Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.2%

Time: 14.0s

Alternatives: 10

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: 98.3% 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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := \frac{l\_m \cdot 2}{Om\_m}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {t\_0}^{2} + 1}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* l_m 2.0) Om_m)))
   (if (<= t_0 2e+109)
     (sqrt
      (*
       (+
        (/
         1.0
         (sqrt
          (+ (* (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0)) (pow t_0 2.0)) 1.0)))
        1.0)
       (/ 1.0 2.0)))
     (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 t_0 = (l_m * 2.0) / Om_m;
	double tmp;
	if (t_0 <= 2e+109) {
		tmp = sqrt((((1.0 / sqrt((((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(t_0, 2.0)) + 1.0))) + 1.0) * (1.0 / 2.0)));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (l_m * 2.0d0) / om_m
    if (t_0 <= 2d+109) then
        tmp = sqrt((((1.0d0 / sqrt(((((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)) * (t_0 ** 2.0d0)) + 1.0d0))) + 1.0d0) * (1.0d0 / 2.0d0)))
    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 t_0 = (l_m * 2.0) / Om_m;
	double tmp;
	if (t_0 <= 2e+109) {
		tmp = Math.sqrt((((1.0 / Math.sqrt((((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)) * Math.pow(t_0, 2.0)) + 1.0))) + 1.0) * (1.0 / 2.0)));
	} 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):
	t_0 = (l_m * 2.0) / Om_m
	tmp = 0
	if t_0 <= 2e+109:
		tmp = math.sqrt((((1.0 / math.sqrt((((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)) * math.pow(t_0, 2.0)) + 1.0))) + 1.0) * (1.0 / 2.0)))
	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)
	t_0 = Float64(Float64(l_m * 2.0) / Om_m)
	tmp = 0.0
	if (t_0 <= 2e+109)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (t_0 ^ 2.0)) + 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	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)
	t_0 = (l_m * 2.0) / Om_m;
	tmp = 0.0;
	if (t_0 <= 2e+109)
		tmp = sqrt((((1.0 / sqrt(((((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (t_0 ^ 2.0)) + 1.0))) + 1.0) * (1.0 / 2.0)));
	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_] := Block[{t$95$0 = N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+109], 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[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.99999999999999996e109

   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

   if 1.99999999999999996e109 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 97.8%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\ell \cdot 2}{Om} \leq 2 \cdot 10^{+109}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 200:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(\left(4 \cdot \frac{l\_m}{Om\_m}\right) \cdot l\_m\right) \cdot \left(0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)\right), \frac{1}{Om\_m}, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{2}{Om\_m} \cdot l\_m\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \end{array} \]

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))
      200.0)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (*
          (* (* 4.0 (/ l_m Om_m)) l_m)
          (- 0.5 (- (* (cos (+ ky ky)) 0.5) (- 0.5 (* (cos (+ kx kx)) 0.5)))))
         (/ 1.0 Om_m)
         1.0)))
      1.0)
     (/ 1.0 2.0)))
   (sqrt
    (*
     (+ (/ 1.0 (* (hypot (sin kx) (sin ky)) (* (/ 2.0 Om_m) l_m))) 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)) <= 200.0) {
		tmp = sqrt((((1.0 / sqrt(fma((((4.0 * (l_m / Om_m)) * l_m) * (0.5 - ((cos((ky + ky)) * 0.5) - (0.5 - (cos((kx + kx)) * 0.5))))), (1.0 / Om_m), 1.0))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = sqrt((((1.0 / (hypot(sin(kx), sin(ky)) * ((2.0 / Om_m) * l_m))) + 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)) <= 200.0)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(Float64(Float64(4.0 * Float64(l_m / Om_m)) * l_m) * Float64(0.5 - Float64(Float64(cos(Float64(ky + ky)) * 0.5) - Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5))))), Float64(1.0 / Om_m), 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(1.0 / Float64(hypot(sin(kx), sin(ky)) * Float64(Float64(2.0 / Om_m) * l_m))) + 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], 200.0], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] * N[(0.5 - N[(N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om$95$m), $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[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[(2.0 / Om$95$m), $MachinePrecision] * l$95$m), $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)))) < 200

   1. Initial program 100.0%

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

   4. Applied rewrites 99.8%

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

   if 200 < (*.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 95.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   5. Applied rewrites 97.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 200:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(\left(4 \cdot \frac{\ell}{Om}\right) \cdot \ell\right) \cdot \left(0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)\right), \frac{1}{Om}, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(\left(4 \cdot \frac{l\_m}{Om\_m}\right) \cdot l\_m\right) \cdot \left(0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)\right), \frac{1}{Om\_m}, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

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))
      5e+24)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (*
          (* (* 4.0 (/ l_m Om_m)) l_m)
          (- 0.5 (- (* (cos (+ ky ky)) 0.5) (- 0.5 (* (cos (+ kx kx)) 0.5)))))
         (/ 1.0 Om_m)
         1.0)))
      1.0)
     (/ 1.0 2.0)))
   (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)) <= 5e+24) {
		tmp = sqrt((((1.0 / sqrt(fma((((4.0 * (l_m / Om_m)) * l_m) * (0.5 - ((cos((ky + ky)) * 0.5) - (0.5 - (cos((kx + kx)) * 0.5))))), (1.0 / Om_m), 1.0))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = 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)) <= 5e+24)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(Float64(Float64(4.0 * Float64(l_m / Om_m)) * l_m) * Float64(0.5 - Float64(Float64(cos(Float64(ky + ky)) * 0.5) - Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5))))), Float64(1.0 / Om_m), 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(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], 5e+24], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] * N[(0.5 - N[(N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $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)))) < 5.00000000000000045e24

   1. Initial program 100.0%

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

   4. Applied rewrites 99.2%

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

   if 5.00000000000000045e24 < (*.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 94.9%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(\left(4 \cdot \frac{\ell}{Om}\right) \cdot \ell\right) \cdot \left(0.5 - \left(\cos \left(ky + ky\right) \cdot 0.5 - \left(0.5 - \cos \left(kx + kx\right) \cdot 0.5\right)\right)\right), \frac{1}{Om}, 1\right)}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{l\_m}{Om\_m}, \left(1 - \left(\cos \left(ky \cdot 2\right) + \cos \left(kx + kx\right)\right) \cdot 0.5\right) \cdot \frac{l\_m}{Om\_m}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

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))
      5e+24)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (* 4.0 (/ l_m Om_m))
         (* (- 1.0 (* (+ (cos (* ky 2.0)) (cos (+ kx kx))) 0.5)) (/ l_m Om_m))
         1.0)))
      1.0)
     0.5))
   (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)) <= 5e+24) {
		tmp = sqrt((((1.0 / sqrt(fma((4.0 * (l_m / Om_m)), ((1.0 - ((cos((ky * 2.0)) + cos((kx + kx))) * 0.5)) * (l_m / Om_m)), 1.0))) + 1.0) * 0.5));
	} else {
		tmp = 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)) <= 5e+24)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(4.0 * Float64(l_m / Om_m)), Float64(Float64(1.0 - Float64(Float64(cos(Float64(ky * 2.0)) + cos(Float64(kx + kx))) * 0.5)) * Float64(l_m / Om_m)), 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(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], 5e+24], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(N[(N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $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)))) < 5.00000000000000045e24

   1. Initial program 100.0%

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

   4. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in kx around inf

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

      1. lower--.f64 N/A

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

      2. distribute-lft-out N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. lower-cos.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

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

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

      14. lower-cos.f64 N/A

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

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

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.2

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

   9. Applied rewrites 99.2%

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

   11. Applied rewrites 99.2%

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

   if 5.00000000000000045e24 < (*.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 94.9%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \left(1 - \left(\cos \left(ky \cdot 2\right) + \cos \left(kx + kx\right)\right) \cdot 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) \cdot \left(\left(4 \cdot \frac{l\_m}{Om\_m}\right) \cdot l\_m\right), \frac{1}{Om\_m}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

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))
      5e+24)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (* (- 0.5 (* (cos (* ky 2.0)) 0.5)) (* (* 4.0 (/ l_m Om_m)) l_m))
         (/ 1.0 Om_m)
         1.0)))
      1.0)
     0.5))
   (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)) <= 5e+24) {
		tmp = sqrt((((1.0 / sqrt(fma(((0.5 - (cos((ky * 2.0)) * 0.5)) * ((4.0 * (l_m / Om_m)) * l_m)), (1.0 / Om_m), 1.0))) + 1.0) * 0.5));
	} else {
		tmp = 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)) <= 5e+24)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(Float64(0.5 - Float64(cos(Float64(ky * 2.0)) * 0.5)) * Float64(Float64(4.0 * Float64(l_m / Om_m)) * l_m)), Float64(1.0 / Om_m), 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(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], 5e+24], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(0.5 - N[(N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / Om$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $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)))) < 5.00000000000000045e24

   1. Initial program 100.0%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-cos.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.7

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

   7. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 96.7

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

   9. Applied rewrites 96.7%

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

   if 5.00000000000000045e24 < (*.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 94.9%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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 99.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) \cdot \left(\left(4 \cdot \frac{\ell}{Om}\right) \cdot \ell\right), \frac{1}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{l\_m}{Om\_m}, \left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) \cdot \frac{l\_m}{Om\_m}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

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))
      5e+24)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (fma
         (* 4.0 (/ l_m Om_m))
         (* (- 0.5 (* (cos (* ky 2.0)) 0.5)) (/ l_m Om_m))
         1.0)))
      1.0)
     0.5))
   (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)) <= 5e+24) {
		tmp = sqrt((((1.0 / sqrt(fma((4.0 * (l_m / Om_m)), ((0.5 - (cos((ky * 2.0)) * 0.5)) * (l_m / Om_m)), 1.0))) + 1.0) * 0.5));
	} else {
		tmp = 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)) <= 5e+24)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(Float64(4.0 * Float64(l_m / Om_m)), Float64(Float64(0.5 - Float64(cos(Float64(ky * 2.0)) * 0.5)) * Float64(l_m / Om_m)), 1.0))) + 1.0) * 0.5));
	else
		tmp = sqrt(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], 5e+24], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $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)))) < 5.00000000000000045e24

   1. Initial program 100.0%

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

   4. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-cos.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.7

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

   9. Applied rewrites 96.7%

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

   if 5.00000000000000045e24 < (*.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 94.9%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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 99.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(4 \cdot \frac{\ell}{Om}, \left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}} + 1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{l\_m \cdot 2}{Om\_m}\right)}^{2} \leq 200:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(4 \cdot \frac{l\_m}{Om\_m}\right) \cdot l\_m, \frac{\mathsf{fma}\left(-0.5, \cos \left(kx + kx\right), 0.5\right)}{Om\_m}, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om\_m}{\sin ky \cdot l\_m}, -0.25, 0.5\right)}\\ \end{array} \end{array} \]

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))
      200.0)
   (sqrt
    (fma
     (sqrt
      (/
       1.0
       (fma
        (* (* 4.0 (/ l_m Om_m)) l_m)
        (/ (fma -0.5 (cos (+ kx kx)) 0.5) Om_m)
        1.0)))
     0.5
     0.5))
   (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)) <= 200.0) {
		tmp = sqrt(fma(sqrt((1.0 / fma(((4.0 * (l_m / Om_m)) * l_m), (fma(-0.5, cos((kx + kx)), 0.5) / Om_m), 1.0))), 0.5, 0.5));
	} 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)) <= 200.0)
		tmp = sqrt(fma(sqrt(Float64(1.0 / fma(Float64(Float64(4.0 * Float64(l_m / Om_m)) * l_m), Float64(fma(-0.5, cos(Float64(kx + kx)), 0.5) / Om_m), 1.0))), 0.5, 0.5));
	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], 200.0], N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(N[(N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] * N[(N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] / Om$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision], 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)))) < 200

   1. Initial program 100.0%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 78.7%

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

   7. Applied rewrites 97.8%

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

   if 200 < (*.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 95.1%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 90.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 200:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(4 \cdot \frac{\ell}{Om}\right) \cdot \ell, \frac{\mathsf{fma}\left(-0.5, \cos \left(kx + kx\right), 0.5\right)}{Om}, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, -0.25, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.7% accurate, 0.9× speedup

Math

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

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))
      5.0)
   (sqrt 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)) <= 5.0) {
		tmp = sqrt(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)) <= 5.0)
		tmp = sqrt(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], 5.0], N[Sqrt[1.0], $MachinePrecision], 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)))) < 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 98.3%

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

   if 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 95.2%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 88.9%

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

4. Final simplification 93.7%

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

Alternative 9: 98.2% accurate, 1.1× speedup

Math

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

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))
      5.0)
   (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)) <= 5.0) {
		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)) <= 5.0d0) 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)) <= 5.0) {
		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)) <= 5.0:
		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)) <= 5.0)
		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)) <= 5.0)
		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], 5.0], 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)))) < 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 98.3%

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

   if 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 95.2%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 96.5%

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

4. Final simplification 97.4%

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

Alternative 10: 56.6% 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 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 57.6%

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

Reproduce

herbie shell --seed 2024236 
(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))))))))))