Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 98.4%

Time: 4.1s

Alternatives: 8

Speedup: 1.4×

Specification

Math

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

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (sqrt
 (*
  (/ 1 2)
  (+
   1
   (/
    1
    (sqrt
     (+
      1
      (*
       (pow (/ (* 2 l) Om) 2)
       (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))

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 / 2), $MachinePrecision] * N[(1 + N[(1 / N[Sqrt[N[(1 + N[(N[Power[N[(N[(2 * l), $MachinePrecision] / Om), $MachinePrecision], 2], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (sqrt
 (*
  (/ 1 2)
  (+
   1
   (/
    1
    (sqrt
     (+
      1
      (*
       (pow (/ (* 2 l) Om) 2)
       (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))

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 / 2), $MachinePrecision] * N[(1 + N[(1 / N[Sqrt[N[(1 + N[(N[Power[N[(N[(2 * l), $MachinePrecision] / Om), $MachinePrecision], 2], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|kx\right|, \left|ky\right|\right)\\ t_1 := \mathsf{min}\left(\left|kx\right|, \left|ky\right|\right)\\ \mathbf{if}\;t\_0 \leq \frac{3022314549036573}{604462909807314587353088}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{\frac{\ell + \ell}{Om} \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left(\ell + \ell\right)\right)}{Om}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\frac{\left(\frac{1}{2} \cdot \left(\left(1 - \cos \left(t\_0 + t\_0\right)\right) - \cos \left(t\_1 + t\_1\right)\right) - \frac{-1}{2}\right) \cdot \left(\left(4 \cdot \frac{\ell}{Om}\right) \cdot \ell\right) + Om}{Om}}} - \frac{-1}{2}}\\ \end{array} \]

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (let* ((t_0 (fmax (fabs kx) (fabs ky)))
       (t_1 (fmin (fabs kx) (fabs ky))))
  (if (<= t_0 3022314549036573/604462909807314587353088)
    (sqrt
     (*
      (/ 1 2)
      (+
       1
       (/
        1
        (sqrt
         (+ 1 (/ (* (/ (+ l l) Om) (* (* t_0 t_0) (+ l l))) Om)))))))
    (sqrt
     (-
      (/
       1/2
       (sqrt
        (/
         (+
          (*
           (-
            (* 1/2 (- (- 1 (cos (+ t_0 t_0))) (cos (+ t_1 t_1))))
            -1/2)
           (* (* 4 (/ l Om)) l))
          Om)
         Om)))
      -1/2)))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = fmax(fabs(kx), fabs(ky));
	double t_1 = fmin(fabs(kx), fabs(ky));
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	} else {
		tmp = sqrt(((0.5 / sqrt((((((0.5 * ((1.0 - cos((t_0 + t_0))) - cos((t_1 + t_1)))) - -0.5) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmax(abs(kx), abs(ky))
    t_1 = fmin(abs(kx), abs(ky))
    if (t_0 <= 5d-9) then
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((l + l) / om) * ((t_0 * t_0) * (l + l))) / om)))))))
    else
        tmp = sqrt(((0.5d0 / sqrt((((((0.5d0 * ((1.0d0 - cos((t_0 + t_0))) - cos((t_1 + t_1)))) - (-0.5d0)) * ((4.0d0 * (l / om)) * l)) + om) / om))) - (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = fmax(Math.abs(kx), Math.abs(ky));
	double t_1 = fmin(Math.abs(kx), Math.abs(ky));
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	} else {
		tmp = Math.sqrt(((0.5 / Math.sqrt((((((0.5 * ((1.0 - Math.cos((t_0 + t_0))) - Math.cos((t_1 + t_1)))) - -0.5) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = fmax(math.fabs(kx), math.fabs(ky))
	t_1 = fmin(math.fabs(kx), math.fabs(ky))
	tmp = 0
	if t_0 <= 5e-9:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))))
	else:
		tmp = math.sqrt(((0.5 / math.sqrt((((((0.5 * ((1.0 - math.cos((t_0 + t_0))) - math.cos((t_1 + t_1)))) - -0.5) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5))
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = fmax(abs(kx), abs(ky))
	t_1 = fmin(abs(kx), abs(ky))
	tmp = 0.0
	if (t_0 <= 5e-9)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(l + l) / Om) * Float64(Float64(t_0 * t_0) * Float64(l + l))) / Om)))))));
	else
		tmp = sqrt(Float64(Float64(0.5 / sqrt(Float64(Float64(Float64(Float64(Float64(0.5 * Float64(Float64(1.0 - cos(Float64(t_0 + t_0))) - cos(Float64(t_1 + t_1)))) - -0.5) * Float64(Float64(4.0 * Float64(l / Om)) * l)) + Om) / Om))) - -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = max(abs(kx), abs(ky));
	t_1 = min(abs(kx), abs(ky));
	tmp = 0.0;
	if (t_0 <= 5e-9)
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	else
		tmp = sqrt(((0.5 / sqrt((((((0.5 * ((1.0 - cos((t_0 + t_0))) - cos((t_1 + t_1)))) - -0.5) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Max[N[Abs[kx], $MachinePrecision], N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[kx], $MachinePrecision], N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 3022314549036573/604462909807314587353088], N[Sqrt[N[(N[(1 / 2), $MachinePrecision] * N[(1 + N[(1 / N[Sqrt[N[(1 + N[(N[(N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1/2 / N[Sqrt[N[(N[(N[(N[(N[(1/2 * N[(N[(1 - N[Cos[N[(t$95$0 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Cos[N[(t$95$1 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1/2), $MachinePrecision] * N[(N[(4 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1/2), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ky < 5.0000000000000001e-9

   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. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 88.4%

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 89.4%

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 89.4%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 80.8%

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

   7. Taylor expanded in ky around 0

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

      1. lower-pow.f64 77.0%

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

   9. Applied rewrites 77.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. associate-*r/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 76.6%

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

   if 5.0000000000000001e-9 < ky

   1. Initial program 98.2%

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

   3. Applied rewrites 79.1%

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

   5. Applied rewrites 91.3%

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

Alternative 2: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|kx\right|, \left|ky\right|\right)\\ \mathbf{if}\;t\_0 \leq \frac{3022314549036573}{604462909807314587353088}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{\frac{\ell + \ell}{Om} \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left(\ell + \ell\right)\right)}{Om}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\frac{\left(\frac{1}{2} \cdot \left(1 - \cos \left(2 \cdot t\_0\right)\right)\right) \cdot \left(\left(4 \cdot \frac{\ell}{Om}\right) \cdot \ell\right) + Om}{Om}}} - \frac{-1}{2}}\\ \end{array} \]

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (let* ((t_0 (fmax (fabs kx) (fabs ky))))
  (if (<= t_0 3022314549036573/604462909807314587353088)
    (sqrt
     (*
      (/ 1 2)
      (+
       1
       (/
        1
        (sqrt
         (+ 1 (/ (* (/ (+ l l) Om) (* (* t_0 t_0) (+ l l))) Om)))))))
    (sqrt
     (-
      (/
       1/2
       (sqrt
        (/
         (+ (* (* 1/2 (- 1 (cos (* 2 t_0)))) (* (* 4 (/ l Om)) l)) Om)
         Om)))
      -1/2)))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = fmax(fabs(kx), fabs(ky));
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	} else {
		tmp = sqrt(((0.5 / sqrt(((((0.5 * (1.0 - cos((2.0 * t_0)))) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmax(abs(kx), abs(ky))
    if (t_0 <= 5d-9) then
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((l + l) / om) * ((t_0 * t_0) * (l + l))) / om)))))))
    else
        tmp = sqrt(((0.5d0 / sqrt(((((0.5d0 * (1.0d0 - cos((2.0d0 * t_0)))) * ((4.0d0 * (l / om)) * l)) + om) / om))) - (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = fmax(Math.abs(kx), Math.abs(ky));
	double tmp;
	if (t_0 <= 5e-9) {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	} else {
		tmp = Math.sqrt(((0.5 / Math.sqrt(((((0.5 * (1.0 - Math.cos((2.0 * t_0)))) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = fmax(math.fabs(kx), math.fabs(ky))
	tmp = 0
	if t_0 <= 5e-9:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))))
	else:
		tmp = math.sqrt(((0.5 / math.sqrt(((((0.5 * (1.0 - math.cos((2.0 * t_0)))) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5))
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = fmax(abs(kx), abs(ky))
	tmp = 0.0
	if (t_0 <= 5e-9)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(l + l) / Om) * Float64(Float64(t_0 * t_0) * Float64(l + l))) / Om)))))));
	else
		tmp = sqrt(Float64(Float64(0.5 / sqrt(Float64(Float64(Float64(Float64(0.5 * Float64(1.0 - cos(Float64(2.0 * t_0)))) * Float64(Float64(4.0 * Float64(l / Om)) * l)) + Om) / Om))) - -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = max(abs(kx), abs(ky));
	tmp = 0.0;
	if (t_0 <= 5e-9)
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	else
		tmp = sqrt(((0.5 / sqrt(((((0.5 * (1.0 - cos((2.0 * t_0)))) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Max[N[Abs[kx], $MachinePrecision], N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 3022314549036573/604462909807314587353088], N[Sqrt[N[(N[(1 / 2), $MachinePrecision] * N[(1 + N[(1 / N[Sqrt[N[(1 + N[(N[(N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1/2 / N[Sqrt[N[(N[(N[(N[(1/2 * N[(1 - N[Cos[N[(2 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(4 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1/2), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 5.0000000000000001e-9

   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. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 88.4%

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 89.4%

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 89.4%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 80.8%

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

   7. Taylor expanded in ky around 0

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

      1. lower-pow.f64 77.0%

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

   9. Applied rewrites 77.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. associate-*r/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 76.6%

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

   if 5.0000000000000001e-9 < ky

   1. Initial program 98.2%

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in ky around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 68.2%

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

   6. Applied rewrites 68.2%

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

   8. Applied rewrites 79.4%

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

   9. Taylor expanded in kx around 0

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-*.f64 79.0%

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

   11. Applied rewrites 79.0%

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

Alternative 3: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|kx\right|, \left|ky\right|\right)\\ t_1 := \mathsf{min}\left(\left|kx\right|, \left|ky\right|\right)\\ \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq \frac{1152921504606847}{576460752303423488}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{2}}{\sqrt{\frac{\left(\frac{1}{2} \cdot \left(1 - \cos \left(t\_1 + t\_1\right)\right)\right) \cdot \left(\left(4 \cdot \frac{\ell}{Om}\right) \cdot \ell\right) + Om}{Om}}} - \frac{-1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{\frac{\ell + \ell}{Om} \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left(\ell + \ell\right)\right)}{Om}}}\right)}\\ \end{array} \]

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (let* ((t_0 (fmax (fabs kx) (fabs ky)))
       (t_1 (fmin (fabs kx) (fabs ky))))
  (if (<= (pow (/ (* 2 l) Om) 2) 1152921504606847/576460752303423488)
    (sqrt
     (-
      (/
       1/2
       (sqrt
        (/
         (+
          (* (* 1/2 (- 1 (cos (+ t_1 t_1)))) (* (* 4 (/ l Om)) l))
          Om)
         Om)))
      -1/2))
    (sqrt
     (*
      (/ 1 2)
      (+
       1
       (/
        1
        (sqrt
         (+
          1
          (/ (* (/ (+ l l) Om) (* (* t_0 t_0) (+ l l))) Om))))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = fmax(fabs(kx), fabs(ky));
	double t_1 = fmin(fabs(kx), fabs(ky));
	double tmp;
	if (pow(((2.0 * l) / Om), 2.0) <= 0.002) {
		tmp = sqrt(((0.5 / sqrt(((((0.5 * (1.0 - cos((t_1 + t_1)))) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	} else {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmax(abs(kx), abs(ky))
    t_1 = fmin(abs(kx), abs(ky))
    if ((((2.0d0 * l) / om) ** 2.0d0) <= 0.002d0) then
        tmp = sqrt(((0.5d0 / sqrt(((((0.5d0 * (1.0d0 - cos((t_1 + t_1)))) * ((4.0d0 * (l / om)) * l)) + om) / om))) - (-0.5d0)))
    else
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((l + l) / om) * ((t_0 * t_0) * (l + l))) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = fmax(Math.abs(kx), Math.abs(ky));
	double t_1 = fmin(Math.abs(kx), Math.abs(ky));
	double tmp;
	if (Math.pow(((2.0 * l) / Om), 2.0) <= 0.002) {
		tmp = Math.sqrt(((0.5 / Math.sqrt(((((0.5 * (1.0 - Math.cos((t_1 + t_1)))) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	} else {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = fmax(math.fabs(kx), math.fabs(ky))
	t_1 = fmin(math.fabs(kx), math.fabs(ky))
	tmp = 0
	if math.pow(((2.0 * l) / Om), 2.0) <= 0.002:
		tmp = math.sqrt(((0.5 / math.sqrt(((((0.5 * (1.0 - math.cos((t_1 + t_1)))) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5))
	else:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = fmax(abs(kx), abs(ky))
	t_1 = fmin(abs(kx), abs(ky))
	tmp = 0.0
	if ((Float64(Float64(2.0 * l) / Om) ^ 2.0) <= 0.002)
		tmp = sqrt(Float64(Float64(0.5 / sqrt(Float64(Float64(Float64(Float64(0.5 * Float64(1.0 - cos(Float64(t_1 + t_1)))) * Float64(Float64(4.0 * Float64(l / Om)) * l)) + Om) / Om))) - -0.5));
	else
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(l + l) / Om) * Float64(Float64(t_0 * t_0) * Float64(l + l))) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = max(abs(kx), abs(ky));
	t_1 = min(abs(kx), abs(ky));
	tmp = 0.0;
	if ((((2.0 * l) / Om) ^ 2.0) <= 0.002)
		tmp = sqrt(((0.5 / sqrt(((((0.5 * (1.0 - cos((t_1 + t_1)))) * ((4.0 * (l / Om)) * l)) + Om) / Om))) - -0.5));
	else
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Max[N[Abs[kx], $MachinePrecision], N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[kx], $MachinePrecision], N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[N[(N[(2 * l), $MachinePrecision] / Om), $MachinePrecision], 2], $MachinePrecision], 1152921504606847/576460752303423488], N[Sqrt[N[(N[(1/2 / N[Sqrt[N[(N[(N[(N[(1/2 * N[(1 - N[Cos[N[(t$95$1 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(4 * N[(l / Om), $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] + Om), $MachinePrecision] / Om), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1/2), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1 / 2), $MachinePrecision] * N[(1 + N[(1 / N[Sqrt[N[(1 + N[(N[(N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) < 2e-3

   1. Initial program 98.2%

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in ky around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 68.2%

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

   6. Applied rewrites 68.2%

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

   8. Applied rewrites 79.4%

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

   if 2e-3 < (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64))

   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. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 88.4%

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 89.4%

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 89.4%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 80.8%

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

   7. Taylor expanded in ky around 0

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

      1. lower-pow.f64 77.0%

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

   9. Applied rewrites 77.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. associate-*r/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 76.6%

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

Alternative 4: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|kx\right|, \left|ky\right|\right)\\ \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq \frac{1152921504606847}{576460752303423488}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \frac{\frac{\ell + \ell}{Om} \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left(\ell + \ell\right)\right)}{Om}}}\right)}\\ \end{array} \]

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (let* ((t_0 (fmax (fabs kx) (fabs ky))))
  (if (<= (pow (/ (* 2 l) Om) 2) 1152921504606847/576460752303423488)
    (sqrt 1)
    (sqrt
     (*
      (/ 1 2)
      (+
       1
       (/
        1
        (sqrt
         (+
          1
          (/ (* (/ (+ l l) Om) (* (* t_0 t_0) (+ l l))) Om))))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = fmax(fabs(kx), fabs(ky));
	double tmp;
	if (pow(((2.0 * l) / Om), 2.0) <= 0.002) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmax(abs(kx), abs(ky))
    if ((((2.0d0 * l) / om) ** 2.0d0) <= 0.002d0) then
        tmp = sqrt(1.0d0)
    else
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((l + l) / om) * ((t_0 * t_0) * (l + l))) / om)))))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = fmax(Math.abs(kx), Math.abs(ky));
	double tmp;
	if (Math.pow(((2.0 * l) / Om), 2.0) <= 0.002) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = fmax(math.fabs(kx), math.fabs(ky))
	tmp = 0
	if math.pow(((2.0 * l) / Om), 2.0) <= 0.002:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = fmax(abs(kx), abs(ky))
	tmp = 0.0
	if ((Float64(Float64(2.0 * l) / Om) ^ 2.0) <= 0.002)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(l + l) / Om) * Float64(Float64(t_0 * t_0) * Float64(l + l))) / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = max(abs(kx), abs(ky));
	tmp = 0.0;
	if ((((2.0 * l) / Om) ^ 2.0) <= 0.002)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((l + l) / Om) * ((t_0 * t_0) * (l + l))) / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Max[N[Abs[kx], $MachinePrecision], N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[N[(N[(2 * l), $MachinePrecision] / Om), $MachinePrecision], 2], $MachinePrecision], 1152921504606847/576460752303423488], N[Sqrt[1], $MachinePrecision], N[Sqrt[N[(N[(1 / 2), $MachinePrecision] * N[(1 + N[(1 / N[Sqrt[N[(1 + N[(N[(N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(l + l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) < 2e-3

   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. Taylor expanded in l around inf

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

   4. Applied rewrites 57.0%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 61.5%

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

   if 2e-3 < (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64))

   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. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 88.4%

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

   4. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 89.4%

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

      10. lift-*.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 89.4%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. sqr-sin-a N/A

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

      18. metadata-eval N/A

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

      19. lower--.f64 N/A

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

   6. Applied rewrites 80.8%

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

   7. Taylor expanded in ky around 0

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

      1. lower-pow.f64 77.0%

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

   9. Applied rewrites 77.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. associate-*r/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 76.6%

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

Alternative 5: 97.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (sqrt
 (*
  (/ 1 2)
  (+
   1
   (/
    1
    (sqrt
     (+
      1
      (*
       (pow (/ (* 2 l) Om) 2)
       (pow (sin (fmax (fabs kx) (fabs ky))) 2)))))))))

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(fmax(fabs(kx), fabs(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(fmax(abs(kx), abs(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(fmax(Math.abs(kx), Math.abs(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(fmax(math.fabs(kx), math.fabs(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) * (sin(fmax(abs(kx), abs(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(max(abs(kx), abs(ky))) ^ 2.0))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1 / 2), $MachinePrecision] * N[(1 + N[(1 / N[Sqrt[N[(1 + N[(N[Power[N[(N[(2 * l), $MachinePrecision] / Om), $MachinePrecision], 2], $MachinePrecision] * N[Power[N[Sin[N[Max[N[Abs[kx], $MachinePrecision], N[Abs[ky], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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. Taylor expanded in kx around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 88.4%

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

4. Applied rewrites 88.4%

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

Alternative 6: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \leq \frac{7926335344172073}{9007199254740992}:\\ \;\;\;\;\sqrt{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \]

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (if (<=
     (sqrt
      (*
       (/ 1 2)
       (+
        1
        (/
         1
         (sqrt
          (+
           1
           (*
            (pow (/ (* 2 l) Om) 2)
            (+ (pow (sin kx) 2) (pow (sin ky) 2)))))))))
     7926335344172073/9007199254740992)
  (sqrt 1/2)
  (sqrt 1)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (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))))))))) <= 0.88) {
		tmp = sqrt(0.5);
	} else {
		tmp = sqrt(1.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (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))))))))) <= 0.88d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = sqrt(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (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))))))))) <= 0.88) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = Math.sqrt(1.0);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if 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))))))))) <= 0.88:
		tmp = math.sqrt(0.5)
	else:
		tmp = math.sqrt(1.0)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (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))))))))) <= 0.88)
		tmp = sqrt(0.5);
	else
		tmp = sqrt(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (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))))))))) <= 0.88)
		tmp = sqrt(0.5);
	else
		tmp = sqrt(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[Sqrt[N[(N[(1 / 2), $MachinePrecision] * N[(1 + N[(1 / N[Sqrt[N[(1 + N[(N[Power[N[(N[(2 * l), $MachinePrecision] / Om), $MachinePrecision], 2], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 7926335344172073/9007199254740992], N[Sqrt[1/2], $MachinePrecision], N[Sqrt[1], $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)))))))))) < 0.88

   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. Taylor expanded in l around inf

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

   4. Applied rewrites 57.0%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]

   if 0.88 < (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 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. Taylor expanded in l around inf

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

   4. Applied rewrites 57.0%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]

   5. Taylor expanded in l around 0

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

   7. Applied rewrites 61.5%

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

Alternative 7: 57.0% accurate, 52.8× speedup

Math

\[\sqrt{\frac{1}{2}} \]

FPCore

(FPCore (l Om kx ky)
  :precision binary64
  (sqrt 1/2))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

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(0.5d0)
end function

Java

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

Python

def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

function code(l, Om, kx, ky)
	return sqrt(0.5)
end

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[1/2], $MachinePrecision]

Derivation

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. Taylor expanded in l around inf

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

4. Applied rewrites 57.0%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))