Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 99.3%

Time: 13.7s

Alternatives: 9

Speedup: 1.1×

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: 98.2% 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: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := {\sin kx}^{2}\\ t_1 := {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2}\\ \mathbf{if}\;\left({\sin ky}^{2} + t\_0\right) \cdot t\_1 \leq 0.2:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left(\left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) + t\_0\right) \cdot t\_1 + 1}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\left(\frac{l\_m}{Om\_m} \cdot 2\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \end{array} \]

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) 0.2)
     (sqrt
      (*
       (+
        (/ 1.0 (sqrt (+ (* (+ (- 0.5 (* (cos (* ky 2.0)) 0.5)) t_0) t_1) 1.0)))
        1.0)
       (/ 1.0 2.0)))
     (sqrt
      (*
       (+ (/ 1.0 (* (* (/ l_m Om_m) 2.0) (hypot (sin kx) (sin ky)))) 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) {
	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) <= 0.2) {
		tmp = sqrt((((1.0 / sqrt(((((0.5 - (cos((ky * 2.0)) * 0.5)) + t_0) * t_1) + 1.0))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = sqrt((((1.0 / (((l_m / Om_m) * 2.0) * hypot(sin(kx), sin(ky)))) + 1.0) * (1.0 / 2.0)));
	}
	return tmp;
}

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 t_0 = Math.pow(Math.sin(kx), 2.0);
	double t_1 = Math.pow(((l_m * 2.0) / Om_m), 2.0);
	double tmp;
	if (((Math.pow(Math.sin(ky), 2.0) + t_0) * t_1) <= 0.2) {
		tmp = Math.sqrt((((1.0 / Math.sqrt(((((0.5 - (Math.cos((ky * 2.0)) * 0.5)) + t_0) * t_1) + 1.0))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = Math.sqrt((((1.0 / (((l_m / Om_m) * 2.0) * Math.hypot(Math.sin(kx), Math.sin(ky)))) + 1.0) * (1.0 / 2.0)));
	}
	return tmp;
}

Python

Om_m = math.fabs(Om)
l_m = math.fabs(l)
def code(l_m, Om_m, kx, ky):
	t_0 = math.pow(math.sin(kx), 2.0)
	t_1 = math.pow(((l_m * 2.0) / Om_m), 2.0)
	tmp = 0
	if ((math.pow(math.sin(ky), 2.0) + t_0) * t_1) <= 0.2:
		tmp = math.sqrt((((1.0 / math.sqrt(((((0.5 - (math.cos((ky * 2.0)) * 0.5)) + t_0) * t_1) + 1.0))) + 1.0) * (1.0 / 2.0)))
	else:
		tmp = math.sqrt((((1.0 / (((l_m / Om_m) * 2.0) * math.hypot(math.sin(kx), math.sin(ky)))) + 1.0) * (1.0 / 2.0)))
	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) <= 0.2)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(ky * 2.0)) * 0.5)) + t_0) * t_1) + 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(1.0 / Float64(Float64(Float64(l_m / Om_m) * 2.0) * hypot(sin(kx), sin(ky)))) + 1.0) * Float64(1.0 / 2.0)));
	end
	return tmp
end

MATLAB

Om_m = abs(Om);
l_m = abs(l);
function tmp_2 = code(l_m, Om_m, kx, ky)
	t_0 = sin(kx) ^ 2.0;
	t_1 = ((l_m * 2.0) / Om_m) ^ 2.0;
	tmp = 0.0;
	if ((((sin(ky) ^ 2.0) + t_0) * t_1) <= 0.2)
		tmp = sqrt((((1.0 / sqrt(((((0.5 - (cos((ky * 2.0)) * 0.5)) + t_0) * t_1) + 1.0))) + 1.0) * (1.0 / 2.0)));
	else
		tmp = sqrt((((1.0 / (((l_m / Om_m) * 2.0) * hypot(sin(kx), sin(ky)))) + 1.0) * (1.0 / 2.0)));
	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_] := 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], 0.2], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(N[(0.5 - N[(N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] * 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[(1.0 / N[(N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.20000000000000001

   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-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. count-2 N/A

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

      17. lower-cos.f64 N/A

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

      18. count-2 N/A

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

      19. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 0.20000000000000001 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 97.6%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.2:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left(\left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\left(\frac{\ell}{Om} \cdot 2\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 0.2:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{l\_m}{Om\_m} \cdot \sin kx\right)}^{2}, 4, 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\left(\frac{l\_m}{Om\_m} \cdot 2\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \end{array} \]

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))
      0.2)
   (sqrt
    (+ (/ 0.5 (sqrt (fma (pow (* (/ l_m Om_m) (sin kx)) 2.0) 4.0 1.0))) 0.5))
   (sqrt
    (*
     (+ (/ 1.0 (* (* (/ l_m Om_m) 2.0) (hypot (sin kx) (sin ky)))) 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) {
	double tmp;
	if (((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l_m * 2.0) / Om_m), 2.0)) <= 0.2) {
		tmp = sqrt(((0.5 / sqrt(fma(pow(((l_m / Om_m) * sin(kx)), 2.0), 4.0, 1.0))) + 0.5));
	} else {
		tmp = sqrt((((1.0 / (((l_m / Om_m) * 2.0) * hypot(sin(kx), sin(ky)))) + 1.0) * (1.0 / 2.0)));
	}
	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)) <= 0.2)
		tmp = sqrt(Float64(Float64(0.5 / sqrt(fma((Float64(Float64(l_m / Om_m) * sin(kx)) ^ 2.0), 4.0, 1.0))) + 0.5));
	else
		tmp = sqrt(Float64(Float64(Float64(1.0 / Float64(Float64(Float64(l_m / Om_m) * 2.0) * hypot(sin(kx), sin(ky)))) + 1.0) * Float64(1.0 / 2.0)));
	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], 0.2], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Power[N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[Sin[kx], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(1.0 / N[(N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.20000000000000001

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 99.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{{\left(\frac{\ell \cdot 2}{Om} \cdot \sin kx\right)}^{2} + 1}}, 0.5, 0.5\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.3%

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

   if 0.20000000000000001 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 97.6%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.2:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om} \cdot \sin kx\right)}^{2}, 4, 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\left(\frac{\ell}{Om} \cdot 2\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := \frac{l\_m}{Om\_m} \cdot 2\\ \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} + 1}} \leq 0.9995:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(t\_0 \cdot \mathsf{fma}\left(\sin kx, \sin kx, ky \cdot ky\right), t\_0, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{l\_m}{Om\_m} \cdot \sin kx\right)}^{2}, 4, 1\right)}} + 0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (* (/ l_m Om_m) 2.0)))
   (if (<=
        (/
         1.0
         (sqrt
          (+
           (*
            (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
            (pow (/ (* l_m 2.0) Om_m) 2.0))
           1.0)))
        0.9995)
     (sqrt
      (*
       (+
        (/ 1.0 (sqrt (fma (* t_0 (fma (sin kx) (sin kx) (* ky ky))) t_0 1.0)))
        1.0)
       (/ 1.0 2.0)))
     (sqrt
      (+
       (/ 0.5 (sqrt (fma (pow (* (/ l_m Om_m) (sin kx)) 2.0) 4.0 1.0)))
       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 = (l_m / Om_m) * 2.0;
	double tmp;
	if ((1.0 / sqrt((((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l_m * 2.0) / Om_m), 2.0)) + 1.0))) <= 0.9995) {
		tmp = sqrt((((1.0 / sqrt(fma((t_0 * fma(sin(kx), sin(kx), (ky * ky))), t_0, 1.0))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = sqrt(((0.5 / sqrt(fma(pow(((l_m / Om_m) * sin(kx)), 2.0), 4.0, 1.0))) + 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 / Om_m) * 2.0)
	tmp = 0.0
	if (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))) <= 0.9995)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(t_0 * fma(sin(kx), sin(kx), Float64(ky * ky))), t_0, 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(Float64(Float64(0.5 / sqrt(fma((Float64(Float64(l_m / Om_m) * sin(kx)) ^ 2.0), 4.0, 1.0))) + 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[(N[(l$95$m / Om$95$m), $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[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], 0.9995], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(t$95$0 * N[(N[Sin[kx], $MachinePrecision] * N[Sin[kx], $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Power[N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[Sin[kx], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))) < 0.99950000000000006

   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

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 98.8%

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

   if 0.99950000000000006 < (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))))

   1. Initial program 97.7%

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

   3. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 87.8%

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

   7. Applied rewrites 99.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{{\left(\frac{\ell \cdot 2}{Om} \cdot \sin kx\right)}^{2} + 1}}, 0.5, 0.5\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.9995:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot 2\right) \cdot \mathsf{fma}\left(\sin kx, \sin kx, ky \cdot ky\right), \frac{\ell}{Om} \cdot 2, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om} \cdot \sin kx\right)}^{2}, 4, 1\right)}} + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 0.2:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{l\_m}{Om\_m} \cdot \sin kx\right)}^{2}, 4, 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky \cdot \left(\frac{l\_m}{Om\_m} \cdot 2\right)} + 0.5}\\ \end{array} \end{array} \]

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))
      0.2)
   (sqrt
    (+ (/ 0.5 (sqrt (fma (pow (* (/ l_m Om_m) (sin kx)) 2.0) 4.0 1.0))) 0.5))
   (sqrt (+ (/ 0.5 (* (sin ky) (* (/ l_m Om_m) 2.0))) 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)) <= 0.2) {
		tmp = sqrt(((0.5 / sqrt(fma(pow(((l_m / Om_m) * sin(kx)), 2.0), 4.0, 1.0))) + 0.5));
	} else {
		tmp = sqrt(((0.5 / (sin(ky) * ((l_m / Om_m) * 2.0))) + 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)) <= 0.2)
		tmp = sqrt(Float64(Float64(0.5 / sqrt(fma((Float64(Float64(l_m / Om_m) * sin(kx)) ^ 2.0), 4.0, 1.0))) + 0.5));
	else
		tmp = sqrt(Float64(Float64(0.5 / Float64(sin(ky) * Float64(Float64(l_m / Om_m) * 2.0))) + 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], 0.2], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Power[N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[Sin[kx], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[(N[Sin[ky], $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.20000000000000001

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 88.2%

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

   7. Applied rewrites 99.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{{\left(\frac{\ell \cdot 2}{Om} \cdot \sin kx\right)}^{2} + 1}}, 0.5, 0.5\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.3%

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

   if 0.20000000000000001 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 97.6%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

   10. Applied rewrites 83.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.2:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om} \cdot \sin kx\right)}^{2}, 4, 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky \cdot \left(\frac{\ell}{Om} \cdot 2\right)} + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky \cdot \left(\frac{l\_m}{Om\_m} \cdot 2\right)} + 0.5}\\ \end{array} \end{array} \]

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))
      0.2)
   1.0
   (sqrt (+ (/ 0.5 (* (sin ky) (* (/ l_m Om_m) 2.0))) 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)) <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = sqrt(((0.5 / (sin(ky) * ((l_m / Om_m) * 2.0))) + 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)) <= 0.2d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(((0.5d0 / (sin(ky) * ((l_m / om_m) * 2.0d0))) + 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)) <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(((0.5 / (Math.sin(ky) * ((l_m / Om_m) * 2.0))) + 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)) <= 0.2:
		tmp = 1.0
	else:
		tmp = math.sqrt(((0.5 / (math.sin(ky) * ((l_m / Om_m) * 2.0))) + 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)) <= 0.2)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(Float64(0.5 / Float64(sin(ky) * Float64(Float64(l_m / Om_m) * 2.0))) + 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)) <= 0.2)
		tmp = 1.0;
	else
		tmp = sqrt(((0.5 / (sin(ky) * ((l_m / Om_m) * 2.0))) + 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], 0.2], 1.0, N[Sqrt[N[(N[(0.5 / N[(N[Sin[ky], $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.20000000000000001

   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 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 2.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

   10. Applied rewrites 2.7%

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

   11. Taylor expanded in Om around inf

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

   13. Applied rewrites 98.2%

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

   if 0.20000000000000001 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 97.6%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

   10. Applied rewrites 83.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sin ky \cdot \left(\frac{\ell}{Om} \cdot 2\right)} + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om\_m}{\sin ky \cdot l\_m}, 0.25, 0.5\right)}\\ \end{array} \end{array} \]

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))
      0.2)
   1.0
   (sqrt (fma (/ Om_m (* (sin ky) l_m)) 0.25 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)) <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma((Om_m / (sin(ky) * l_m)), 0.25, 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)) <= 0.2)
		tmp = 1.0;
	else
		tmp = sqrt(fma(Float64(Om_m / Float64(sin(ky) * l_m)), 0.25, 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], 0.2], 1.0, N[Sqrt[N[(N[(Om$95$m / N[(N[Sin[ky], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.20000000000000001

   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 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 2.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

   10. Applied rewrites 2.7%

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

   11. Taylor expanded in Om around inf

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

   13. Applied rewrites 98.2%

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

   if 0.20000000000000001 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 97.6%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 75.1%

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

   9. Taylor expanded in Om around 0

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

   11. Applied rewrites 83.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% 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 98.8%

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

3. Final simplification 98.8%

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

Alternative 8: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

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))
      0.2)
   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)) <= 0.2) {
		tmp = 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)) <= 0.2d0) then
        tmp = 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)) <= 0.2) {
		tmp = 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)) <= 0.2:
		tmp = 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)) <= 0.2)
		tmp = 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)) <= 0.2)
		tmp = 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], 0.2], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 0.20000000000000001

   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 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 2.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

   10. Applied rewrites 2.7%

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

   11. Taylor expanded in Om around inf

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

   13. Applied rewrites 98.2%

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

   if 0.20000000000000001 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 97.6%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 581.0× speedup

Math

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

FPCore

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

C

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

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 = 1.0d0
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 1.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in Om around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-hypot.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-sin.f64 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in kx around 0

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

8. Applied rewrites 42.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-rgt-in N/A

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

10. Applied rewrites 42.4%

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

11. Taylor expanded in Om around inf

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

13. Applied rewrites 60.0%

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

Reproduce

herbie shell --seed 2024244 
(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))))))))))