Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 98.2%

Time: 14.0s

Alternatives: 6

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

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: 98.2% 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: 98.2% 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]

Derivation

1. Initial program 98.6%

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

Alternative 2: 93.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+123}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 1e+123)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (*
         (/ (* (/ l Om) 4.0) (/ Om l))
         (+ 1.0 (* -0.5 (+ (cos (* 2.0 kx)) (cos (* 2.0 ky)))))))))))
   (sqrt (+ 0.5 (* 0.25 (/ Om (* l (sin ky))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+123) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((((l / Om) * 4.0) / (Om / l)) * (1.0 + (-0.5 * (cos((2.0 * kx)) + cos((2.0 * ky)))))))))));
	} else {
		tmp = sqrt((0.5 + (0.25 * (Om / (l * sin(ky))))));
	}
	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 (((2.0d0 * l) / om) <= 1d+123) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((((l / om) * 4.0d0) / (om / l)) * (1.0d0 + ((-0.5d0) * (cos((2.0d0 * kx)) + cos((2.0d0 * ky)))))))))))
    else
        tmp = sqrt((0.5d0 + (0.25d0 * (om / (l * sin(ky))))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+123) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((((l / Om) * 4.0) / (Om / l)) * (1.0 + (-0.5 * (Math.cos((2.0 * kx)) + Math.cos((2.0 * ky)))))))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.25 * (Om / (l * Math.sin(ky))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((2.0 * l) / Om) <= 1e+123:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((((l / Om) * 4.0) / (Om / l)) * (1.0 + (-0.5 * (math.cos((2.0 * kx)) + math.cos((2.0 * ky)))))))))))
	else:
		tmp = math.sqrt((0.5 + (0.25 * (Om / (l * math.sin(ky))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 1e+123)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(l / Om) * 4.0) / Float64(Om / l)) * Float64(1.0 + Float64(-0.5 * Float64(cos(Float64(2.0 * kx)) + cos(Float64(2.0 * ky)))))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(Om / Float64(l * sin(ky))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((2.0 * l) / Om) <= 1e+123)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((((l / Om) * 4.0) / (Om / l)) * (1.0 + (-0.5 * (cos((2.0 * kx)) + cos((2.0 * ky)))))))))));
	else
		tmp = sqrt((0.5 + (0.25 * (Om / (l * sin(ky))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 1e+123], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(N[(l / Om), $MachinePrecision] * 4.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.25 * N[(Om / N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 9.99999999999999978e122

   1. Initial program 99.7%

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

   4. Applied egg-rr 95.0%

      \[\leadsto \sqrt{\color{blue}{\frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}} + 0.5}} \]

   5. Taylor expanded in kx around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \color{blue}{\left(1 + \left(\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right)}\right)\right)\right)\right), \frac{1}{2}\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \cos \left(2 \cdot kx\right) + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(\cos \left(2 \cdot kx\right), \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(2 \cdot kx\right)\right), \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \mathsf{cos.f64}\left(\left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      8. *-lowering-*.f64 95.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, kx\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

   7. Simplified 95.0%

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)}}} + 0.5} \]

   if 9.99999999999999978e122 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(1 + \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 81.4%

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

   6. Taylor expanded in l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{Om}{\ell \cdot \sin ky}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(Om, \left(\ell \cdot \sin ky\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(\ell, \sin ky\right)\right)\right)\right)\right) \]

      5. sin-lowering-sin.f64 90.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(\ell, \mathsf{sin.f64}\left(ky\right)\right)\right)\right)\right)\right) \]

   8. Simplified 90.6%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+123}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+123}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 1e+123)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (*
         (/ (* (/ l Om) 4.0) (/ Om l))
         (+ 0.5 (* -0.5 (cos (* 2.0 ky))))))))))
   (sqrt (+ 0.5 (* 0.25 (/ Om (* l (sin ky))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+123) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((((l / Om) * 4.0) / (Om / l)) * (0.5 + (-0.5 * cos((2.0 * ky))))))))));
	} else {
		tmp = sqrt((0.5 + (0.25 * (Om / (l * sin(ky))))));
	}
	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 (((2.0d0 * l) / om) <= 1d+123) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((((l / om) * 4.0d0) / (om / l)) * (0.5d0 + ((-0.5d0) * cos((2.0d0 * ky))))))))))
    else
        tmp = sqrt((0.5d0 + (0.25d0 * (om / (l * sin(ky))))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 1e+123) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((((l / Om) * 4.0) / (Om / l)) * (0.5 + (-0.5 * Math.cos((2.0 * ky))))))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.25 * (Om / (l * Math.sin(ky))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((2.0 * l) / Om) <= 1e+123:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((((l / Om) * 4.0) / (Om / l)) * (0.5 + (-0.5 * math.cos((2.0 * ky))))))))))
	else:
		tmp = math.sqrt((0.5 + (0.25 * (Om / (l * math.sin(ky))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 1e+123)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(l / Om) * 4.0) / Float64(Om / l)) * Float64(0.5 + Float64(-0.5 * cos(Float64(2.0 * ky))))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(Om / Float64(l * sin(ky))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((2.0 * l) / Om) <= 1e+123)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((((l / Om) * 4.0) / (Om / l)) * (0.5 + (-0.5 * cos((2.0 * ky))))))))));
	else
		tmp = sqrt((0.5 + (0.25 * (Om / (l * sin(ky))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 1e+123], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(N[(l / Om), $MachinePrecision] * 4.0), $MachinePrecision] / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.25 * N[(Om / N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 9.99999999999999978e122

   1. Initial program 99.7%

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

   4. Applied egg-rr 95.0%

      \[\leadsto \sqrt{\color{blue}{\frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}} + 0.5}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}\right)\right)\right)\right), \frac{1}{2}\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      4. *-lowering-*.f64 86.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

   7. Simplified 86.7%

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \color{blue}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} + 0.5} \]

   if 9.99999999999999978e122 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 92.9%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(1 + \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}\right)\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\left(\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)}{{Om}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(4 \cdot \left({\ell}^{2} \cdot {\sin ky}^{2}\right)\right), \left({Om}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 81.4%

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

   6. Taylor expanded in l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \left(\frac{Om}{\ell \cdot \sin ky}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(Om, \left(\ell \cdot \sin ky\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(\ell, \sin ky\right)\right)\right)\right)\right) \]

      5. sin-lowering-sin.f64 90.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(\ell, \mathsf{sin.f64}\left(ky\right)\right)\right)\right)\right)\right) \]

   8. Simplified 90.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 10^{+123}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-76}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + 4 \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{ky \cdot ky}{Om}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 5e-76)
   1.0
   (sqrt
    (+
     0.5
     (/ 0.5 (sqrt (+ 1.0 (* 4.0 (* (/ (* l l) Om) (/ (* ky ky) Om))))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5e-76) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (4.0 * (((l * l) / Om) * ((ky * ky) / Om))))))));
	}
	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 <= 5d-76) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (4.0d0 * (((l * l) / om) * ((ky * ky) / om))))))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5e-76) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (4.0 * (((l * l) / Om) * ((ky * ky) / Om))))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 5e-76:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (4.0 * (((l * l) / Om) * ((ky * ky) / Om))))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 5e-76)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(4.0 * Float64(Float64(Float64(l * l) / Om) * Float64(Float64(ky * ky) / Om))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 5e-76)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (4.0 * (((l * l) / Om) * ((ky * ky) / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 5e-76], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(4.0 * N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999998e-76

   1. Initial program 98.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 71.4%

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

      1. metadata-eval 71.4%

         \[\leadsto 1 \]

   7. Applied egg-rr 71.4%

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

   if 4.9999999999999998e-76 < l

   1. Initial program 98.1%

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

   4. Applied egg-rr 88.6%

      \[\leadsto \sqrt{\color{blue}{\frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)\right)}} + 0.5}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}\right)\right)\right)\right), \frac{1}{2}\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \cos \left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      4. *-lowering-*.f64 73.3%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, Om\right), 4\right), \mathsf{/.f64}\left(Om, \ell\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, ky\right)\right)\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \sqrt{\frac{0.5}{\sqrt{1 + \frac{\frac{\ell}{Om} \cdot 4}{\frac{Om}{\ell}} \cdot \color{blue}{\left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} + 0.5} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\left(4 \cdot \frac{{ky}^{2} \cdot {\ell}^{2}}{{Om}^{2}}\right)}\right)\right)\right), \frac{1}{2}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\frac{{ky}^{2} \cdot {\ell}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\frac{{\ell}^{2} \cdot {ky}^{2}}{{Om}^{2}}\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\frac{{\ell}^{2} \cdot {ky}^{2}}{Om \cdot Om}\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      4. times-frac N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \left(\frac{{\ell}^{2}}{Om} \cdot \frac{{ky}^{2}}{Om}\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\left(\frac{{\ell}^{2}}{Om}\right), \left(\frac{{ky}^{2}}{Om}\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({\ell}^{2}\right), Om\right), \left(\frac{{ky}^{2}}{Om}\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), Om\right), \left(\frac{{ky}^{2}}{Om}\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), \left(\frac{{ky}^{2}}{Om}\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), \mathsf{/.f64}\left(\left({ky}^{2}\right), Om\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), \mathsf{/.f64}\left(\left(ky \cdot ky\right), Om\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

      11. *-lowering-*.f64 71.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), Om\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(ky, ky\right), Om\right)\right)\right)\right)\right)\right), \frac{1}{2}\right)\right) \]

   10. Simplified 71.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-76}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + 4 \cdot \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{ky \cdot ky}{Om}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+26}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 1e+26) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1e+26) {
		tmp = 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 (l <= 1d+26) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1e+26) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1e+26:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1e+26)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1e+26)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1e+26], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.00000000000000005e26

   1. Initial program 98.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{1}\right) \]
   4. Step-by-step derivation

   5. Simplified 70.8%

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

      1. metadata-eval 70.8%

         \[\leadsto 1 \]

   7. Applied egg-rr 70.8%

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

   if 1.00000000000000005e26 < l

   1. Initial program 98.1%

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

   3. Taylor expanded in l around inf

      \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\frac{1}{2}}\right) \]
   4. Step-by-step derivation

   5. Simplified 74.7%

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

Alternative 6: 62.6% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{1}\right) \]
4. Step-by-step derivation

5. Simplified 64.8%

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

   1. metadata-eval 64.8%

      \[\leadsto 1 \]

7. Applied egg-rr 64.8%

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

Reproduce

herbie shell --seed 2024191 
(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))))))))))