Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 7.0s

Alternatives: 6

Speedup: 1.3×

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: 100.0% accurate, 1.2× speedup

Math

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

FPCore

ky_m = (fabs.f64 ky)
(FPCore (l Om kx ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l l) Om)))
   (if (<= ky_m 2e-8)
     (sqrt
      (*
       (/ 1.0 2.0)
       (+
        1.0
        (/
         1.0
         (sqrt
          (+ 1.0 (+ (pow (* t_0 (sin kx)) 2.0) (pow (* t_0 ky_m) 2.0))))))))
     (sqrt
      (+
       0.5
       (*
        (/
         1.0
         (sqrt
          (+
           1.0
           (*
            (* t_0 t_0)
            (- 1.0 (* 0.5 (+ (cos (+ kx kx)) (cos (+ ky_m ky_m)))))))))
        0.5))))))

C

ky_m = fabs(ky);
double code(double l, double Om, double kx, double ky_m) {
	double t_0 = (l + l) / Om;
	double tmp;
	if (ky_m <= 2e-8) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow((t_0 * sin(kx)), 2.0) + pow((t_0 * ky_m), 2.0))))))));
	} else {
		tmp = sqrt((0.5 + ((1.0 / sqrt((1.0 + ((t_0 * t_0) * (1.0 - (0.5 * (cos((kx + kx)) + cos((ky_m + ky_m))))))))) * 0.5)));
	}
	return tmp;
}

Fortran

ky_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l + l) / om
    if (ky_m <= 2d-8) then
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + (((t_0 * sin(kx)) ** 2.0d0) + ((t_0 * ky_m) ** 2.0d0))))))))
    else
        tmp = sqrt((0.5d0 + ((1.0d0 / sqrt((1.0d0 + ((t_0 * t_0) * (1.0d0 - (0.5d0 * (cos((kx + kx)) + cos((ky_m + ky_m))))))))) * 0.5d0)))
    end if
    code = tmp
end function

Java

ky_m = Math.abs(ky);
public static double code(double l, double Om, double kx, double ky_m) {
	double t_0 = (l + l) / Om;
	double tmp;
	if (ky_m <= 2e-8) {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow((t_0 * Math.sin(kx)), 2.0) + Math.pow((t_0 * ky_m), 2.0))))))));
	} else {
		tmp = Math.sqrt((0.5 + ((1.0 / Math.sqrt((1.0 + ((t_0 * t_0) * (1.0 - (0.5 * (Math.cos((kx + kx)) + Math.cos((ky_m + ky_m))))))))) * 0.5)));
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
def code(l, Om, kx, ky_m):
	t_0 = (l + l) / Om
	tmp = 0
	if ky_m <= 2e-8:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow((t_0 * math.sin(kx)), 2.0) + math.pow((t_0 * ky_m), 2.0))))))))
	else:
		tmp = math.sqrt((0.5 + ((1.0 / math.sqrt((1.0 + ((t_0 * t_0) * (1.0 - (0.5 * (math.cos((kx + kx)) + math.cos((ky_m + ky_m))))))))) * 0.5)))
	return tmp

Julia

ky_m = abs(ky)
function code(l, Om, kx, ky_m)
	t_0 = Float64(Float64(l + l) / Om)
	tmp = 0.0
	if (ky_m <= 2e-8)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(t_0 * sin(kx)) ^ 2.0) + (Float64(t_0 * ky_m) ^ 2.0))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(t_0 * t_0) * Float64(1.0 - Float64(0.5 * Float64(cos(Float64(kx + kx)) + cos(Float64(ky_m + ky_m))))))))) * 0.5)));
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
function tmp_2 = code(l, Om, kx, ky_m)
	t_0 = (l + l) / Om;
	tmp = 0.0;
	if (ky_m <= 2e-8)
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (((t_0 * sin(kx)) ^ 2.0) + ((t_0 * ky_m) ^ 2.0))))))));
	else
		tmp = sqrt((0.5 + ((1.0 / sqrt((1.0 + ((t_0 * t_0) * (1.0 - (0.5 * (cos((kx + kx)) + cos((ky_m + ky_m))))))))) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
code[l_, Om_, kx_, ky$95$m_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[ky$95$m, 2e-8], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(t$95$0 * N[Sin[kx], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(t$95$0 * ky$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 2e-8

   1. Initial program 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. lower-+.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 90.7%

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

   if 2e-8 < ky

   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 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. unpow2 N/A

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

      2. sqr-sin-a N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in kx around inf

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

      1. unpow2 N/A

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

      2. sqr-sin-a-rev N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. unpow2 N/A

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

      16. sqr-sin-a-rev N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-*.f64 N/A

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

   7. Applied rewrites 91.5%

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

   9. Applied rewrites 91.5%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\left(t\_0 \cdot \sin kx\right)}^{2} + {\left(t\_0 \cdot \sin ky\_m\right)}^{2}\right)}}\right)} \end{array} \end{array} \]

FPCore

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

C

ky_m = fabs(ky);
double code(double l, double Om, double kx, double ky_m) {
	double t_0 = (l + l) / Om;
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow((t_0 * sin(kx)), 2.0) + pow((t_0 * sin(ky_m)), 2.0))))))));
}

Fortran

ky_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    t_0 = (l + l) / om
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + (((t_0 * sin(kx)) ** 2.0d0) + ((t_0 * sin(ky_m)) ** 2.0d0))))))))
end function

Java

ky_m = Math.abs(ky);
public static double code(double l, double Om, double kx, double ky_m) {
	double t_0 = (l + l) / Om;
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow((t_0 * Math.sin(kx)), 2.0) + Math.pow((t_0 * Math.sin(ky_m)), 2.0))))))));
}

Python

ky_m = math.fabs(ky)
def code(l, Om, kx, ky_m):
	t_0 = (l + l) / Om
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow((t_0 * math.sin(kx)), 2.0) + math.pow((t_0 * math.sin(ky_m)), 2.0))))))))

Julia

ky_m = abs(ky)
function code(l, Om, kx, ky_m)
	t_0 = Float64(Float64(l + l) / Om)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(t_0 * sin(kx)) ^ 2.0) + (Float64(t_0 * sin(ky_m)) ^ 2.0))))))))
end

MATLAB

ky_m = abs(ky);
function tmp = code(l, Om, kx, ky_m)
	t_0 = (l + l) / Om;
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (((t_0 * sin(kx)) ^ 2.0) + ((t_0 * sin(ky_m)) ^ 2.0))))))));
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
code[l_, Om_, kx_, ky$95$m_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]}, N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(t$95$0 * N[Sin[kx], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(t$95$0 * N[Sin[ky$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. distribute-lft-in N/A

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

   11. lower-+.f64 N/A

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

3. Applied rewrites 100.0%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;ky\_m \leq 4 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\left(t\_0 \cdot \sin kx\right)}^{2} + {\left(t\_0 \cdot ky\_m\right)}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{1 + \left(t\_0 \cdot t\_0\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(ky\_m + ky\_m\right)\right)}} \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
(FPCore (l Om kx ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l l) Om)))
   (if (<= ky_m 4e-9)
     (sqrt
      (*
       (/ 1.0 2.0)
       (+
        1.0
        (/
         1.0
         (sqrt
          (+ 1.0 (+ (pow (* t_0 (sin kx)) 2.0) (pow (* t_0 ky_m) 2.0))))))))
     (sqrt
      (+
       0.5
       (*
        (/
         1.0
         (sqrt (+ 1.0 (* (* t_0 t_0) (- 0.5 (* 0.5 (cos (+ ky_m ky_m))))))))
        0.5))))))

C

ky_m = fabs(ky);
double code(double l, double Om, double kx, double ky_m) {
	double t_0 = (l + l) / Om;
	double tmp;
	if (ky_m <= 4e-9) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow((t_0 * sin(kx)), 2.0) + pow((t_0 * ky_m), 2.0))))))));
	} else {
		tmp = sqrt((0.5 + ((1.0 / sqrt((1.0 + ((t_0 * t_0) * (0.5 - (0.5 * cos((ky_m + ky_m)))))))) * 0.5)));
	}
	return tmp;
}

Fortran

ky_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l + l) / om
    if (ky_m <= 4d-9) then
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + (((t_0 * sin(kx)) ** 2.0d0) + ((t_0 * ky_m) ** 2.0d0))))))))
    else
        tmp = sqrt((0.5d0 + ((1.0d0 / sqrt((1.0d0 + ((t_0 * t_0) * (0.5d0 - (0.5d0 * cos((ky_m + ky_m)))))))) * 0.5d0)))
    end if
    code = tmp
end function

Java

ky_m = Math.abs(ky);
public static double code(double l, double Om, double kx, double ky_m) {
	double t_0 = (l + l) / Om;
	double tmp;
	if (ky_m <= 4e-9) {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow((t_0 * Math.sin(kx)), 2.0) + Math.pow((t_0 * ky_m), 2.0))))))));
	} else {
		tmp = Math.sqrt((0.5 + ((1.0 / Math.sqrt((1.0 + ((t_0 * t_0) * (0.5 - (0.5 * Math.cos((ky_m + ky_m)))))))) * 0.5)));
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
def code(l, Om, kx, ky_m):
	t_0 = (l + l) / Om
	tmp = 0
	if ky_m <= 4e-9:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow((t_0 * math.sin(kx)), 2.0) + math.pow((t_0 * ky_m), 2.0))))))))
	else:
		tmp = math.sqrt((0.5 + ((1.0 / math.sqrt((1.0 + ((t_0 * t_0) * (0.5 - (0.5 * math.cos((ky_m + ky_m)))))))) * 0.5)))
	return tmp

Julia

ky_m = abs(ky)
function code(l, Om, kx, ky_m)
	t_0 = Float64(Float64(l + l) / Om)
	tmp = 0.0
	if (ky_m <= 4e-9)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(t_0 * sin(kx)) ^ 2.0) + (Float64(t_0 * ky_m) ^ 2.0))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(t_0 * t_0) * Float64(0.5 - Float64(0.5 * cos(Float64(ky_m + ky_m)))))))) * 0.5)));
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
function tmp_2 = code(l, Om, kx, ky_m)
	t_0 = (l + l) / Om;
	tmp = 0.0;
	if (ky_m <= 4e-9)
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (((t_0 * sin(kx)) ^ 2.0) + ((t_0 * ky_m) ^ 2.0))))))));
	else
		tmp = sqrt((0.5 + ((1.0 / sqrt((1.0 + ((t_0 * t_0) * (0.5 - (0.5 * cos((ky_m + ky_m)))))))) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
code[l_, Om_, kx_, ky$95$m_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[ky$95$m, 4e-9], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(t$95$0 * N[Sin[kx], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(t$95$0 * ky$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 4.00000000000000025e-9

   1. Initial program 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. lower-+.f64 N/A

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 90.7%

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

   if 4.00000000000000025e-9 < ky

   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 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. unpow2 N/A

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

      2. sqr-sin-a N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   6. Applied rewrites 80.1%

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

Alternative 4: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky\_m}^{2}\right)} \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{1 + \left(t\_0 \cdot t\_0\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(ky\_m + ky\_m\right)\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

ky_m = (fabs.f64 ky)
(FPCore (l Om kx ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l l) Om)))
   (if (<=
        (sqrt
         (+
          1.0
          (*
           (pow (/ (* 2.0 l) Om) 2.0)
           (+ (pow (sin kx) 2.0) (pow (sin ky_m) 2.0)))))
        3e+15)
     (sqrt
      (+
       0.5
       (*
        (/
         1.0
         (sqrt (+ 1.0 (* (* t_0 t_0) (- 0.5 (* 0.5 (cos (+ ky_m ky_m))))))))
        0.5)))
     (sqrt 0.5))))

C

ky_m = fabs(ky);
double code(double l, double Om, double kx, double ky_m) {
	double t_0 = (l + l) / Om;
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky_m), 2.0))))) <= 3e+15) {
		tmp = sqrt((0.5 + ((1.0 / sqrt((1.0 + ((t_0 * t_0) * (0.5 - (0.5 * cos((ky_m + ky_m)))))))) * 0.5)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

ky_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l + l) / om
    if (sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky_m) ** 2.0d0))))) <= 3d+15) then
        tmp = sqrt((0.5d0 + ((1.0d0 / sqrt((1.0d0 + ((t_0 * t_0) * (0.5d0 - (0.5d0 * cos((ky_m + ky_m)))))))) * 0.5d0)))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

ky_m = Math.abs(ky);
public static double code(double l, double Om, double kx, double ky_m) {
	double t_0 = (l + l) / Om;
	double tmp;
	if (Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky_m), 2.0))))) <= 3e+15) {
		tmp = Math.sqrt((0.5 + ((1.0 / Math.sqrt((1.0 + ((t_0 * t_0) * (0.5 - (0.5 * Math.cos((ky_m + ky_m)))))))) * 0.5)));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

ky_m = math.fabs(ky)
def code(l, Om, kx, ky_m):
	t_0 = (l + l) / Om
	tmp = 0
	if math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky_m), 2.0))))) <= 3e+15:
		tmp = math.sqrt((0.5 + ((1.0 / math.sqrt((1.0 + ((t_0 * t_0) * (0.5 - (0.5 * math.cos((ky_m + ky_m)))))))) * 0.5)))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

ky_m = abs(ky)
function code(l, Om, kx, ky_m)
	t_0 = Float64(Float64(l + l) / Om)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 3e+15)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(t_0 * t_0) * Float64(0.5 - Float64(0.5 * cos(Float64(ky_m + ky_m)))))))) * 0.5)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

ky_m = abs(ky);
function tmp_2 = code(l, Om, kx, ky_m)
	t_0 = (l + l) / Om;
	tmp = 0.0;
	if (sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 3e+15)
		tmp = sqrt((0.5 + ((1.0 / sqrt((1.0 + ((t_0 * t_0) * (0.5 - (0.5 * cos((ky_m + ky_m)))))))) * 0.5)));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
code[l_, Om_, kx_, ky$95$m_] := Block[{t$95$0 = N[(N[(l + l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[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$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3e+15], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(ky$95$m + ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 3e15

   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 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. unpow2 N/A

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

      2. sqr-sin-a N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 80.1

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

   4. Applied rewrites 80.1%

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

   6. Applied rewrites 80.1%

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

   if 3e15 < (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 98.4%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 55.8%

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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

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

FPCore

ky_m = (fabs.f64 ky)
(FPCore (l Om kx ky_m)
 :precision binary64
 (if (<=
      (sqrt
       (+
        1.0
        (*
         (pow (/ (* 2.0 l) Om) 2.0)
         (+ (pow (sin kx) 2.0) (pow (sin ky_m) 2.0)))))
      2.0)
   1.0
   (sqrt 0.5)))

C

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

Fortran

ky_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky_m) ** 2.0d0))))) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
code[l_, Om_, kx_, ky$95$m_] := If[LessEqual[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$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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 62.7

         \[\leadsto 1 \]

   4. Applied rewrites 62.7%

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

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

   1. Initial program 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 inf

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

   4. Applied rewrites 55.8%

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

Alternative 6: 62.7% accurate, 142.7× speedup

Math

\[\begin{array}{l} ky_m = \left|ky\right| \\ 1 \end{array} \]

FPCore

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

C

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

Fortran

ky_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky_m
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

ky_m = N[Abs[ky], $MachinePrecision]
code[l_, Om_, kx_, 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 62.7

      \[\leadsto 1 \]

4. Applied rewrites 62.7%

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

Reproduce

herbie shell --seed 2025129 
(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))))))))))