eval 62.8

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

7. Applied egg-rr 62.8%

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

Reproduce

herbie shell --seed 2024197 
(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))))))))))