Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 11.5s

Alternatives: 8

Speedup: 1.0×

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + {\left(\sqrt{\mathsf{fma}\left(4, {\left(\sin kx \cdot \frac{\ell}{Om}\right)}^{2} + {\left(\sin ky \cdot \frac{\ell}{Om}\right)}^{2}, 1\right)}\right)}^{-1}\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   0.5
   (+
    1.0
    (pow
     (sqrt
      (fma
       4.0
       (+ (pow (* (sin kx) (/ l Om)) 2.0) (pow (* (sin ky) (/ l Om)) 2.0))
       1.0))
     -1.0)))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 * (1.0 + pow(sqrt(fma(4.0, (pow((sin(kx) * (l / Om)), 2.0) + pow((sin(ky) * (l / Om)), 2.0)), 1.0)), -1.0))));
}

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 * Float64(1.0 + (sqrt(fma(4.0, Float64((Float64(sin(kx) * Float64(l / Om)) ^ 2.0) + (Float64(sin(ky) * Float64(l / Om)) ^ 2.0)), 1.0)) ^ -1.0))))
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 * N[(1.0 + N[Power[N[Sqrt[N[(4.0 * N[(N[Power[N[(N[Sin[kx], $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[ky], $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-+l+ N/A

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

   7. lower-+.f64 N/A

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

4. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 100.0

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

6. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{0.5} \cdot \left(1 + \frac{1}{\sqrt{{\left(\frac{\ell \cdot 2}{Om} \cdot \sin kx\right)}^{2} + \left({\left(\frac{\ell \cdot 2}{Om} \cdot \sin ky\right)}^{2} + 1\right)}}\right)} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. associate-*l* N/A

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

   9. unpow-prod-down N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 100.0

       \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(4, {\left(\color{blue}{\frac{\ell}{Om}} \cdot \sin kx\right)}^{2}, {\left(\frac{\ell \cdot 2}{Om} \cdot \sin ky\right)}^{2} + 1\right)}}\right)} \]

   15. lift-+.f64 N/A

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

8. Applied rewrites 100.0%

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(4, {\left(\frac{\ell}{Om} \cdot \sin kx\right)}^{2}, \mathsf{fma}\left(4, {\left(\frac{\ell}{Om} \cdot \sin ky\right)}^{2}, 1\right)\right)}}}\right)} \]
9. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+r+ N/A

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

   4. distribute-lft-out N/A

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

   5. lower-fma.f64 N/A

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

10. Applied rewrites 100.0%

    \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(4, {\left(\sin kx \cdot \frac{\ell}{Om}\right)}^{2} + {\left(\sin ky \cdot \frac{\ell}{Om}\right)}^{2}, 1\right)}}}\right)} \]

11. Final simplification 100.0%

    \[\leadsto \sqrt{0.5 \cdot \left(1 + {\left(\sqrt{\mathsf{fma}\left(4, {\left(\sin kx \cdot \frac{\ell}{Om}\right)}^{2} + {\left(\sin ky \cdot \frac{\ell}{Om}\right)}^{2}, 1\right)}\right)}^{-1}\right)} \]
12. Add Preprocessing

Alternative 2: 91.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;\sqrt{1 + {t\_0}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 1.2:\\ \;\;\;\;\sqrt{{2}^{-1} \cdot \left(1 + {\left(\sqrt{\mathsf{fma}\left(t\_0, \frac{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot \ell}{2 \cdot Om} \cdot 2, 1\right)}\right)}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l) Om)))
   (if (<=
        (sqrt
         (+ 1.0 (* (pow t_0 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        1.2)
     (sqrt
      (*
       (pow 2.0 -1.0)
       (+
        1.0
        (pow
         (sqrt
          (fma t_0 (* (/ (* (- 1.0 (cos (* 2.0 kx))) l) (* 2.0 Om)) 2.0) 1.0))
         -1.0))))
     (sqrt (fma (/ Om (* (sin ky) l)) 0.25 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = (2.0 * l) / Om;
	double tmp;
	if (sqrt((1.0 + (pow(t_0, 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 1.2) {
		tmp = sqrt((pow(2.0, -1.0) * (1.0 + pow(sqrt(fma(t_0, ((((1.0 - cos((2.0 * kx))) * l) / (2.0 * Om)) * 2.0), 1.0)), -1.0))));
	} else {
		tmp = sqrt(fma((Om / (sin(ky) * l)), 0.25, 0.5));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	t_0 = Float64(Float64(2.0 * l) / Om)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((t_0 ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 1.2)
		tmp = sqrt(Float64((2.0 ^ -1.0) * Float64(1.0 + (sqrt(fma(t_0, Float64(Float64(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) * l) / Float64(2.0 * Om)) * 2.0), 1.0)) ^ -1.0))));
	else
		tmp = sqrt(fma(Float64(Om / Float64(sin(ky) * l)), 0.25, 0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[t$95$0, 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], 1.2], N[Sqrt[N[(N[Power[2.0, -1.0], $MachinePrecision] * N[(1.0 + N[Power[N[Sqrt[N[(t$95$0 * N[(N[(N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * l), $MachinePrecision] / N[(2.0 * Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(Om / N[(N[Sin[ky], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.25 + 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)))))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 99.3

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

   10. Applied rewrites 99.3%

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

   12. Applied rewrites 99.3%

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

   if 1.19999999999999996 < (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 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]

   6. Taylor expanded in l around inf

      \[\leadsto \sqrt{\frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 1.2:\\ \;\;\;\;\sqrt{{2}^{-1} \cdot \left(1 + {\left(\sqrt{\mathsf{fma}\left(\frac{2 \cdot \ell}{Om}, \frac{\left(1 - \cos \left(2 \cdot kx\right)\right) \cdot \ell}{2 \cdot Om} \cdot 2, 1\right)}\right)}^{-1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 1.2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.25, 0.5\right)}\\ \end{array} \end{array} \]

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)))))
      1.2)
   (sqrt 1.0)
   (sqrt (fma (/ Om (* (sin ky) l)) 0.25 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))))) <= 1.2) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(fma((Om / (sin(ky) * l)), 0.25, 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))))) <= 1.2)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(fma(Float64(Om / Float64(sin(ky) * l)), 0.25, 0.5));
	end
	return 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], 1.2], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(Om / N[(N[Sin[ky], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.25 + 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)))))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 99.3%

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

   if 1.19999999999999996 < (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 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]

   6. Taylor expanded in l around inf

      \[\leadsto \sqrt{\frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.3%

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

Alternative 4: 91.8% accurate, 0.9× speedup

Math

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

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)
   (sqrt 1.0)
   (sqrt (fma (/ Om (* (sin ky) l)) -0.25 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 = sqrt(1.0);
	} else {
		tmp = sqrt(fma((Om / (sin(ky) * l)), -0.25, 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 = sqrt(1.0);
	else
		tmp = sqrt(fma(Float64(Om / Float64(sin(ky) * l)), -0.25, 0.5));
	end
	return 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], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(Om / N[(N[Sin[ky], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.25 + 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%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 98.7%

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

   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 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 75.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]

   6. Taylor expanded in l around -inf

      \[\leadsto \sqrt{\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.0%

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

Alternative 5: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 1.2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(Om, \frac{0.25}{ky \cdot \ell}, 0.5\right)}\\ \end{array} \end{array} \]

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)))))
      1.2)
   (sqrt 1.0)
   (sqrt (fma Om (/ 0.25 (* ky l)) 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))))) <= 1.2) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(fma(Om, (0.25 / (ky * l)), 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))))) <= 1.2)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(fma(Om, Float64(0.25 / Float64(ky * l)), 0.5));
	end
	return 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], 1.2], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(Om * N[(0.25 / N[(ky * l), $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)))))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 99.3%

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

   if 1.19999999999999996 < (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 94.1%

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

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 74.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]

   6. Taylor expanded in l around inf

      \[\leadsto \sqrt{\frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.3%

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

   9. Taylor expanded in ky around 0

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

   11. Applied rewrites 81.1%

       \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, 0.25, 0.5\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 81.1%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \leq 2.2:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

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)))))
      2.2)
   (sqrt 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))))) <= 2.2) {
		tmp = sqrt(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))))) <= 2.2d0) then
        tmp = sqrt(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))))) <= 2.2) {
		tmp = Math.sqrt(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))))) <= 2.2:
		tmp = math.sqrt(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))))) <= 2.2)
		tmp = sqrt(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))))) <= 2.2)
		tmp = sqrt(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], 2.2], N[Sqrt[1.0], $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)))))) < 2.2000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

   5. Applied rewrites 98.7%

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

   if 2.2000000000000002 < (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 94.1%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 96.8%

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

Alternative 7: 93.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{\sin ky \cdot \ell}{Om}\right)}^{2}, 4, 1\right)}} + 0.5} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt (+ (/ 0.5 (sqrt (fma (pow (/ (* (sin ky) l) Om) 2.0) 4.0 1.0))) 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((0.5 / sqrt(fma(pow(((sin(ky) * l) / Om), 2.0), 4.0, 1.0))) + 0.5));
}

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(0.5 / sqrt(fma((Float64(Float64(sin(ky) * l) / Om) ^ 2.0), 4.0, 1.0))) + 0.5))
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Power[N[(N[(N[Sin[ky], $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in kx around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 84.1%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 83.2%

   \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot Om}, 4, 1\right)}}, 0.5, 0.5\right)} \]

9. Applied rewrites 93.4%

   \[\leadsto \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{\sin ky \cdot \ell}{Om}\right)}^{2}, 4, 1\right)}} + \color{blue}{0.5}} \]
10. Add Preprocessing

Alternative 8: 55.7% accurate, 52.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (sqrt 0.5))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(l, om, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in l around inf

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

5. Applied rewrites 57.2%

   \[\leadsto \sqrt{\color{blue}{0.5}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(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))))))))))