Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 98.1%

Time: 16.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% 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: 98.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 10^{+68}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{l\_m \cdot \left(1 - 0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)}{Om\_m} \cdot \frac{l\_m \cdot 4}{Om\_m}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 1e+68)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (*
         (/ (* l_m (- 1.0 (* 0.5 (+ (cos (* 2.0 kx)) (cos (* 2.0 ky)))))) Om_m)
         (/ (* l_m 4.0) Om_m)))))))
   (sqrt 0.5)))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 1e+68) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l_m * (1.0 - (0.5 * (cos((2.0 * kx)) + cos((2.0 * ky)))))) / Om_m) * ((l_m * 4.0) / Om_m)))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
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 (((2.0d0 * l_m) / om_m) <= 1d+68) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((l_m * (1.0d0 - (0.5d0 * (cos((2.0d0 * kx)) + cos((2.0d0 * ky)))))) / om_m) * ((l_m * 4.0d0) / om_m)))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 1e+68) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((l_m * (1.0 - (0.5 * (Math.cos((2.0 * kx)) + Math.cos((2.0 * ky)))))) / Om_m) * ((l_m * 4.0) / Om_m)))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 1e+68:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((l_m * (1.0 - (0.5 * (math.cos((2.0 * kx)) + math.cos((2.0 * ky)))))) / Om_m) * ((l_m * 4.0) / Om_m)))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 1e+68)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(l_m * Float64(1.0 - Float64(0.5 * Float64(cos(Float64(2.0 * kx)) + cos(Float64(2.0 * ky)))))) / Om_m) * Float64(Float64(l_m * 4.0) / Om_m)))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 1e+68)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l_m * (1.0 - (0.5 * (cos((2.0 * kx)) + cos((2.0 * ky)))))) / Om_m) * ((l_m * 4.0) / Om_m)))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 1e+68], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(l$95$m * N[(1.0 - N[(0.5 * N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision] * N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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) < 9.99999999999999953e67

   1. Initial program 98.6%

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 83.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell\right) \cdot \left(\ell \cdot 4\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om} \cdot \frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om}\right), \left(\frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 95.3%

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

   7. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\ell \cdot \left(1 - \left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right)\right)}, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(1 - \left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right) \cdot \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(1 - \left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      4. distribute-lft-out N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{1}{2} \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\cos \left(2 \cdot kx\right), \cos \left(2 \cdot ky\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot kx\right)\right), \cos \left(2 \cdot ky\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \cos \left(2 \cdot ky\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \mathsf{cos.f64}\left(\left(2 \cdot ky\right)\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 95.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 95.3%

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

   if 9.99999999999999953e67 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   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 l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right) \]
   4. Step-by-step derivation

   5. Simplified 96.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+68}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot \left(1 - 0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)}{Om} \cdot \frac{\ell \cdot 4}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

l_m = abs(l)
Om_m = abs(om)
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 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l_m) / om_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, 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$95$m), $MachinePrecision] / Om$95$m), $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]

Derivation

1. Initial program 98.4%

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

Alternative 3: 89.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 10000000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left(\left(l\_m \cdot 4\right) \cdot \frac{l\_m}{Om\_m}\right) \cdot \left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right)}{Om\_m}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{l\_m \cdot 4}{Om\_m} \cdot \frac{l\_m \cdot \left(ky \cdot ky\right)}{Om\_m}}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 10000000000000.0)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (/
         (* (* (* l_m 4.0) (/ l_m Om_m)) (+ 0.5 (* (cos (* 2.0 ky)) -0.5)))
         Om_m))))))
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt (+ 1.0 (* (/ (* l_m 4.0) Om_m) (/ (* l_m (* ky ky)) Om_m)))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 10000000000000.0) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((((l_m * 4.0) * (l_m / Om_m)) * (0.5 + (cos((2.0 * ky)) * -0.5))) / Om_m))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l_m * 4.0) / Om_m) * ((l_m * (ky * ky)) / Om_m)))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
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 (((2.0d0 * l_m) / om_m) <= 10000000000000.0d0) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((((l_m * 4.0d0) * (l_m / om_m)) * (0.5d0 + (cos((2.0d0 * ky)) * (-0.5d0)))) / om_m))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((l_m * 4.0d0) / om_m) * ((l_m * (ky * ky)) / om_m)))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 10000000000000.0) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((((l_m * 4.0) * (l_m / Om_m)) * (0.5 + (Math.cos((2.0 * ky)) * -0.5))) / Om_m))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((l_m * 4.0) / Om_m) * ((l_m * (ky * ky)) / Om_m)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 10000000000000.0:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((((l_m * 4.0) * (l_m / Om_m)) * (0.5 + (math.cos((2.0 * ky)) * -0.5))) / Om_m))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((l_m * 4.0) / Om_m) * ((l_m * (ky * ky)) / Om_m)))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 10000000000000.0)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(l_m * 4.0) * Float64(l_m / Om_m)) * Float64(0.5 + Float64(cos(Float64(2.0 * ky)) * -0.5))) / Om_m))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(l_m * 4.0) / Om_m) * Float64(Float64(l_m * Float64(ky * ky)) / Om_m)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 10000000000000.0)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((((l_m * 4.0) * (l_m / Om_m)) * (0.5 + (cos((2.0 * ky)) * -0.5))) / Om_m))))));
	else
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l_m * 4.0) / Om_m) * ((l_m * (ky * ky)) / Om_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 10000000000000.0], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(N[(l$95$m * 4.0), $MachinePrecision] * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision] * N[(N[(l$95$m * N[(ky * ky), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1e13

   1. Initial program 98.5%

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 85.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\sqrt{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(4 \cdot {\ell}^{2}\right) \cdot {\sin ky}^{2}\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(4 \cdot {\ell}^{2}\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left({\ell}^{2}\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \left(\ell \cdot \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \left({\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\sin ky, 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \left(Om \cdot Om\right)\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 79.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(ky\right), 2\right)\right), \mathsf{*.f64}\left(Om, Om\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 79.3%

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

      1. pow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\sin ky \cdot \sin ky\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\sin ky \cdot \sin ky}{Om}\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{4 \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \left(\sin ky \cdot \sin ky\right)}{Om}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{4 \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \left(\sin ky \cdot \sin ky\right)\right), Om\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 86.3%

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

   if 1e13 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 98.2%

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 76.4%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell\right) \cdot \left(\ell \cdot 4\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om} \cdot \frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om}\right), \left(\frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 91.1%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \color{blue}{\left({ky}^{2}\right)}\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \left(ky \cdot ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 97.1%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \mathsf{*.f64}\left(ky, ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 97.1%

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

   10. Taylor expanded in kx around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left({ky}^{2} \cdot \ell\right)}, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({ky}^{2}\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(ky \cdot ky\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 83.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, ky\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 83.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10000000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left(\left(\ell \cdot 4\right) \cdot \frac{\ell}{Om}\right) \cdot \left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right)}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot 4}{Om} \cdot \frac{\ell \cdot \left(ky \cdot ky\right)}{Om}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} t_0 := \frac{l\_m \cdot 4}{Om\_m}\\ \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 40000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + t\_0 \cdot \frac{l\_m \cdot \left(0.5 + \cos \left(2 \cdot kx\right) \cdot -0.5\right)}{Om\_m}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + t\_0 \cdot \frac{l\_m \cdot \left(ky \cdot ky\right)}{Om\_m}}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* l_m 4.0) Om_m)))
   (if (<= (/ (* 2.0 l_m) Om_m) 40000.0)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+ 1.0 (* t_0 (/ (* l_m (+ 0.5 (* (cos (* 2.0 kx)) -0.5))) Om_m)))))))
     (sqrt
      (+ 0.5 (/ 0.5 (sqrt (+ 1.0 (* t_0 (/ (* l_m (* ky ky)) Om_m))))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = (l_m * 4.0) / Om_m;
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 40000.0) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (t_0 * ((l_m * (0.5 + (cos((2.0 * kx)) * -0.5))) / Om_m)))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (t_0 * ((l_m * (ky * ky)) / Om_m)))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
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 * 4.0d0) / om_m
    if (((2.0d0 * l_m) / om_m) <= 40000.0d0) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (t_0 * ((l_m * (0.5d0 + (cos((2.0d0 * kx)) * (-0.5d0)))) / om_m)))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (t_0 * ((l_m * (ky * ky)) / om_m)))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = (l_m * 4.0) / Om_m;
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 40000.0) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (t_0 * ((l_m * (0.5 + (Math.cos((2.0 * kx)) * -0.5))) / Om_m)))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (t_0 * ((l_m * (ky * ky)) / Om_m)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	t_0 = (l_m * 4.0) / Om_m
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 40000.0:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (t_0 * ((l_m * (0.5 + (math.cos((2.0 * kx)) * -0.5))) / Om_m)))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (t_0 * ((l_m * (ky * ky)) / Om_m)))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(l_m * 4.0) / Om_m)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 40000.0)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(t_0 * Float64(Float64(l_m * Float64(0.5 + Float64(cos(Float64(2.0 * kx)) * -0.5))) / Om_m)))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(t_0 * Float64(Float64(l_m * Float64(ky * ky)) / Om_m)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	t_0 = (l_m * 4.0) / Om_m;
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 40000.0)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (t_0 * ((l_m * (0.5 + (cos((2.0 * kx)) * -0.5))) / Om_m)))))));
	else
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (t_0 * ((l_m * (ky * ky)) / Om_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := Block[{t$95$0 = N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 40000.0], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(t$95$0 * N[(N[(l$95$m * N[(0.5 + N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(t$95$0 * N[(N[(l$95$m * N[(ky * ky), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 4e4

   1. Initial program 98.5%

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell\right) \cdot \left(\ell \cdot 4\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om} \cdot \frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om}\right), \left(\frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 95.3%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right)}, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right) \cdot \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot kx\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot kx\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \cos \left(2 \cdot kx\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot kx\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 87.4%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 87.4%

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

   if 4e4 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 98.2%

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell\right) \cdot \left(\ell \cdot 4\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om} \cdot \frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om}\right), \left(\frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 91.4%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \color{blue}{\left({ky}^{2}\right)}\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \left(ky \cdot ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 97.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \mathsf{*.f64}\left(ky, ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 97.0%

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

   10. Taylor expanded in kx around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left({ky}^{2} \cdot \ell\right)}, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({ky}^{2}\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(ky \cdot ky\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 82.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, ky\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 82.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 40000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot 4}{Om} \cdot \frac{\ell \cdot \left(0.5 + \cos \left(2 \cdot kx\right) \cdot -0.5\right)}{Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\ell \cdot 4}{Om} \cdot \frac{\ell \cdot \left(ky \cdot ky\right)}{Om}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 3.2× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 1.0)
   1.0
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt (+ 1.0 (* (/ (* l_m 4.0) Om_m) (/ (* l_m (* ky ky)) Om_m)))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l_m * 4.0) / Om_m) * ((l_m * (ky * ky)) / Om_m)))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
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 (((2.0d0 * l_m) / om_m) <= 1.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((l_m * 4.0d0) / om_m) * ((l_m * (ky * ky)) / om_m)))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((l_m * 4.0) / Om_m) * ((l_m * (ky * ky)) / Om_m)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if ((2.0 * l_m) / Om_m) <= 1.0:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((l_m * 4.0) / Om_m) * ((l_m * (ky * ky)) / Om_m)))))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 1.0)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(l_m * 4.0) / Om_m) * Float64(Float64(l_m * Float64(ky * ky)) / Om_m)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (((2.0 * l_m) / Om_m) <= 1.0)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((l_m * 4.0) / Om_m) * ((l_m * (ky * ky)) / Om_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 1.0], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision] * N[(N[(l$95$m * N[(ky * ky), $MachinePrecision]), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1

   1. Initial program 98.5%

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 85.5%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0

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

   7. Simplified 75.1%

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

   if 1 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 98.3%

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

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 77.4%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left(\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell\right) \cdot \left(\ell \cdot 4\right)}{Om \cdot Om}\right)\right)\right)\right)\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om} \cdot \frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \ell}{Om}\right), \left(\frac{\ell \cdot 4}{Om}\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 91.6%

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

   7. Taylor expanded in ky around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \color{blue}{\left({ky}^{2}\right)}\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \left(ky \cdot ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 96.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right)\right)\right), \mathsf{*.f64}\left(ky, ky\right)\right)\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 96.0%

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

   10. Taylor expanded in kx around 0

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left({ky}^{2} \cdot \ell\right)}, Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

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

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({ky}^{2}\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(ky \cdot ky\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 81.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(ky, ky\right), \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, 4\right), Om\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 81.9%

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

4. Final simplification 76.7%

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

Alternative 6: 79.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 3.1 \cdot 10^{-44}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= Om_m 3.1e-44) (sqrt 0.5) 1.0))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 3.1e-44) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
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 (om_m <= 3.1d-44) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 3.1e-44) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if Om_m <= 3.1e-44:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Om_m <= 3.1e-44)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (Om_m <= 3.1e-44)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[Om$95$m, 3.1e-44], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 3.09999999999999984e-44

   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 l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right) \]
   4. Step-by-step derivation

   5. Simplified 61.8%

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

   if 3.09999999999999984e-44 < Om

   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. Step-by-step derivation

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

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   3. Simplified 79.4%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0

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

   7. Simplified 82.2%

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

Alternative 7: 62.8% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

l_m = abs(l)
Om_m = abs(om)
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

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	return 1.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

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

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right) \cdot \frac{1}{2}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} + 1\right) \cdot \frac{1}{2}\right)\right) \]

   4. distribute-rgt1-in N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2}\right), \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}} \cdot \frac{1}{2}\right)\right)\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1 \cdot \frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)\right)\right) \]

3. Simplified 83.6%

   \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \left(\ell \cdot \left(\ell \cdot 4\right)\right)}{Om \cdot Om}}}}} \]
4. Add Preprocessing

5. Taylor expanded in l around 0

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

7. Simplified 64.3%

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

Reproduce

herbie shell --seed 2024161 
(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))))))))))