Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.6% → 99.7%

Time: 4.8s

Alternatives: 3

Speedup: N/A×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx, ky)
use fmin_fmax_functions
    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: 98.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, N/A× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0
         (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)))))))))))
   (if (<= t_0 2.0)
     t_0
     (sqrt
      (*
       (/ 1.0 2.0)
       (+
        1.0
        (/
         1.0
         (*
          (/ (* l_m 2.0) Om_m)
          (hypot (pow (sin ky) 1.0) (pow (sin kx) 1.0))))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = 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)))))))));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (((l_m * 2.0) / Om_m) * hypot(pow(sin(ky), 1.0), pow(sin(kx), 1.0)))))));
	}
	return tmp;
}

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

Python

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

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	t_0 = 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)))))))))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(l_m * 2.0) / Om_m) * hypot((sin(ky) ^ 1.0), (sin(kx) ^ 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)
	t_0 = 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)))))))));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (((l_m * 2.0) / Om_m) * hypot((sin(ky) ^ 1.0), (sin(kx) ^ 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_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[(N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision] * N[Sqrt[N[Power[N[Sin[ky], $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[Power[N[Sin[kx], $MachinePrecision], 1.0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. sqr-pow N/A

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

      9. sqr-pow N/A

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

      10. lower-hypot.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 69.3

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

   5. Applied rewrites 69.3%

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

Alternative 2: 99.0% accurate, N/A× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (sqrt
       (+
        1.0
        (*
         (pow (/ (* 2.0 l_m) Om_m) 2.0)
         (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      2.0)
   (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt 1.0)))))
   (sqrt
    (*
     (/ 1.0 2.0)
     (+
      1.0
      (/
       1.0
       (*
        (/ (* l_m 2.0) Om_m)
        (hypot (pow (sin ky) 1.0) (pow (sin kx) 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 (sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 2.0) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt(1.0)))));
	} else {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (((l_m * 2.0) / Om_m) * hypot(pow(sin(ky), 1.0), pow(sin(kx), 1.0)))))));
	}
	return tmp;
}

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 (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))))) <= 2.0) {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt(1.0)))));
	} else {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (((l_m * 2.0) / Om_m) * Math.hypot(Math.pow(Math.sin(ky), 1.0), Math.pow(Math.sin(kx), 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 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))))) <= 2.0:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt(1.0)))))
	else:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (((l_m * 2.0) / Om_m) * math.hypot(math.pow(math.sin(ky), 1.0), math.pow(math.sin(kx), 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 (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2.0)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(1.0)))));
	else
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / Float64(Float64(Float64(l_m * 2.0) / Om_m) * hypot((sin(ky) ^ 1.0), (sin(kx) ^ 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 (sqrt((1.0 + ((((2.0 * l_m) / Om_m) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2.0)
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt(1.0)))));
	else
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (((l_m * 2.0) / Om_m) * hypot((sin(ky) ^ 1.0), (sin(kx) ^ 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[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], 2.0], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[(N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision] * N[Sqrt[N[Power[N[Sin[ky], $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[Power[N[Sin[kx], $MachinePrecision], 1.0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 97.7%

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

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

   1. Initial program 95.5%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. sqr-pow N/A

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

      9. sqr-pow N/A

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

      10. lower-hypot.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-sin.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 3: 49.1% accurate, N/A× speedup

Math

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

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
     (*
      (/ (* l_m 2.0) Om_m)
      (hypot (pow (sin ky) 1.0) (pow (sin kx) 1.0))))))))

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 / (((l_m * 2.0) / Om_m) * hypot(pow(sin(ky), 1.0), pow(sin(kx), 1.0)))))));
}

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 / (((l_m * 2.0) / Om_m) * Math.hypot(Math.pow(Math.sin(ky), 1.0), Math.pow(Math.sin(kx), 1.0)))))));
}

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 / (((l_m * 2.0) / Om_m) * math.hypot(math.pow(math.sin(ky), 1.0), math.pow(math.sin(kx), 1.0)))))))

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 / Float64(Float64(Float64(l_m * 2.0) / Om_m) * hypot((sin(ky) ^ 1.0), (sin(kx) ^ 1.0)))))))
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 / (((l_m * 2.0) / Om_m) * hypot((sin(ky) ^ 1.0), (sin(kx) ^ 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_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[(N[(N[(l$95$m * 2.0), $MachinePrecision] / Om$95$m), $MachinePrecision] * N[Sqrt[N[Power[N[Sin[ky], $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[Power[N[Sin[kx], $MachinePrecision], 1.0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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

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

   1. associate-*r* N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. sqr-pow N/A

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

   9. sqr-pow N/A

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

   10. lower-hypot.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-sin.f64 52.9

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

5. Applied rewrites 52.9%

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

Reproduce

herbie shell --seed 2025066 
(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))))))))))