Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 99.6%

Time: 16.5s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} t_0 := \frac{2 \cdot l\_m}{Om\_m}\\ \mathbf{if}\;t\_0 \leq 10^{+137}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {t\_0}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l_m) Om_m)))
   (if (<= t_0 1e+137)
     (sqrt
      (*
       (/ 1.0 2.0)
       (+
        1.0
        (/
         1.0
         (sqrt
          (+
           1.0
           (* (pow t_0 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))))))
     (sqrt 0.5))))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = (2.0 * l_m) / Om_m;
	double tmp;
	if (t_0 <= 1e+137) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(t_0, 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 * l_m) / om_m
    if (t_0 <= 1d+137) then
        tmp = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((t_0 ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = (2.0 * l_m) / Om_m;
	double tmp;
	if (t_0 <= 1e+137) {
		tmp = Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(t_0, 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	t_0 = (2.0 * l_m) / Om_m
	tmp = 0
	if t_0 <= 1e+137:
		tmp = math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(t_0, 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(2.0 * l_m) / Om_m)
	tmp = 0.0
	if (t_0 <= 1e+137)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((t_0 ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	t_0 = (2.0 * l_m) / Om_m;
	tmp = 0.0;
	if (t_0 <= 1e+137)
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((t_0 ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := Block[{t$95$0 = N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+137], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1e137

   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

   if 1e137 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 91.7%

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

   3. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 7 \cdot 10^{-250}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om\_m \leq 9 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{4 \cdot \frac{l\_m \cdot l\_m}{Om\_m}}{Om\_m} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1}{0.5 + {\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}{\frac{\frac{Om\_m}{l\_m}}{l\_m} \cdot \frac{Om\_m}{4}}\right)}^{-0.5} \cdot 0.5}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= Om_m 7e-250)
   (sqrt 0.5)
   (if (<= Om_m 9e+149)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (*
           (/ (* 4.0 (/ (* l_m l_m) Om_m)) Om_m)
           (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))
     (/
      1.0
      (sqrt
       (/
        1.0
        (+
         0.5
         (*
          (pow
           (+
            1.0
            (/
             (+
              (- 0.5 (* 0.5 (cos (* 2.0 kx))))
              (- 0.5 (* 0.5 (cos (* 2.0 ky)))))
             (* (/ (/ Om_m l_m) l_m) (/ Om_m 4.0))))
           -0.5)
          0.5))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 7e-250) {
		tmp = sqrt(0.5);
	} else if (Om_m <= 9e+149) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((4.0 * ((l_m * l_m) / Om_m)) / Om_m) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))))));
	} else {
		tmp = 1.0 / sqrt((1.0 / (0.5 + (pow((1.0 + (((0.5 - (0.5 * cos((2.0 * kx)))) + (0.5 - (0.5 * cos((2.0 * ky))))) / (((Om_m / l_m) / l_m) * (Om_m / 4.0)))), -0.5) * 0.5))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om_m <= 7d-250) then
        tmp = sqrt(0.5d0)
    else if (om_m <= 9d+149) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((4.0d0 * ((l_m * l_m) / om_m)) / om_m) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))))))))
    else
        tmp = 1.0d0 / sqrt((1.0d0 / (0.5d0 + (((1.0d0 + (((0.5d0 - (0.5d0 * cos((2.0d0 * kx)))) + (0.5d0 - (0.5d0 * cos((2.0d0 * ky))))) / (((om_m / l_m) / l_m) * (om_m / 4.0d0)))) ** (-0.5d0)) * 0.5d0))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 7e-250) {
		tmp = Math.sqrt(0.5);
	} else if (Om_m <= 9e+149) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((4.0 * ((l_m * l_m) / Om_m)) / Om_m) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))))))));
	} else {
		tmp = 1.0 / Math.sqrt((1.0 / (0.5 + (Math.pow((1.0 + (((0.5 - (0.5 * Math.cos((2.0 * kx)))) + (0.5 - (0.5 * Math.cos((2.0 * ky))))) / (((Om_m / l_m) / l_m) * (Om_m / 4.0)))), -0.5) * 0.5))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if Om_m <= 7e-250:
		tmp = math.sqrt(0.5)
	elif Om_m <= 9e+149:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((4.0 * ((l_m * l_m) / Om_m)) / Om_m) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))))))))
	else:
		tmp = 1.0 / math.sqrt((1.0 / (0.5 + (math.pow((1.0 + (((0.5 - (0.5 * math.cos((2.0 * kx)))) + (0.5 - (0.5 * math.cos((2.0 * ky))))) / (((Om_m / l_m) / l_m) * (Om_m / 4.0)))), -0.5) * 0.5))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Om_m <= 7e-250)
		tmp = sqrt(0.5);
	elseif (Om_m <= 9e+149)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(4.0 * Float64(Float64(l_m * l_m) / Om_m)) / Om_m) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))))));
	else
		tmp = Float64(1.0 / sqrt(Float64(1.0 / Float64(0.5 + Float64((Float64(1.0 + Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky))))) / Float64(Float64(Float64(Om_m / l_m) / l_m) * Float64(Om_m / 4.0)))) ^ -0.5) * 0.5)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (Om_m <= 7e-250)
		tmp = sqrt(0.5);
	elseif (Om_m <= 9e+149)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((4.0 * ((l_m * l_m) / Om_m)) / Om_m) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))))));
	else
		tmp = 1.0 / sqrt((1.0 / (0.5 + (((1.0 + (((0.5 - (0.5 * cos((2.0 * kx)))) + (0.5 - (0.5 * cos((2.0 * ky))))) / (((Om_m / l_m) / l_m) * (Om_m / 4.0)))) ^ -0.5) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[Om$95$m, 7e-250], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om$95$m, 9e+149], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(4.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision] / Om$95$m), $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], N[(1.0 / N[Sqrt[N[(1.0 / N[(0.5 + N[(N[Power[N[(1.0 + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Om$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(Om$95$m / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < 6.9999999999999998e-250

   1. Initial program 97.3%

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

   3. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 6.9999999999999998e-250 < Om < 8.99999999999999965e149

   1. Initial program 97.4%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   if 8.99999999999999965e149 < Om

   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

   4. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om\_m \leq 3 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{4 \cdot \left(\left(l\_m \cdot l\_m\right) \cdot {\sin ky}^{2}\right)}{Om\_m \cdot Om\_m}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1}{0.5 + {\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}{\frac{\frac{Om\_m}{l\_m}}{l\_m} \cdot \frac{Om\_m}{4}}\right)}^{-0.5} \cdot 0.5}}}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= Om_m 9e-163)
   (sqrt 0.5)
   (if (<= Om_m 3e+149)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (/ (* 4.0 (* (* l_m l_m) (pow (sin ky) 2.0))) (* Om_m Om_m)))))))
     (/
      1.0
      (sqrt
       (/
        1.0
        (+
         0.5
         (*
          (pow
           (+
            1.0
            (/
             (+
              (- 0.5 (* 0.5 (cos (* 2.0 kx))))
              (- 0.5 (* 0.5 (cos (* 2.0 ky)))))
             (* (/ (/ Om_m l_m) l_m) (/ Om_m 4.0))))
           -0.5)
          0.5))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 9e-163) {
		tmp = sqrt(0.5);
	} else if (Om_m <= 3e+149) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((4.0 * ((l_m * l_m) * pow(sin(ky), 2.0))) / (Om_m * Om_m)))))));
	} else {
		tmp = 1.0 / sqrt((1.0 / (0.5 + (pow((1.0 + (((0.5 - (0.5 * cos((2.0 * kx)))) + (0.5 - (0.5 * cos((2.0 * ky))))) / (((Om_m / l_m) / l_m) * (Om_m / 4.0)))), -0.5) * 0.5))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om_m <= 9d-163) then
        tmp = sqrt(0.5d0)
    else if (om_m <= 3d+149) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((4.0d0 * ((l_m * l_m) * (sin(ky) ** 2.0d0))) / (om_m * om_m)))))))
    else
        tmp = 1.0d0 / sqrt((1.0d0 / (0.5d0 + (((1.0d0 + (((0.5d0 - (0.5d0 * cos((2.0d0 * kx)))) + (0.5d0 - (0.5d0 * cos((2.0d0 * ky))))) / (((om_m / l_m) / l_m) * (om_m / 4.0d0)))) ** (-0.5d0)) * 0.5d0))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 9e-163) {
		tmp = Math.sqrt(0.5);
	} else if (Om_m <= 3e+149) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((4.0 * ((l_m * l_m) * Math.pow(Math.sin(ky), 2.0))) / (Om_m * Om_m)))))));
	} else {
		tmp = 1.0 / Math.sqrt((1.0 / (0.5 + (Math.pow((1.0 + (((0.5 - (0.5 * Math.cos((2.0 * kx)))) + (0.5 - (0.5 * Math.cos((2.0 * ky))))) / (((Om_m / l_m) / l_m) * (Om_m / 4.0)))), -0.5) * 0.5))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if Om_m <= 9e-163:
		tmp = math.sqrt(0.5)
	elif Om_m <= 3e+149:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((4.0 * ((l_m * l_m) * math.pow(math.sin(ky), 2.0))) / (Om_m * Om_m)))))))
	else:
		tmp = 1.0 / math.sqrt((1.0 / (0.5 + (math.pow((1.0 + (((0.5 - (0.5 * math.cos((2.0 * kx)))) + (0.5 - (0.5 * math.cos((2.0 * ky))))) / (((Om_m / l_m) / l_m) * (Om_m / 4.0)))), -0.5) * 0.5))))
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Om_m <= 9e-163)
		tmp = sqrt(0.5);
	elseif (Om_m <= 3e+149)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(l_m * l_m) * (sin(ky) ^ 2.0))) / Float64(Om_m * Om_m)))))));
	else
		tmp = Float64(1.0 / sqrt(Float64(1.0 / Float64(0.5 + Float64((Float64(1.0 + Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx)))) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky))))) / Float64(Float64(Float64(Om_m / l_m) / l_m) * Float64(Om_m / 4.0)))) ^ -0.5) * 0.5)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (Om_m <= 9e-163)
		tmp = sqrt(0.5);
	elseif (Om_m <= 3e+149)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((4.0 * ((l_m * l_m) * (sin(ky) ^ 2.0))) / (Om_m * Om_m)))))));
	else
		tmp = 1.0 / sqrt((1.0 / (0.5 + (((1.0 + (((0.5 - (0.5 * cos((2.0 * kx)))) + (0.5 - (0.5 * cos((2.0 * ky))))) / (((Om_m / l_m) / l_m) * (Om_m / 4.0)))) ^ -0.5) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[Om$95$m, 9e-163], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om$95$m, 3e+149], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(4.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Sqrt[N[(1.0 / N[(0.5 + N[(N[Power[N[(1.0 + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(Om$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(Om$95$m / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if Om < 8.9999999999999995e-163

   1. Initial program 97.6%

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

   3. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 8.9999999999999995e-163 < Om < 3.00000000000000003e149

   1. Initial program 96.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 3.00000000000000003e149 < Om

   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

   4. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om\_m \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{4 \cdot \left(\left(l\_m \cdot l\_m\right) \cdot {\sin ky}^{2}\right)}{Om\_m \cdot Om\_m}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= Om_m 9e-163)
   (sqrt 0.5)
   (if (<= Om_m 5e+149)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (/ (* 4.0 (* (* l_m l_m) (pow (sin ky) 2.0))) (* Om_m Om_m)))))))
     1.0)))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 9e-163) {
		tmp = sqrt(0.5);
	} else if (Om_m <= 5e+149) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((4.0 * ((l_m * l_m) * pow(sin(ky), 2.0))) / (Om_m * Om_m)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om_m <= 9d-163) then
        tmp = sqrt(0.5d0)
    else if (om_m <= 5d+149) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + ((4.0d0 * ((l_m * l_m) * (sin(ky) ** 2.0d0))) / (om_m * om_m)))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 9e-163) {
		tmp = Math.sqrt(0.5);
	} else if (Om_m <= 5e+149) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + ((4.0 * ((l_m * l_m) * Math.pow(Math.sin(ky), 2.0))) / (Om_m * Om_m)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if Om_m <= 9e-163:
		tmp = math.sqrt(0.5)
	elif Om_m <= 5e+149:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + ((4.0 * ((l_m * l_m) * math.pow(math.sin(ky), 2.0))) / (Om_m * Om_m)))))))
	else:
		tmp = 1.0
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Om_m <= 9e-163)
		tmp = sqrt(0.5);
	elseif (Om_m <= 5e+149)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(l_m * l_m) * (sin(ky) ^ 2.0))) / Float64(Om_m * Om_m)))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (Om_m <= 9e-163)
		tmp = sqrt(0.5);
	elseif (Om_m <= 5e+149)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + ((4.0 * ((l_m * l_m) * (sin(ky) ^ 2.0))) / (Om_m * Om_m)))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[Om$95$m, 9e-163], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om$95$m, 5e+149], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(4.0 * N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 8.9999999999999995e-163

   1. Initial program 97.6%

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

   3. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 8.9999999999999995e-163 < Om < 4.9999999999999999e149

   1. Initial program 96.6%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in kx around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 4.9999999999999999e149 < Om

   1. Initial program 100.0%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 4.8 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om\_m \leq 4.8 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt{\frac{1}{1 + \frac{4 \cdot \left(\left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right) \cdot \left(l\_m \cdot l\_m\right)\right)}{Om\_m \cdot Om\_m}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= Om_m 4.8e-149)
   (sqrt 0.5)
   (if (<= Om_m 4.8e+149)
     (sqrt
      (+
       0.5
       (*
        0.5
        (sqrt
         (/
          1.0
          (+
           1.0
           (/
            (* 4.0 (* (+ 0.5 (* (cos (* 2.0 ky)) -0.5)) (* l_m l_m)))
            (* Om_m Om_m))))))))
     1.0)))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 4.8e-149) {
		tmp = sqrt(0.5);
	} else if (Om_m <= 4.8e+149) {
		tmp = sqrt((0.5 + (0.5 * sqrt((1.0 / (1.0 + ((4.0 * ((0.5 + (cos((2.0 * ky)) * -0.5)) * (l_m * l_m))) / (Om_m * Om_m))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om_m <= 4.8d-149) then
        tmp = sqrt(0.5d0)
    else if (om_m <= 4.8d+149) then
        tmp = sqrt((0.5d0 + (0.5d0 * sqrt((1.0d0 / (1.0d0 + ((4.0d0 * ((0.5d0 + (cos((2.0d0 * ky)) * (-0.5d0))) * (l_m * l_m))) / (om_m * om_m))))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 4.8e-149) {
		tmp = Math.sqrt(0.5);
	} else if (Om_m <= 4.8e+149) {
		tmp = Math.sqrt((0.5 + (0.5 * Math.sqrt((1.0 / (1.0 + ((4.0 * ((0.5 + (Math.cos((2.0 * ky)) * -0.5)) * (l_m * l_m))) / (Om_m * Om_m))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if Om_m <= 4.8e-149:
		tmp = math.sqrt(0.5)
	elif Om_m <= 4.8e+149:
		tmp = math.sqrt((0.5 + (0.5 * math.sqrt((1.0 / (1.0 + ((4.0 * ((0.5 + (math.cos((2.0 * ky)) * -0.5)) * (l_m * l_m))) / (Om_m * Om_m))))))))
	else:
		tmp = 1.0
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Om_m <= 4.8e-149)
		tmp = sqrt(0.5);
	elseif (Om_m <= 4.8e+149)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * sqrt(Float64(1.0 / Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(0.5 + Float64(cos(Float64(2.0 * ky)) * -0.5)) * Float64(l_m * l_m))) / Float64(Om_m * Om_m))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (Om_m <= 4.8e-149)
		tmp = sqrt(0.5);
	elseif (Om_m <= 4.8e+149)
		tmp = sqrt((0.5 + (0.5 * sqrt((1.0 / (1.0 + ((4.0 * ((0.5 + (cos((2.0 * ky)) * -0.5)) * (l_m * l_m))) / (Om_m * Om_m))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[Om$95$m, 4.8e-149], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om$95$m, 4.8e+149], N[Sqrt[N[(0.5 + N[(0.5 * N[Sqrt[N[(1.0 / N[(1.0 + N[(N[(4.0 * N[(N[(0.5 + N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 4.8000000000000002e-149

   1. Initial program 97.7%

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

   3. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 4.8000000000000002e-149 < Om < 4.80000000000000024e149

   1. Initial program 96.4%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in kx around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 4.80000000000000024e149 < Om

   1. Initial program 100.0%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 83.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 9.5 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om\_m \leq 1.26 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{4 \cdot \left(\left(0.5 + \cos \left(2 \cdot ky\right) \cdot -0.5\right) \cdot \frac{l\_m \cdot l\_m}{Om\_m \cdot Om\_m}\right) + 1}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= Om_m 9.5e-149)
   (sqrt 0.5)
   (if (<= Om_m 1.26e+150)
     (sqrt
      (+
       (/
        0.5
        (sqrt
         (+
          (*
           4.0
           (* (+ 0.5 (* (cos (* 2.0 ky)) -0.5)) (/ (* l_m l_m) (* Om_m Om_m))))
          1.0)))
       0.5))
     1.0)))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 9.5e-149) {
		tmp = sqrt(0.5);
	} else if (Om_m <= 1.26e+150) {
		tmp = sqrt(((0.5 / sqrt(((4.0 * ((0.5 + (cos((2.0 * ky)) * -0.5)) * ((l_m * l_m) / (Om_m * Om_m)))) + 1.0))) + 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om_m <= 9.5d-149) then
        tmp = sqrt(0.5d0)
    else if (om_m <= 1.26d+150) then
        tmp = sqrt(((0.5d0 / sqrt(((4.0d0 * ((0.5d0 + (cos((2.0d0 * ky)) * (-0.5d0))) * ((l_m * l_m) / (om_m * om_m)))) + 1.0d0))) + 0.5d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (Om_m <= 9.5e-149) {
		tmp = Math.sqrt(0.5);
	} else if (Om_m <= 1.26e+150) {
		tmp = Math.sqrt(((0.5 / Math.sqrt(((4.0 * ((0.5 + (Math.cos((2.0 * ky)) * -0.5)) * ((l_m * l_m) / (Om_m * Om_m)))) + 1.0))) + 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if Om_m <= 9.5e-149:
		tmp = math.sqrt(0.5)
	elif Om_m <= 1.26e+150:
		tmp = math.sqrt(((0.5 / math.sqrt(((4.0 * ((0.5 + (math.cos((2.0 * ky)) * -0.5)) * ((l_m * l_m) / (Om_m * Om_m)))) + 1.0))) + 0.5))
	else:
		tmp = 1.0
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (Om_m <= 9.5e-149)
		tmp = sqrt(0.5);
	elseif (Om_m <= 1.26e+150)
		tmp = sqrt(Float64(Float64(0.5 / sqrt(Float64(Float64(4.0 * Float64(Float64(0.5 + Float64(cos(Float64(2.0 * ky)) * -0.5)) * Float64(Float64(l_m * l_m) / Float64(Om_m * Om_m)))) + 1.0))) + 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (Om_m <= 9.5e-149)
		tmp = sqrt(0.5);
	elseif (Om_m <= 1.26e+150)
		tmp = sqrt(((0.5 / sqrt(((4.0 * ((0.5 + (cos((2.0 * ky)) * -0.5)) * ((l_m * l_m) / (Om_m * Om_m)))) + 1.0))) + 0.5));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[Om$95$m, 9.5e-149], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om$95$m, 1.26e+150], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[(4.0 * N[(N[(0.5 + N[(N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(Om$95$m * Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 9.50000000000000034e-149

   1. Initial program 97.7%

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

   3. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 9.50000000000000034e-149 < Om < 1.26e150

   1. Initial program 96.4%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in kx around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if 1.26e150 < Om

   1. Initial program 100.0%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= l_m 1.3e-9) 1.0 (sqrt 0.5)))

C

l_m = fabs(l);
Om_m = fabs(Om);
double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (l_m <= 1.3e-9) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l_m <= 1.3d-9) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
public static double code(double l_m, double Om_m, double kx, double ky) {
	double tmp;
	if (l_m <= 1.3e-9) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
def code(l_m, Om_m, kx, ky):
	tmp = 0
	if l_m <= 1.3e-9:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	tmp = 0.0
	if (l_m <= 1.3e-9)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
function tmp_2 = code(l_m, Om_m, kx, ky)
	tmp = 0.0;
	if (l_m <= 1.3e-9)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := If[LessEqual[l$95$m, 1.3e-9], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3000000000000001e-9

   1. Initial program 98.5%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 1.3000000000000001e-9 < l

   1. Initial program 94.9%

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

   3. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.1% accurate, 722.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky) :precision binary64 1.0)

C

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

Fortran

l_m = abs(l)
Om_m = abs(om)
real(8) function code(l_m, om_m, kx, ky)
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0
end function

Java

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

Python

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

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, kx, ky)
	return 1.0
end

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, kx_, ky_] := 1.0

Derivation

1. Initial program 97.7%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in l around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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))))))))))