Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 12.6s

Alternatives: 10

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(l, om, kx, ky)
    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: 98.5% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
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. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lower-+.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 10^{-19}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left({\left(\frac{kx \cdot \ell}{Om} \cdot 2\right)}^{2} + 1\right) + {\left(\left(ky \cdot \frac{\ell}{Om}\right) \cdot 2\right)}^{2}}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left(\left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (pow (sin ky) 2.0) 1e-19)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (+
         (+ (pow (* (/ (* kx l) Om) 2.0) 2.0) 1.0)
         (pow (* (* ky (/ l Om)) 2.0) 2.0))))
      1.0)
     (/ 1.0 2.0)))
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (+
         (*
          (+ (- 0.5 (* (cos (* ky 2.0)) 0.5)) (pow (sin kx) 2.0))
          (pow (/ (* l 2.0) Om) 2.0))
         1.0)))
      1.0)
     (/ 1.0 2.0)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (pow(sin(ky), 2.0) <= 1e-19) {
		tmp = sqrt((((1.0 / sqrt(((pow((((kx * l) / Om) * 2.0), 2.0) + 1.0) + pow(((ky * (l / Om)) * 2.0), 2.0)))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = sqrt((((1.0 / sqrt(((((0.5 - (cos((ky * 2.0)) * 0.5)) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) + 1.0))) + 1.0) * (1.0 / 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if ((sin(ky) ** 2.0d0) <= 1d-19) then
        tmp = sqrt((((1.0d0 / sqrt(((((((kx * l) / om) * 2.0d0) ** 2.0d0) + 1.0d0) + (((ky * (l / om)) * 2.0d0) ** 2.0d0)))) + 1.0d0) * (1.0d0 / 2.0d0)))
    else
        tmp = sqrt((((1.0d0 / sqrt(((((0.5d0 - (cos((ky * 2.0d0)) * 0.5d0)) + (sin(kx) ** 2.0d0)) * (((l * 2.0d0) / om) ** 2.0d0)) + 1.0d0))) + 1.0d0) * (1.0d0 / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Math.pow(Math.sin(ky), 2.0) <= 1e-19) {
		tmp = Math.sqrt((((1.0 / Math.sqrt(((Math.pow((((kx * l) / Om) * 2.0), 2.0) + 1.0) + Math.pow(((ky * (l / Om)) * 2.0), 2.0)))) + 1.0) * (1.0 / 2.0)));
	} else {
		tmp = Math.sqrt((((1.0 / Math.sqrt(((((0.5 - (Math.cos((ky * 2.0)) * 0.5)) + Math.pow(Math.sin(kx), 2.0)) * Math.pow(((l * 2.0) / Om), 2.0)) + 1.0))) + 1.0) * (1.0 / 2.0)));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if math.pow(math.sin(ky), 2.0) <= 1e-19:
		tmp = math.sqrt((((1.0 / math.sqrt(((math.pow((((kx * l) / Om) * 2.0), 2.0) + 1.0) + math.pow(((ky * (l / Om)) * 2.0), 2.0)))) + 1.0) * (1.0 / 2.0)))
	else:
		tmp = math.sqrt((((1.0 / math.sqrt(((((0.5 - (math.cos((ky * 2.0)) * 0.5)) + math.pow(math.sin(kx), 2.0)) * math.pow(((l * 2.0) / Om), 2.0)) + 1.0))) + 1.0) * (1.0 / 2.0)))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if ((sin(ky) ^ 2.0) <= 1e-19)
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(Float64(Float64((Float64(Float64(Float64(kx * l) / Om) * 2.0) ^ 2.0) + 1.0) + (Float64(Float64(ky * Float64(l / Om)) * 2.0) ^ 2.0)))) + 1.0) * Float64(1.0 / 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(ky * 2.0)) * 0.5)) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) + 1.0))) + 1.0) * Float64(1.0 / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((sin(ky) ^ 2.0) <= 1e-19)
		tmp = sqrt((((1.0 / sqrt(((((((kx * l) / Om) * 2.0) ^ 2.0) + 1.0) + (((ky * (l / Om)) * 2.0) ^ 2.0)))) + 1.0) * (1.0 / 2.0)));
	else
		tmp = sqrt((((1.0 / sqrt(((((0.5 - (cos((ky * 2.0)) * 0.5)) + (sin(kx) ^ 2.0)) * (((l * 2.0) / Om) ^ 2.0)) + 1.0))) + 1.0) * (1.0 / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision], 1e-19], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[Power[N[(N[(N[(kx * l), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + N[Power[N[(N[(ky * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[(N[(0.5 - N[(N[Cos[N[(ky * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 ky) #s(literal 2 binary64)) < 9.9999999999999998e-20

   1. Initial program 95.6%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.2

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

   10. Applied rewrites 90.2%

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

   if 9.9999999999999998e-20 < (pow.f64 (sin.f64 ky) #s(literal 2 binary64))

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. count-2 N/A

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

      17. lower-cos.f64 N/A

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

      18. count-2 N/A

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

      19. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 10^{-19}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left({\left(\frac{kx \cdot \ell}{Om} \cdot 2\right)}^{2} + 1\right) + {\left(\left(ky \cdot \frac{\ell}{Om}\right) \cdot 2\right)}^{2}}} + 1\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left(\left(0.5 - \cos \left(ky \cdot 2\right) \cdot 0.5\right) + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      2e-6)
   (sqrt 1.0)
   (* (sqrt 0.5) (sqrt (fma (/ Om (* (sin ky) l)) 0.5 1.0)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 2e-6) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(0.5) * sqrt(fma((Om / (sin(ky) * l)), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 2e-6)
		tmp = sqrt(1.0);
	else
		tmp = Float64(sqrt(0.5) * sqrt(fma(Float64(Om / Float64(sin(ky) * l)), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-6], N[Sqrt[1.0], $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(N[(Om / N[(N[Sin[ky], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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)))) < 1.99999999999999991e-6

   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 Om around inf

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

   5. Applied rewrites 99.7%

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

   if 1.99999999999999991e-6 < (*.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 95.3%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in l around -inf

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

   8. Applied rewrites 80.6%

      \[\leadsto \sqrt{\frac{Om}{\sin ky \cdot \ell} \cdot -0.5 + 1} \cdot \sqrt{0.5} \]

   9. Taylor expanded in Om around 0

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

   11. Applied rewrites 80.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, 0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{\mathsf{fma}\left(\frac{ky}{Om} \cdot ky, \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 4, 1\right)}} + 1} \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      5e-17)
   (sqrt 1.0)
   (*
    (sqrt
     (+ (sqrt (/ 1.0 (fma (* (/ ky Om) ky) (* (* (/ l Om) l) 4.0) 1.0))) 1.0))
    (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 5e-17) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt((sqrt((1.0 / fma(((ky / Om) * ky), (((l / Om) * l) * 4.0), 1.0))) + 1.0)) * sqrt(0.5);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 5e-17)
		tmp = sqrt(1.0);
	else
		tmp = Float64(sqrt(Float64(sqrt(Float64(1.0 / fma(Float64(Float64(ky / Om) * ky), Float64(Float64(Float64(l / Om) * l) * 4.0), 1.0))) + 1.0)) * sqrt(0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 5e-17], N[Sqrt[1.0], $MachinePrecision], N[(N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(N[(N[(ky / Om), $MachinePrecision] * ky), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * 4.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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)))) < 4.9999999999999999e-17

   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 Om around inf

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

   5. Applied rewrites 100.0%

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

   if 4.9999999999999999e-17 < (*.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 95.3%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.2%

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 71.1%

       \[\leadsto \sqrt{\sqrt{\frac{1}{\mathsf{fma}\left(\frac{ky \cdot ky}{Om}, 4 \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 1\right)}} + 1} \cdot \sqrt{0.5} \]
   11. Step-by-step derivation

   12. Applied rewrites 75.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{1}{\mathsf{fma}\left(\frac{ky}{Om} \cdot ky, \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 4, 1\right)}} + 1} \cdot \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin kx}^{2}\\ \mathbf{if}\;t\_0 \leq 10^{-99}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{{\left(\left(kx \cdot \frac{\ell}{Om}\right) \cdot 2\right)}^{2} + \left({\left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \sin ky\right)}^{2} + 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t\_0}{Om}, \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 4, 1\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (pow (sin kx) 2.0)))
   (if (<= t_0 1e-99)
     (sqrt
      (+
       (/
        0.5
        (sqrt
         (+
          (pow (* (* kx (/ l Om)) 2.0) 2.0)
          (+ (pow (* (* (/ 2.0 Om) l) (sin ky)) 2.0) 1.0))))
       0.5))
     (sqrt
      (fma
       (sqrt (/ 1.0 (fma (/ t_0 Om) (* (* (/ l Om) l) 4.0) 1.0)))
       0.5
       0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = pow(sin(kx), 2.0);
	double tmp;
	if (t_0 <= 1e-99) {
		tmp = sqrt(((0.5 / sqrt((pow(((kx * (l / Om)) * 2.0), 2.0) + (pow((((2.0 / Om) * l) * sin(ky)), 2.0) + 1.0)))) + 0.5));
	} else {
		tmp = sqrt(fma(sqrt((1.0 / fma((t_0 / Om), (((l / Om) * l) * 4.0), 1.0))), 0.5, 0.5));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	t_0 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_0 <= 1e-99)
		tmp = sqrt(Float64(Float64(0.5 / sqrt(Float64((Float64(Float64(kx * Float64(l / Om)) * 2.0) ^ 2.0) + Float64((Float64(Float64(Float64(2.0 / Om) * l) * sin(ky)) ^ 2.0) + 1.0)))) + 0.5));
	else
		tmp = sqrt(fma(sqrt(Float64(1.0 / fma(Float64(t_0 / Om), Float64(Float64(Float64(l / Om) * l) * 4.0), 1.0))), 0.5, 0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, 1e-99], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[Power[N[(N[(kx * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[(N[(N[(2.0 / Om), $MachinePrecision] * l), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(N[(t$95$0 / Om), $MachinePrecision] * N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * 4.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 1e-99

   1. Initial program 94.6%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1e-99 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 93.4%

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

   7. Applied rewrites 99.5%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin kx}^{2}}{Om}, 4 \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 1\right)}}, 0.5, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 10^{-99}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{{\left(\left(kx \cdot \frac{\ell}{Om}\right) \cdot 2\right)}^{2} + \left({\left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \sin ky\right)}^{2} + 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin kx}^{2}}{Om}, \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 4, 1\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{Om}{\ell}, 0.25, ky\right)}{ky} \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      2e-6)
   (sqrt 1.0)
   (* (/ (fma (/ Om l) 0.25 ky) ky) (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 2e-6) {
		tmp = sqrt(1.0);
	} else {
		tmp = (fma((Om / l), 0.25, ky) / ky) * sqrt(0.5);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 2e-6)
		tmp = sqrt(1.0);
	else
		tmp = Float64(Float64(fma(Float64(Om / l), 0.25, ky) / ky) * sqrt(0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-6], N[Sqrt[1.0], $MachinePrecision], N[(N[(N[(N[(Om / l), $MachinePrecision] * 0.25 + ky), $MachinePrecision] / ky), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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)))) < 1.99999999999999991e-6

   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 Om around inf

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

   5. Applied rewrites 99.7%

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

   if 1.99999999999999991e-6 < (*.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 95.3%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.4%

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 80.6%

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

   11. Taylor expanded in ky around 0

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

   13. Applied rewrites 80.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{Om}{\ell}, 0.25, ky\right)}{ky} \cdot \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, 0.25, 1\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      2e-6)
   (sqrt 1.0)
   (* (fma (/ Om (* ky l)) 0.25 1.0) (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 2e-6) {
		tmp = sqrt(1.0);
	} else {
		tmp = fma((Om / (ky * l)), 0.25, 1.0) * sqrt(0.5);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 2e-6)
		tmp = sqrt(1.0);
	else
		tmp = Float64(fma(Float64(Om / Float64(ky * l)), 0.25, 1.0) * sqrt(0.5));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-6], N[Sqrt[1.0], $MachinePrecision], N[(N[(N[(Om / N[(ky * l), $MachinePrecision]), $MachinePrecision] * 0.25 + 1.0), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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)))) < 1.99999999999999991e-6

   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 Om around inf

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

   5. Applied rewrites 99.7%

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

   if 1.99999999999999991e-6 < (*.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 95.3%

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

   3. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.4%

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 80.6%

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

   11. Taylor expanded in ky around 0

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

   13. Applied rewrites 80.4%

       \[\leadsto \mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, 0.25, 1\right) \cdot \sqrt{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, 0.25, 1\right) \cdot \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 3.8:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)) (pow (/ (* l 2.0) Om) 2.0))
      3.8)
   (sqrt 1.0)
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * pow(((l * 2.0) / Om), 2.0)) <= 3.8) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if ((((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)) * (((l * 2.0d0) / om) ** 2.0d0)) <= 3.8d0) 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.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)) * Math.pow(((l * 2.0) / Om), 2.0)) <= 3.8) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)) * math.pow(((l * 2.0) / Om), 2.0)) <= 3.8:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) <= 3.8)
		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 ((((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (((l * 2.0) / Om) ^ 2.0)) <= 3.8)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.8], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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)))) < 3.7999999999999998

   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 Om around inf

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

   5. Applied rewrites 99.7%

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

   if 3.7999999999999998 < (*.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 95.3%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 95.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} \leq 3.8:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\ell \cdot 2}{Om} \leq 10^{-8}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left({\left(\frac{kx \cdot \ell}{Om} \cdot 2\right)}^{2} + 1\right) + {\left(\left(ky \cdot \frac{\ell}{Om}\right) \cdot 2\right)}^{2}}} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* l 2.0) Om) 1e-8)
   (sqrt 1.0)
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt
        (+
         (+ (pow (* (/ (* kx l) Om) 2.0) 2.0) 1.0)
         (pow (* (* ky (/ l Om)) 2.0) 2.0))))
      1.0)
     (/ 1.0 2.0)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((l * 2.0) / Om) <= 1e-8) {
		tmp = sqrt(1.0);
	} else {
		tmp = sqrt((((1.0 / sqrt(((pow((((kx * l) / Om) * 2.0), 2.0) + 1.0) + pow(((ky * (l / Om)) * 2.0), 2.0)))) + 1.0) * (1.0 / 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((l * 2.0d0) / om) <= 1d-8) then
        tmp = sqrt(1.0d0)
    else
        tmp = sqrt((((1.0d0 / sqrt(((((((kx * l) / om) * 2.0d0) ** 2.0d0) + 1.0d0) + (((ky * (l / om)) * 2.0d0) ** 2.0d0)))) + 1.0d0) * (1.0d0 / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((l * 2.0) / Om) <= 1e-8) {
		tmp = Math.sqrt(1.0);
	} else {
		tmp = Math.sqrt((((1.0 / Math.sqrt(((Math.pow((((kx * l) / Om) * 2.0), 2.0) + 1.0) + Math.pow(((ky * (l / Om)) * 2.0), 2.0)))) + 1.0) * (1.0 / 2.0)));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((l * 2.0) / Om) <= 1e-8:
		tmp = math.sqrt(1.0)
	else:
		tmp = math.sqrt((((1.0 / math.sqrt(((math.pow((((kx * l) / Om) * 2.0), 2.0) + 1.0) + math.pow(((ky * (l / Om)) * 2.0), 2.0)))) + 1.0) * (1.0 / 2.0)))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(l * 2.0) / Om) <= 1e-8)
		tmp = sqrt(1.0);
	else
		tmp = sqrt(Float64(Float64(Float64(1.0 / sqrt(Float64(Float64((Float64(Float64(Float64(kx * l) / Om) * 2.0) ^ 2.0) + 1.0) + (Float64(Float64(ky * Float64(l / Om)) * 2.0) ^ 2.0)))) + 1.0) * Float64(1.0 / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((l * 2.0) / Om) <= 1e-8)
		tmp = sqrt(1.0);
	else
		tmp = sqrt((((1.0 / sqrt(((((((kx * l) / Om) * 2.0) ^ 2.0) + 1.0) + (((ky * (l / Om)) * 2.0) ^ 2.0)))) + 1.0) * (1.0 / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 1e-8], N[Sqrt[1.0], $MachinePrecision], N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(N[(N[Power[N[(N[(N[(kx * l), $MachinePrecision] / Om), $MachinePrecision] * 2.0), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + N[Power[N[(N[(ky * N[(l / Om), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1e-8

   1. Initial program 98.5%

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

   3. Taylor expanded in Om around inf

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

   5. Applied rewrites 77.3%

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

   if 1e-8 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 95.9%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   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. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 98.9

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

   10. Applied rewrites 98.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\ell \cdot 2}{Om} \leq 10^{-8}:\\ \;\;\;\;\sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{\sqrt{\left({\left(\frac{kx \cdot \ell}{Om} \cdot 2\right)}^{2} + 1\right) + {\left(\left(ky \cdot \frac{\ell}{Om}\right) \cdot 2\right)}^{2}}} + 1\right) \cdot \frac{1}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.2% 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

real(8) function code(l, om, kx, ky)
    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 97.9%

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

3. Taylor expanded in Om around 0

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

5. Applied rewrites 54.4%

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

Reproduce

herbie shell --seed 2024254 
(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))))))))))