Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.7%

Time: 6.2s

Alternatives: 9

Speedup: 1.1×

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0
         (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)))))))))))
   (if (<= t_0 2.0) t_0 (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = 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)))))))));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(l, om, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 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)))))))))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

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

Python

def code(l, Om, kx, ky):
	t_0 = 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)))))))))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = 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)))))))));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $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]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 82.6%

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

Alternative 2: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 100000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :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) 2.0)))))
        100000000000.0)
     (sqrt
      (+
       0.5
       (*
        (/
         1.0
         (sqrt
          (fma
           (+
            (- 0.5 (* 0.5 (cos (* 2.0 ky))))
            (- 0.5 (* 0.5 (cos (* 2.0 kx)))))
           (* t_0 t_0)
           1.0)))
        0.5)))
     (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	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), 2.0))))) <= 100000000000.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma(((0.5 - (0.5 * cos((2.0 * ky)))) + (0.5 - (0.5 * cos((2.0 * kx))))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	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) ^ 2.0))))) <= 100000000000.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky)))) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx))))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 100000000000.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $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)))))) < 1e11

   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 99.4%

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

   if 1e11 < (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 96.2%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 99.2%

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

Alternative 3: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 100000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(1 - \mathsf{fma}\left(0.5, \cos \left(2 \cdot kx\right), 0.5 \cdot \cos \left(2 \cdot ky\right)\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :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) 2.0)))))
        100000000000.0)
     (sqrt
      (+
       0.5
       (*
        (/
         1.0
         (sqrt
          (fma
           (- 1.0 (fma 0.5 (cos (* 2.0 kx)) (* 0.5 (cos (* 2.0 ky)))))
           (* t_0 t_0)
           1.0)))
        0.5)))
     (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	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), 2.0))))) <= 100000000000.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((1.0 - fma(0.5, cos((2.0 * kx)), (0.5 * cos((2.0 * ky))))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	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) ^ 2.0))))) <= 100000000000.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(1.0 - fma(0.5, cos(Float64(2.0 * kx)), Float64(0.5 * cos(Float64(2.0 * ky))))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 100000000000.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(1.0 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $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)))))) < 1e11

   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 99.4%

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

   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 99.4

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

   6. Applied rewrites 99.4%

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

   if 1e11 < (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 96.2%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 99.2%

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

Alternative 4: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 2:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25 \cdot \frac{\sqrt{0.5} \cdot Om}{\ell}, \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :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) 2.0)))))
        2.0)
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 ky)))) (* t_0 t_0) 1.0)))
        0.5)))
     (fma
      (* 0.25 (/ (* (sqrt 0.5) Om) l))
      (/ 1.0 (hypot (sin ky) (sin kx)))
      (sqrt 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	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), 2.0))))) <= 2.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * ky)))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = fma((0.25 * ((sqrt(0.5) * Om) / l)), (1.0 / hypot(sin(ky), sin(kx))), sqrt(0.5));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	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) ^ 2.0))))) <= 2.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky)))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = fma(Float64(0.25 * Float64(Float64(sqrt(0.5) * Om) / l)), Float64(1.0 / hypot(sin(ky), sin(kx))), sqrt(0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.25 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * Om), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Applied rewrites 99.7%

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

   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 99.3

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

   6. Applied rewrites 99.3%

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

   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 96.3%

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

   2. Taylor expanded in l around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 \cdot \frac{\sqrt{0.5} \cdot Om}{\ell}, \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, \sqrt{0.5}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 2:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :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) 2.0)))))
        2.0)
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 ky)))) (* t_0 t_0) 1.0)))
        0.5)))
     (sqrt (fma 0.25 (* (/ Om l) (/ 1.0 (hypot (sin ky) (sin kx)))) 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	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), 2.0))))) <= 2.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * ky)))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(fma(0.25, ((Om / l) * (1.0 / hypot(sin(ky), sin(kx)))), 0.5));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	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) ^ 2.0))))) <= 2.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky)))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(fma(0.25, Float64(Float64(Om / l) * Float64(1.0 / hypot(sin(ky), sin(kx)))), 0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.25 * N[(N[(Om / l), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Applied rewrites 99.7%

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

   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 99.3

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

   6. Applied rewrites 99.3%

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

   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 96.3%

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

   2. 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 98.0%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 100000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :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) 2.0)))))
        100000000000.0)
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 ky)))) (* t_0 t_0) 1.0)))
        0.5)))
     (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	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), 2.0))))) <= 100000000000.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * ky)))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	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) ^ 2.0))))) <= 100000000000.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky)))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 100000000000.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $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)))))) < 1e11

   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 99.4%

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

   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 98.1

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

   6. Applied rewrites 98.1%

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

   if 1e11 < (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 96.2%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 99.2%

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

Alternative 7: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell + \ell}{Om}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 100000000:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :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) 2.0)))))
        100000000.0)
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 kx)))) (* t_0 t_0) 1.0)))
        0.5)))
     (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	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), 2.0))))) <= 100000000.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * kx)))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	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) ^ 2.0))))) <= 100000000.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := 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], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 100000000.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $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)))))) < 1e8

   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 99.5%

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

   4. Taylor expanded in ky around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 98.3

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

   6. Applied rewrites 98.3%

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

   if 1e8 < (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 96.2%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 98.8%

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

Alternative 8: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(l, Om, kx, ky):
	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), 2.0))))) <= 2.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := 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], $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 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 99.1

         \[\leadsto 1 \]

   4. Applied rewrites 99.1%

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

   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 96.3%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 97.5%

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

Alternative 9: 63.7% accurate, 142.7× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 1.0)

C

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

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 = 1.0d0
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

Python

def code(l, Om, kx, ky):
	return 1.0

Julia

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

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := 1.0

Derivation

1. Initial program 98.3%

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

2. 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 63.7

      \[\leadsto 1 \]

4. Applied rewrites 63.7%

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

Reproduce

herbie shell --seed 2025114 
(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))))))))))