Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 99.7%

Time: 15.8s

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} t_0 := \frac{2 \cdot l\_m}{Om\_m}\\ \mathbf{if}\;t\_0 \leq 10^{+153}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {t\_0}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

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+153)
     (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+153) {
		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+153) 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+153) {
		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+153:
		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+153)
		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+153)
		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+153], 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) < 1e153

   1. Initial program 98.2%

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

   if 1e153 < (/.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

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

   5. Simplified 100.0%

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

Alternative 2: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} t_0 := \frac{\frac{Om\_m}{l\_m}}{l\_m \cdot 4}\\ \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \mathsf{fma}\left(\frac{\sin ky}{t\_0}, \frac{\sin ky}{Om\_m}, \frac{1 - \cos \left(2 \cdot kx\right)}{t\_0} \cdot \frac{0.5}{Om\_m}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (/ Om_m l_m) (* l_m 4.0))))
   (if (<= (/ (* 2.0 l_m) Om_m) 2e+38)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (+
          1.0
          (fma
           (/ (sin ky) t_0)
           (/ (sin ky) Om_m)
           (* (/ (- 1.0 (cos (* 2.0 kx))) t_0) (/ 0.5 Om_m))))))))
     (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 = (Om_m / l_m) / (l_m * 4.0);
	double tmp;
	if (((2.0 * l_m) / Om_m) <= 2e+38) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + fma((sin(ky) / t_0), (sin(ky) / Om_m), (((1.0 - cos((2.0 * kx))) / t_0) * (0.5 / Om_m))))))));
	} else {
		tmp = 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(Om_m / l_m) / Float64(l_m * 4.0))
	tmp = 0.0
	if (Float64(Float64(2.0 * l_m) / Om_m) <= 2e+38)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + fma(Float64(sin(ky) / t_0), Float64(sin(ky) / Om_m), Float64(Float64(Float64(1.0 - cos(Float64(2.0 * kx))) / t_0) * Float64(0.5 / Om_m))))))));
	else
		tmp = sqrt(0.5);
	end
	return 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[(Om$95$m / l$95$m), $MachinePrecision] / N[(l$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2e+38], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[Sin[ky], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / Om$95$m), $MachinePrecision] + N[(N[(N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(0.5 / Om$95$m), $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) < 1.99999999999999995e38

   1. Initial program 98.0%

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

   4. Applied egg-rr 93.5%

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

   6. Applied egg-rr 89.9%

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

   if 1.99999999999999995e38 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 94.1%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 98.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \mathsf{fma}\left(\frac{\sin ky}{\frac{\frac{Om}{\ell}}{\ell \cdot 4}}, \frac{\sin ky}{Om}, \frac{1 - \cos \left(2 \cdot kx\right)}{\frac{\frac{Om}{\ell}}{\ell \cdot 4}} \cdot \frac{0.5}{Om}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{4 \cdot \frac{l\_m}{Om\_m}}{\frac{Om\_m}{l\_m}} \cdot \left(1 + -0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 2e+38)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (*
         (/ (* 4.0 (/ l_m Om_m)) (/ Om_m l_m))
         (+ 1.0 (* -0.5 (+ (cos (* 2.0 kx)) (cos (* 2.0 ky)))))))))))
   (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 (((2.0 * l_m) / Om_m) <= 2e+38) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((4.0 * (l_m / Om_m)) / (Om_m / l_m)) * (1.0 + (-0.5 * (cos((2.0 * kx)) + cos((2.0 * ky)))))))))));
	} 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 (((2.0d0 * l_m) / om_m) <= 2d+38) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((4.0d0 * (l_m / om_m)) / (om_m / l_m)) * (1.0d0 + ((-0.5d0) * (cos((2.0d0 * kx)) + cos((2.0d0 * ky)))))))))))
    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 (((2.0 * l_m) / Om_m) <= 2e+38) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((4.0 * (l_m / Om_m)) / (Om_m / l_m)) * (1.0 + (-0.5 * (Math.cos((2.0 * kx)) + Math.cos((2.0 * ky)))))))))));
	} 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 ((2.0 * l_m) / Om_m) <= 2e+38:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((4.0 * (l_m / Om_m)) / (Om_m / l_m)) * (1.0 + (-0.5 * (math.cos((2.0 * kx)) + math.cos((2.0 * ky)))))))))))
	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 (Float64(Float64(2.0 * l_m) / Om_m) <= 2e+38)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(4.0 * Float64(l_m / Om_m)) / Float64(Om_m / l_m)) * Float64(1.0 + Float64(-0.5 * Float64(cos(Float64(2.0 * kx)) + cos(Float64(2.0 * ky)))))))))));
	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 (((2.0 * l_m) / Om_m) <= 2e+38)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((4.0 * (l_m / Om_m)) / (Om_m / l_m)) * (1.0 + (-0.5 * (cos((2.0 * kx)) + cos((2.0 * ky)))))))))));
	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[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2e+38], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m / l$95$m), $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[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.99999999999999995e38

   1. Initial program 98.0%

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

   4. Applied egg-rr 93.5%

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

   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 93.5%

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

   7. Simplified 93.5%

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

   if 1.99999999999999995e38 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 94.1%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 98.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{4 \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(1 + -0.5 \cdot \left(\cos \left(2 \cdot kx\right) + \cos \left(2 \cdot ky\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{e^{0.5 \cdot \mathsf{log1p}\left(\frac{0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)}{\frac{Om\_m}{l\_m \cdot \frac{4}{\frac{Om\_m}{l\_m}}}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 2e+38)
   (sqrt
    (+
     0.5
     (/
      0.5
      (exp
       (*
        0.5
        (log1p
         (/
          (+ 0.5 (* -0.5 (cos (* 2.0 ky))))
          (/ Om_m (* l_m (/ 4.0 (/ Om_m l_m)))))))))))
   (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 (((2.0 * l_m) / Om_m) <= 2e+38) {
		tmp = sqrt((0.5 + (0.5 / exp((0.5 * log1p(((0.5 + (-0.5 * cos((2.0 * ky)))) / (Om_m / (l_m * (4.0 / (Om_m / l_m)))))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

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 (((2.0 * l_m) / Om_m) <= 2e+38) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.exp((0.5 * Math.log1p(((0.5 + (-0.5 * Math.cos((2.0 * ky)))) / (Om_m / (l_m * (4.0 / (Om_m / l_m)))))))))));
	} 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 ((2.0 * l_m) / Om_m) <= 2e+38:
		tmp = math.sqrt((0.5 + (0.5 / math.exp((0.5 * math.log1p(((0.5 + (-0.5 * math.cos((2.0 * ky)))) / (Om_m / (l_m * (4.0 / (Om_m / l_m)))))))))))
	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 (Float64(Float64(2.0 * l_m) / Om_m) <= 2e+38)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / exp(Float64(0.5 * log1p(Float64(Float64(0.5 + Float64(-0.5 * cos(Float64(2.0 * ky)))) / Float64(Om_m / Float64(l_m * Float64(4.0 / Float64(Om_m / l_m)))))))))));
	else
		tmp = sqrt(0.5);
	end
	return 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[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2e+38], N[Sqrt[N[(0.5 + N[(0.5 / N[Exp[N[(0.5 * N[Log[1 + N[(N[(0.5 + N[(-0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m / N[(l$95$m * N[(4.0 / N[(Om$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $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) < 1.99999999999999995e38

   1. Initial program 98.0%

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

   4. Applied egg-rr 93.5%

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

   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 85.5%

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

   7. Simplified 85.5%

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

      1. pow1/2 N/A

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

      2. pow-to-exp N/A

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

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

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

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

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

   9. Applied egg-rr 85.3%

      \[\leadsto \sqrt{\frac{0.5}{\color{blue}{e^{\mathsf{log1p}\left(\frac{0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)}{\frac{Om}{\frac{4}{\frac{Om}{\ell}} \cdot \ell}}\right) \cdot 0.5}}} + 0.5} \]

   if 1.99999999999999995e38 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 94.1%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 98.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{e^{0.5 \cdot \mathsf{log1p}\left(\frac{0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)}{\frac{Om}{\ell \cdot \frac{4}{\frac{Om}{\ell}}}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot l\_m}{Om\_m} \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{4 \cdot \frac{l\_m}{Om\_m}}{\frac{Om\_m}{l\_m}} \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l_m) Om_m) 2e+38)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (+
        1.0
        (*
         (/ (* 4.0 (/ l_m Om_m)) (/ Om_m l_m))
         (+ 0.5 (* -0.5 (cos (* 2.0 ky))))))))))
   (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 (((2.0 * l_m) / Om_m) <= 2e+38) {
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((4.0 * (l_m / Om_m)) / (Om_m / l_m)) * (0.5 + (-0.5 * cos((2.0 * ky))))))))));
	} 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 (((2.0d0 * l_m) / om_m) <= 2d+38) then
        tmp = sqrt((0.5d0 + (0.5d0 / sqrt((1.0d0 + (((4.0d0 * (l_m / om_m)) / (om_m / l_m)) * (0.5d0 + ((-0.5d0) * cos((2.0d0 * ky))))))))))
    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 (((2.0 * l_m) / Om_m) <= 2e+38) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.sqrt((1.0 + (((4.0 * (l_m / Om_m)) / (Om_m / l_m)) * (0.5 + (-0.5 * Math.cos((2.0 * ky))))))))));
	} 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 ((2.0 * l_m) / Om_m) <= 2e+38:
		tmp = math.sqrt((0.5 + (0.5 / math.sqrt((1.0 + (((4.0 * (l_m / Om_m)) / (Om_m / l_m)) * (0.5 + (-0.5 * math.cos((2.0 * ky))))))))))
	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 (Float64(Float64(2.0 * l_m) / Om_m) <= 2e+38)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(Float64(1.0 + Float64(Float64(Float64(4.0 * Float64(l_m / Om_m)) / Float64(Om_m / l_m)) * Float64(0.5 + Float64(-0.5 * cos(Float64(2.0 * ky))))))))));
	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 (((2.0 * l_m) / Om_m) <= 2e+38)
		tmp = sqrt((0.5 + (0.5 / sqrt((1.0 + (((4.0 * (l_m / Om_m)) / (Om_m / l_m)) * (0.5 + (-0.5 * cos((2.0 * ky))))))))));
	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[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2e+38], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(1.0 + N[(N[(N[(4.0 * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] / N[(Om$95$m / l$95$m), $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[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 1.99999999999999995e38

   1. Initial program 98.0%

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

   4. Applied egg-rr 93.5%

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

   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 85.5%

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

   7. Simplified 85.5%

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

   if 1.99999999999999995e38 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 94.1%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 98.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 2 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{1 + \frac{4 \cdot \frac{\ell}{Om}}{\frac{Om}{\ell}} \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 10^{-133}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om\_m \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right) \cdot \left(l\_m \cdot l\_m\right)\right)}{Om\_m \cdot Om\_m}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= Om_m 1e-133)
   (sqrt 0.5)
   (if (<= Om_m 4.2e+152)
     (sqrt
      (+
       0.5
       (/
        0.5
        (+
         1.0
         (/
          (* 2.0 (* (+ 0.5 (* -0.5 (cos (* 2.0 ky)))) (* 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 <= 1e-133) {
		tmp = sqrt(0.5);
	} else if (Om_m <= 4.2e+152) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((0.5 + (-0.5 * cos((2.0 * ky)))) * (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 <= 1d-133) then
        tmp = sqrt(0.5d0)
    else if (om_m <= 4.2d+152) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + ((2.0d0 * ((0.5d0 + ((-0.5d0) * cos((2.0d0 * ky)))) * (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 <= 1e-133) {
		tmp = Math.sqrt(0.5);
	} else if (Om_m <= 4.2e+152) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((0.5 + (-0.5 * Math.cos((2.0 * ky)))) * (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 <= 1e-133:
		tmp = math.sqrt(0.5)
	elif Om_m <= 4.2e+152:
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((0.5 + (-0.5 * math.cos((2.0 * ky)))) * (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 <= 1e-133)
		tmp = sqrt(0.5);
	elseif (Om_m <= 4.2e+152)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(0.5 + Float64(-0.5 * cos(Float64(2.0 * ky)))) * 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 <= 1e-133)
		tmp = sqrt(0.5);
	elseif (Om_m <= 4.2e+152)
		tmp = sqrt((0.5 + (0.5 / (1.0 + ((2.0 * ((0.5 + (-0.5 * cos((2.0 * ky)))) * (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, 1e-133], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om$95$m, 4.2e+152], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(N[(2.0 * N[(N[(0.5 + N[(-0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $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], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 1.0000000000000001e-133

   1. Initial program 95.8%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 63.1%

      \[\leadsto \sqrt{\color{blue}{0.5}} \]

   if 1.0000000000000001e-133 < Om < 4.2000000000000003e152

   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 94.1%

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

   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 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in l around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 83.5%

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

   10. Simplified 83.5%

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

   if 4.2000000000000003e152 < 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

   3. Taylor expanded in l around 0

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

   5. Simplified 93.9%

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

      1. metadata-eval 93.9%

         \[\leadsto 1 \]

   7. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 10^{-133}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 4.2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + \frac{2 \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(2 \cdot ky\right)\right) \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;Om\_m \leq 7 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<= Om_m 7e-30) (sqrt 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 <= 7e-30) {
		tmp = sqrt(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 <= 7d-30) then
        tmp = sqrt(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 <= 7e-30) {
		tmp = Math.sqrt(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 <= 7e-30:
		tmp = math.sqrt(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 <= 7e-30)
		tmp = sqrt(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 <= 7e-30)
		tmp = sqrt(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, 7e-30], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 7.0000000000000006e-30

   1. Initial program 96.3%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 63.8%

      \[\leadsto \sqrt{\color{blue}{0.5}} \]

   if 7.0000000000000006e-30 < 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

   3. Taylor expanded in l around 0

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

   5. Simplified 78.4%

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

      1. metadata-eval 78.4%

         \[\leadsto 1 \]

   7. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.3% 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 97.3%

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

3. Taylor expanded in l around 0

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

5. Simplified 61.6%

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

   1. metadata-eval 61.6%

      \[\leadsto 1 \]

7. Applied egg-rr 61.6%

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

Reproduce

herbie shell --seed 2024185 
(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))))))))))