Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.1%

Time: 12.8s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2}\\ \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot t\_0 \leq 10^{+15}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\frac{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(1 - \cos \left(kx \cdot 2\right)\right) \cdot 2\right)}{4} \cdot t\_0 + 1}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{l\_m} \cdot Om\_m, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (pow (/ (* l_m 2.0) Om_m) 2.0)))
   (if (<= (* (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)) t_0) 1e+15)
     (sqrt
      (*
       (+
        (/
         1.0
         (sqrt
          (+
           (*
            (/
             (fma
              (- 1.0 (cos (* ky 2.0)))
              2.0
              (* (- 1.0 (cos (* kx 2.0))) 2.0))
             4.0)
            t_0)
           1.0)))
        1.0)
       0.5))
     (sqrt
      (fma
       (* (/ 0.25 l_m) Om_m)
       (sqrt (/ 1.0 (hypot (sin kx) (sin ky))))
       0.5)))))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = pow(((l_m * 2.0) / Om_m), 2.0);
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * t_0) <= 1e+15) {
		tmp = sqrt((((1.0 / sqrt((((fma((1.0 - cos((ky * 2.0))), 2.0, ((1.0 - cos((kx * 2.0))) * 2.0)) / 4.0) * t_0) + 1.0))) + 1.0) * 0.5));
	} else {
		tmp = sqrt(fma(((0.25 / l_m) * Om_m), sqrt((1.0 / hypot(sin(kx), sin(ky)))), 0.5));
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(l_m * 2.0) / Om_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * t_0) <= 1e+15)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(Float64(fma(Float64(1.0 - cos(Float64(ky * 2.0))), 2.0, Float64(Float64(1.0 - cos(Float64(kx * 2.0))) * 2.0)) / 4.0) * t_0) + 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(fma(Float64(Float64(0.25 / l_m) * Om_m), sqrt(Float64(1.0 / hypot(sin(kx), sin(ky)))), 0.5));
	end
	return tmp
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := Block[{t$95$0 = N[Power[N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 1e+15], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(N[(N[(1.0 - N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(1.0 - N[Cos[N[(kx * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(0.25 / l$95$m), $MachinePrecision] * Om$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1e15

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. sqr-sin-a N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-+l- N/A

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

      11. lower--.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. count-2 N/A

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

      21. lower-cos.f64 N/A

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

      22. count-2 N/A

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

      23. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqr-sin-a N/A

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

      8. sin-mult N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. sin-mult N/A

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

      14. frac-add N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

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

   if 1e15 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 96.8%

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

   3. Taylor expanded in Om around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   5. Applied rewrites 99.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(Om \cdot \frac{0.25}{\ell}, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 10^{+15}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\frac{\mathsf{fma}\left(1 - \cos \left(ky \cdot 2\right), 2, \left(1 - \cos \left(kx \cdot 2\right)\right) \cdot 2\right)}{4} \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{\ell} \cdot Om, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := {\sin kx}^{2}\\ t_1 := {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2}\\ \mathbf{if}\;\left({\sin ky}^{2} + t\_0\right) \cdot t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{t\_0 \cdot t\_1 + 1}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{l\_m} \cdot Om\_m, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (pow (sin kx) 2.0)) (t_1 (pow (/ (* l_m 2.0) Om_m) 2.0)))
   (if (<= (* (+ (pow (sin ky) 2.0) t_0) t_1) 2e-5)
     (sqrt (* (+ (/ 1.0 (sqrt (+ (* t_0 t_1) 1.0))) 1.0) (/ 1.0 2.0)))
     (sqrt
      (fma
       (* (/ 0.25 l_m) Om_m)
       (sqrt (/ 1.0 (hypot (sin kx) (sin ky))))
       0.5)))))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = pow(sin(kx), 2.0);
	double t_1 = pow(((l_m * 2.0) / Om_m), 2.0);
	double tmp;
	if (((pow(sin(ky), 2.0) + t_0) * t_1) <= 2e-5) {
		tmp = sqrt((((1.0 / sqrt(((t_0 * t_1) + 1.0))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = sqrt(fma(((0.25 / l_m) * Om_m), sqrt((1.0 / hypot(sin(kx), sin(ky)))), 0.5));
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	t_0 = sin(kx) ^ 2.0
	t_1 = Float64(Float64(l_m * 2.0) / Om_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + t_0) * t_1) <= 2e-5)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(t_0 * t_1) + 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(fma(Float64(Float64(0.25 / l_m) * Om_m), sqrt(Float64(1.0 / hypot(sin(kx), sin(ky)))), 0.5));
	end
	return tmp
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := Block[{t$95$0 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], 2e-5], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(t$95$0 * t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(0.25 / l$95$m), $MachinePrecision] * Om$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in ky around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 2.00000000000000016e-5 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 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 Om around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   5. Applied rewrites 98.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(Om \cdot \frac{0.25}{\ell}, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{{\sin kx}^{2} \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{\ell} \cdot Om, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{l\_m} \cdot Om\_m, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
       (pow (/ (* l_m 2.0) Om_m) 2.0))
      2e-5)
   (sqrt 1.0)
   (sqrt
    (fma (* (/ 0.25 l_m) Om_m) (sqrt (/ 1.0 (hypot (sin kx) (sin ky)))) 0.5))))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l_m * 2.0) / Om_m), 2.0)) <= 2e-5) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(fma(((0.25 / l_m) * Om_m), sqrt((1.0 / hypot(sin(kx), sin(ky)))), 0.5));
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l_m * 2.0) / Om_m) ^ 2.0)) <= 2e-5)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(fma(Float64(Float64(0.25 / l_m) * Om_m), sqrt(Float64(1.0 / hypot(sin(kx), sin(ky)))), 0.5));
	end
	return tmp
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-5], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(N[(0.25 / l$95$m), $MachinePrecision] * Om$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in Om around inf

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

   5. Applied rewrites 99.6%

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

   if 2.00000000000000016e-5 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 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 Om around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

   5. Applied rewrites 98.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(Om \cdot \frac{0.25}{\ell}, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{0.25}{\ell} \cdot Om, \sqrt{\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
       (pow (/ (* l_m 2.0) Om_m) 2.0))
      2e-5)
   (sqrt 1.0)
   (sqrt 0.5)))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l_m * 2.0) / Om_m), 2.0)) <= 2e-5) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

Om_m = abs(om)
l_m = abs(l)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if ((((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)) * (((l_m * 2.0d0) / om_m) ** 2.0d0)) <= 2d-5) then
        tmp = sqrt(1.0d0)
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

Om_m = Math.abs(Om);
l_m = Math.abs(l);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)) * Math.pow(((l_m * 2.0) / Om_m), 2.0)) <= 2e-5) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

Om_m = math.fabs(Om)
l_m = math.fabs(l)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if ((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)) * math.pow(((l_m * 2.0) / Om_m), 2.0)) <= 2e-5:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l_m * 2.0) / Om_m) ^ 2.0)) <= 2e-5)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

Om_m = abs(Om);
l_m = abs(l);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if ((((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (((l_m * 2.0) / Om_m) ^ 2.0)) <= 2e-5)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 2.00000000000000016e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in Om around inf

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

   5. Applied rewrites 99.6%

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

   if 2.00000000000000016e-5 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 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 Om around 0

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

   5. Applied rewrites 97.8%

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

4. Final simplification 98.7%

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

Alternative 6: 88.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $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. Taylor expanded in kx around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 86.4

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

5. Applied rewrites 86.4%

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

6. Final simplification 86.4%

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

Alternative 7: 57.0% accurate, 52.8× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \sqrt{0.5} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky) :precision binary64 (sqrt 0.5))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	return sqrt(0.5);
}

Fortran

Om_m = abs(om)
l_m = abs(l)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(0.5d0)
end function

Java

Om_m = Math.abs(Om);
l_m = Math.abs(l);
public static double code(double l_m, double Om_m, double kx, double ky) {
	return Math.sqrt(0.5);
}

Python

Om_m = math.fabs(Om)
l_m = math.fabs(l)
def code(l_m, Om_m, kx, ky):
	return math.sqrt(0.5)

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	return sqrt(0.5)
end

MATLAB

Om_m = abs(Om);
l_m = abs(l);
function tmp = code(l_m, Om_m, kx, ky)
	tmp = sqrt(0.5);
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := N[Sqrt[0.5], $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. Taylor expanded in Om around 0

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

5. Applied rewrites 58.6%

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

Reproduce

herbie shell --seed 2024249 
(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))))))))))