Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 98.8%

Time: 5.8s

Alternatives: 7

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: 98.3% 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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ [l, Om_m, kx, ky] = \mathsf{sort}([l, Om_m, kx, ky])\\ \\ \begin{array}{l} t_0 := {\sin kx}^{2}\\ t_1 := {\left(\frac{2 \cdot \ell}{Om\_m}\right)}^{2}\\ \mathbf{if}\;\sqrt{1 + t\_1 \cdot \left(t\_0 + {\sin ky}^{2}\right)} \leq 2:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + t\_1 \cdot t\_0}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25 \cdot \frac{Om\_m}{\ell}, \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
(FPCore (l Om_m kx ky)
 :precision binary64
 (let* ((t_0 (pow (sin kx) 2.0)) (t_1 (pow (/ (* 2.0 l) Om_m) 2.0)))
   (if (<= (sqrt (+ 1.0 (* t_1 (+ t_0 (pow (sin ky) 2.0))))) 2.0)
     (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* t_1 t_0)))))))
     (sqrt (fma (* 0.25 (/ Om_m l)) (/ 1.0 (hypot (sin ky) (sin kx))) 0.5)))))

C

Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx && kx < ky);
double code(double l, double Om_m, double kx, double ky) {
	double t_0 = pow(sin(kx), 2.0);
	double t_1 = pow(((2.0 * l) / Om_m), 2.0);
	double tmp;
	if (sqrt((1.0 + (t_1 * (t_0 + pow(sin(ky), 2.0))))) <= 2.0) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (t_1 * t_0)))))));
	} else {
		tmp = sqrt(fma((0.25 * (Om_m / l)), (1.0 / hypot(sin(ky), sin(kx))), 0.5));
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l, Om_m, kx, ky = sort([l, Om_m, kx, ky])
function code(l, Om_m, kx, ky)
	t_0 = sin(kx) ^ 2.0
	t_1 = Float64(Float64(2.0 * l) / Om_m) ^ 2.0
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64(t_1 * Float64(t_0 + (sin(ky) ^ 2.0))))) <= 2.0)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_0)))))));
	else
		tmp = sqrt(fma(Float64(0.25 * Float64(Om_m / l)), Float64(1.0 / hypot(sin(ky), sin(kx))), 0.5));
	end
	return tmp
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx_, ky_] := Block[{t$95$0 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 + N[(t$95$1 * N[(t$95$0 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(0.25 * N[(Om$95$m / l), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $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%

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

   2. Taylor expanded in ky around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 99.4

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

   4. Applied rewrites 99.4%

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

   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 96.4%

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

   2. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}} \]
   6. 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. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.1%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

Om_m = abs(Om)
l, Om_m, kx, ky = sort([l, Om_m, kx, ky])
function code(l, Om_m, kx, ky)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt(fma(Float64(0.25 * Float64(Om_m / l)), Float64(1.0 / hypot(sin(ky), sin(kx))), 0.5));
	end
	return tmp
end

Wolfram

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

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

   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 96.4%

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

   2. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}} \]
   6. 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. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.1%

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

Alternative 3: 91.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx && kx < ky);
double code(double l, double Om_m, double kx, double ky) {
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 - (-0.25 * (Om_m / (l * sin(ky))))));
	}
	return tmp;
}

Fortran

Om_m =     private
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(l, om_m, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (sqrt((1.0d0 + ((((2.0d0 * l) / om_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))))) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 - ((-0.25d0) * (om_m / (l * sin(ky))))))
    end if
    code = tmp
end function

Java

Om_m = Math.abs(Om);
assert l < Om_m && Om_m < kx && kx < ky;
public static double code(double l, double Om_m, double kx, double ky) {
	double tmp;
	if (Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om_m), 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 - (-0.25 * (Om_m / (l * Math.sin(ky))))));
	}
	return tmp;
}

Python

Om_m = math.fabs(Om)
[l, Om_m, kx, ky] = sort([l, Om_m, kx, ky])
def code(l, Om_m, kx, ky):
	tmp = 0
	if math.sqrt((1.0 + (math.pow(((2.0 * l) / Om_m), 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 - (-0.25 * (Om_m / (l * math.sin(ky))))))
	return tmp

Julia

Om_m = abs(Om)
l, Om_m, kx, ky = sort([l, Om_m, kx, ky])
function code(l, Om_m, kx, ky)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 - Float64(-0.25 * Float64(Om_m / Float64(l * sin(ky))))));
	end
	return tmp
end

MATLAB

Om_m = abs(Om);
l, Om_m, kx, ky = num2cell(sort([l, Om_m, kx, ky])){:}
function tmp_2 = code(l, Om_m, kx, ky)
	tmp = 0.0;
	if (sqrt((1.0 + ((((2.0 * l) / Om_m) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 - (-0.25 * (Om_m / (l * sin(ky))))));
	end
	tmp_2 = tmp;
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx_, ky_] := If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om$95$m), $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[N[(0.5 - N[(-0.25 * N[(Om$95$m / N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. 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 96.4%

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

   2. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}} \]
   6. 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. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in kx around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \frac{-1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}} \]

      7. lift-sin.f64 83.7

         \[\leadsto \sqrt{0.5 - -0.25 \cdot \frac{Om}{\ell \cdot \sin ky}} \]

   10. Applied rewrites 83.7%

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

Alternative 4: 91.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx && kx < ky);
double code(double l, double Om_m, double kx, double ky) {
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (-0.25 * (Om_m / (l * sin(ky))))));
	}
	return tmp;
}

Fortran

Om_m =     private
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(l, om_m, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (sqrt((1.0d0 + ((((2.0d0 * l) / om_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))))) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + ((-0.25d0) * (om_m / (l * sin(ky))))))
    end if
    code = tmp
end function

Java

Om_m = Math.abs(Om);
assert l < Om_m && Om_m < kx && kx < ky;
public static double code(double l, double Om_m, double kx, double ky) {
	double tmp;
	if (Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om_m), 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 + (-0.25 * (Om_m / (l * Math.sin(ky))))));
	}
	return tmp;
}

Python

Om_m = math.fabs(Om)
[l, Om_m, kx, ky] = sort([l, Om_m, kx, ky])
def code(l, Om_m, kx, ky):
	tmp = 0
	if math.sqrt((1.0 + (math.pow(((2.0 * l) / Om_m), 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 + (-0.25 * (Om_m / (l * math.sin(ky))))))
	return tmp

Julia

Om_m = abs(Om)
l, Om_m, kx, ky = sort([l, Om_m, kx, ky])
function code(l, Om_m, kx, ky)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(-0.25 * Float64(Om_m / Float64(l * sin(ky))))));
	end
	return tmp
end

MATLAB

Om_m = abs(Om);
l, Om_m, kx, ky = num2cell(sort([l, Om_m, kx, ky])){:}
function tmp_2 = code(l, Om_m, kx, ky)
	tmp = 0.0;
	if (sqrt((1.0 + ((((2.0 * l) / Om_m) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (-0.25 * (Om_m / (l * sin(ky))))));
	end
	tmp_2 = tmp;
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx_, ky_] := If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om$95$m), $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[N[(0.5 + N[(-0.25 * N[(Om$95$m / N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. 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 96.4%

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

   2. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in l around -inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}} \]
   6. 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. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in kx around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 83.7

         \[\leadsto \sqrt{0.5 + -0.25 \cdot \frac{Om}{\ell \cdot \sin ky}} \]

   10. Applied rewrites 83.7%

       \[\leadsto \sqrt{0.5 + \color{blue}{-0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

Om_m =     private
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(l, om_m, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om_m
    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_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

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

Python

Om_m = math.fabs(Om)
[l, Om_m, kx, ky] = sort([l, Om_m, kx, ky])
def code(l, Om_m, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om_m), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

Om_m = abs(Om)
l, Om_m, kx, ky = sort([l, Om_m, kx, ky])
function code(l, Om_m, 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_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

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

Wolfram

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

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. Add Preprocessing

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

Om_m = fabs(Om);
assert(l < Om_m && Om_m < kx && kx < ky);
double code(double l, double Om_m, double kx, double ky) {
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l) / Om_m), 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

Om_m =     private
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(l, om_m, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (sqrt((1.0d0 + ((((2.0d0 * l) / om_m) ** 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

Om_m = Math.abs(Om);
assert l < Om_m && Om_m < kx && kx < ky;
public static double code(double l, double Om_m, double kx, double ky) {
	double tmp;
	if (Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om_m), 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

Om_m = math.fabs(Om)
[l, Om_m, kx, ky] = sort([l, Om_m, kx, ky])
def code(l, Om_m, kx, ky):
	tmp = 0
	if math.sqrt((1.0 + (math.pow(((2.0 * l) / Om_m), 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

Om_m = abs(Om)
l, Om_m, kx, ky = sort([l, Om_m, kx, ky])
function code(l, Om_m, kx, ky)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om_m) ^ 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

Om_m = abs(Om);
l, Om_m, kx, ky = num2cell(sort([l, Om_m, kx, ky])){:}
function tmp_2 = code(l, Om_m, kx, ky)
	tmp = 0.0;
	if (sqrt((1.0 + ((((2.0 * l) / Om_m) ^ 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

Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, kx_, ky_] := If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om$95$m), $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 96.4%

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

   2. Taylor expanded in l around inf

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

   4. Applied rewrites 97.5%

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

Alternative 7: 62.4% accurate, 581.0× speedup

Math

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

FPCore

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

C

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

Fortran

Om_m =     private
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(l, om_m, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
NOTE: l, Om_m, kx, and ky should be sorted in increasing order before calling this function.
code[l_, Om$95$m_, 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 62.4

      \[\leadsto 1 \]

4. Applied rewrites 62.4%

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

Reproduce

herbie shell --seed 2025100 
(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))))))))))