Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 98.4%

Time: 7.0s

Alternatives: 8

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;ky\_m \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(ky\_m \cdot ky\_m, t\_1, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(1 - \mathsf{fma}\left(0.5, \cos \left(2 \cdot kx\_m\right), 0.5 \cdot \cos \left(2 \cdot ky\_m\right)\right), t\_1, 1\right)}} \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l l) Om)) (t_1 (* t_0 t_0)))
   (if (<= ky_m 5e-9)
     (sqrt (+ 0.5 (* (/ 1.0 (sqrt (fma (* ky_m ky_m) t_1 1.0))) 0.5)))
     (sqrt
      (+
       0.5
       (*
        (/
         1.0
         (sqrt
          (fma
           (- 1.0 (fma 0.5 (cos (* 2.0 kx_m)) (* 0.5 (cos (* 2.0 ky_m)))))
           t_1
           1.0)))
        0.5))))))

C

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

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	t_0 = Float64(Float64(l + l) / Om)
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (ky_m <= 5e-9)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(ky_m * ky_m), t_1, 1.0))) * 0.5)));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(1.0 - fma(0.5, cos(Float64(2.0 * kx_m)), Float64(0.5 * cos(Float64(2.0 * ky_m))))), t_1, 1.0))) * 0.5)));
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[ky$95$m, 5e-9], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(ky$95$m * ky$95$m), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(1.0 - N[(0.5 * N[Cos[N[(2.0 * kx$95$m), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ky < 5.0000000000000001e-9

   1. Initial program 93.3%

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in kx around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 66.8

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

   6. Applied rewrites 66.8%

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

   7. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 93.3

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

   9. Applied rewrites 93.3%

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

   if 5.0000000000000001e-9 < ky

   1. Initial program 100.0%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around inf

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0))))))))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0)))))))));
}

Fortran

kx_m =     private
ky_m =     private
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
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_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0)))))))))
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	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_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0)))))))));
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	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_m), 2.0) + math.pow(math.sin(ky_m), 2.0)))))))))

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	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_m) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))))
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp = code(l, Om, kx_m, ky_m)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))));
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

Alternative 3: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;ky\_m \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(ky\_m \cdot ky\_m, t\_1, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\_m\right), t\_1, 1\right)}} \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l l) Om)) (t_1 (* t_0 t_0)))
   (if (<= ky_m 5e-8)
     (sqrt (+ 0.5 (* (/ 1.0 (sqrt (fma (* ky_m ky_m) t_1 1.0))) 0.5)))
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 ky_m)))) t_1 1.0)))
        0.5))))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double t_0 = (l + l) / Om;
	double t_1 = t_0 * t_0;
	double tmp;
	if (ky_m <= 5e-8) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((ky_m * ky_m), t_1, 1.0))) * 0.5)));
	} else {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * ky_m)))), t_1, 1.0))) * 0.5)));
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	t_0 = Float64(Float64(l + l) / Om)
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (ky_m <= 5e-8)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(ky_m * ky_m), t_1, 1.0))) * 0.5)));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky_m)))), t_1, 1.0))) * 0.5)));
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[ky$95$m, 5e-8], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(ky$95$m * ky$95$m), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ky < 4.9999999999999998e-8

   1. Initial program 93.4%

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

   3. Applied rewrites 66.9%

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

   4. Taylor expanded in kx around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 66.9

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

   6. Applied rewrites 66.9%

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

   7. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 93.4

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

   9. Applied rewrites 93.4%

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

   if 4.9999999999999998e-8 < ky

   1. Initial program 100.0%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in kx around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 4: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky\_m, kx\_m\right)}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (/ 1.0 2.0)
        (+
         1.0
         (/
          1.0
          (sqrt
           (+
            1.0
            (*
             (pow (/ (* 2.0 l) Om) 2.0)
             (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))))))
      0.8)
   (sqrt (fma 0.25 (* (/ Om l) (/ 1.0 (hypot (sin ky_m) kx_m))) 0.5))
   1.0))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	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_m), 2.0) + pow(sin(ky_m), 2.0))))))))) <= 0.8) {
		tmp = sqrt(fma(0.25, ((Om / l) * (1.0 / hypot(sin(ky_m), kx_m))), 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	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_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))))))) <= 0.8)
		tmp = sqrt(fma(0.25, Float64(Float64(Om / l) * Float64(1.0 / hypot(sin(ky_m), kx_m))), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[Sqrt[N[(0.25 * N[(N[(Om / l), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + kx$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], 1.0]

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)))))))))) < 0.80000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in l around inf

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 98.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, 0.5\right)}} \]

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 98.6%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, kx\right)}, 0.5\right)} \]

   if 0.80000000000000004 < (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 97.2%

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

   2. Taylor expanded in l around 0

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

      1. metadata-eval N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \sqrt{1} \]

      5. metadata-eval 97.3

         \[\leadsto 1 \]

   4. Applied rewrites 97.3%

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

Alternative 5: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)}}\right)} \leq 0.8:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\ell \cdot \sin ky\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (/ 1.0 2.0)
        (+
         1.0
         (/
          1.0
          (sqrt
           (+
            1.0
            (*
             (pow (/ (* 2.0 l) Om) 2.0)
             (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))))))
      0.8)
   (sqrt (fma 0.25 (/ Om (* l (sin ky_m))) 0.5))
   1.0))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	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_m), 2.0) + pow(sin(ky_m), 2.0))))))))) <= 0.8) {
		tmp = sqrt(fma(0.25, (Om / (l * sin(ky_m))), 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	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_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))))))) <= 0.8)
		tmp = sqrt(fma(0.25, Float64(Om / Float64(l * sin(ky_m))), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[Sqrt[N[(0.25 * N[(Om / N[(l * N[Sin[ky$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], 1.0]

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)))))))))) < 0.80000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in l around inf

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

      1. metadata-eval N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 98.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, 0.5\right)}} \]

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 98.7

         \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\ell \cdot \sin ky}, 0.5\right)} \]

   7. Applied rewrites 98.7%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\color{blue}{\ell \cdot \sin ky}}, 0.5\right)} \]

   if 0.80000000000000004 < (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 97.2%

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

   2. Taylor expanded in l around 0

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

      1. metadata-eval N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \sqrt{1} \]

      5. metadata-eval 97.3

         \[\leadsto 1 \]

   4. Applied rewrites 97.3%

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

Alternative 6: 97.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(ky\_m \cdot ky\_m, t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l l) Om)))
   (if (<= (pow (/ (* 2.0 l) Om) 2.0) 0.2)
     1.0
     (sqrt
      (+ 0.5 (* (/ 1.0 (sqrt (fma (* ky_m ky_m) (* t_0 t_0) 1.0))) 0.5))))))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	double t_0 = (l + l) / Om;
	double tmp;
	if (pow(((2.0 * l) / Om), 2.0) <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((ky_m * ky_m), (t_0 * t_0), 1.0))) * 0.5)));
	}
	return tmp;
}

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	t_0 = Float64(Float64(l + l) / Om)
	tmp = 0.0
	if ((Float64(Float64(2.0 * l) / Om) ^ 2.0) <= 0.2)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(ky_m * ky_m), Float64(t_0 * t_0), 1.0))) * 0.5)));
	end
	return tmp
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision], 0.2], 1.0, N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(ky$95$m * ky$95$m), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $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)) < 0.20000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0

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

      1. metadata-eval N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \sqrt{1} \]

      5. metadata-eval 99.2

         \[\leadsto 1 \]

   4. Applied rewrites 99.2%

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

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

   1. Initial program 96.9%

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in kx around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 84.2

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

   6. Applied rewrites 84.2%

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

   7. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 95.3

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

   9. Applied rewrites 95.3%

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

Alternative 7: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)}}\right)} \leq 0.85:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m)
 :precision binary64
 (if (<=
      (sqrt
       (*
        (/ 1.0 2.0)
        (+
         1.0
         (/
          1.0
          (sqrt
           (+
            1.0
            (*
             (pow (/ (* 2.0 l) Om) 2.0)
             (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))))))
      0.85)
   (sqrt 0.5)
   1.0))

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	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_m), 2.0) + pow(sin(ky_m), 2.0))))))))) <= 0.85) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

kx_m =     private
ky_m =     private
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
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_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))))))))) <= 0.85d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	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_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))))))))) <= 0.85) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	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_m), 2.0) + math.pow(math.sin(ky_m), 2.0))))))))) <= 0.85:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	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_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))))))) <= 0.85)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp_2 = code(l, Om, kx_m, ky_m)
	tmp = 0.0;
	if (sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))))))) <= 0.85)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := If[LessEqual[N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.85], N[Sqrt[0.5], $MachinePrecision], 1.0]

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)))))))))) < 0.849999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 98.1%

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

   if 0.849999999999999978 < (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 97.2%

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

   2. Taylor expanded in l around 0

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

      1. metadata-eval N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

         \[\leadsto \sqrt{1} \]

      5. metadata-eval 97.3

         \[\leadsto 1 \]

   4. Applied rewrites 97.3%

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

Alternative 8: 62.2% accurate, 142.7× speedup

Math

\[\begin{array}{l} kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l, Om, kx_m, ky_m] = \mathsf{sort}([l, Om, kx_m, ky_m])\\ \\ 1 \end{array} \]

FPCore

kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l Om kx_m ky_m) :precision binary64 1.0)

C

kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l < Om && Om < kx_m && kx_m < ky_m);
double code(double l, double Om, double kx_m, double ky_m) {
	return 1.0;
}

Fortran

kx_m =     private
ky_m =     private
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
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_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = 1.0d0
end function

Java

kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l < Om && Om < kx_m && kx_m < ky_m;
public static double code(double l, double Om, double kx_m, double ky_m) {
	return 1.0;
}

Python

kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l, Om, kx_m, ky_m] = sort([l, Om, kx_m, ky_m])
def code(l, Om, kx_m, ky_m):
	return 1.0

Julia

kx_m = abs(kx)
ky_m = abs(ky)
l, Om, kx_m, ky_m = sort([l, Om, kx_m, ky_m])
function code(l, Om, kx_m, ky_m)
	return 1.0
end

MATLAB

kx_m = abs(kx);
ky_m = abs(ky);
l, Om, kx_m, ky_m = num2cell(sort([l, Om, kx_m, ky_m])){:}
function tmp = code(l, Om, kx_m, ky_m)
	tmp = 1.0;
end

Wolfram

kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l, Om, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l_, Om_, kx$95$m_, ky$95$m_] := 1.0

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in l around 0

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

   1. metadata-eval N/A

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

   2. sqrt-unprod N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   5. metadata-eval 62.2

      \[\leadsto 1 \]

4. Applied rewrites 62.2%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))