Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.6%

Time: 15.6s

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: 98.4% 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

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := \frac{2 \cdot l\_m}{Om\_m}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {t\_0}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l_m) Om_m)))
   (if (<= t_0 2e+129)
     (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

Om_m = fabs(Om);
l_m = fabs(l);
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 <= 2e+129) {
		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

Om_m = abs(om)
l_m = abs(l)
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 <= 2d+129) 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

Om_m = Math.abs(Om);
l_m = Math.abs(l);
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 <= 2e+129) {
		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

Om_m = math.fabs(Om)
l_m = math.fabs(l)
def code(l_m, Om_m, kx, ky):
	t_0 = (2.0 * l_m) / Om_m
	tmp = 0
	if t_0 <= 2e+129:
		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

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(2.0 * l_m) / Om_m)
	tmp = 0.0
	if (t_0 <= 2e+129)
		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

Om_m = abs(Om);
l_m = abs(l);
function tmp_2 = code(l_m, Om_m, kx, ky)
	t_0 = (2.0 * l_m) / Om_m;
	tmp = 0.0;
	if (t_0 <= 2e+129)
		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

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $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, 2e+129], 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) < 2e129

   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

   if 2e129 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 93.9%

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

   3. Taylor expanded in l around inf

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

   5. Applied rewrites 99.2%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} Om_m = \left|Om\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_0 := \frac{2 \cdot l\_m}{Om\_m}\\ \mathbf{if}\;{t\_0}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 2000000:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{l\_m}{Om\_m} \cdot 4, \frac{l\_m}{Om\_m} \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right)\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{t\_0 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}\right)}\\ \end{array} \end{array} \]

FPCore

Om_m = (fabs.f64 Om)
l_m = (fabs.f64 l)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l_m) Om_m)))
   (if (<=
        (* (pow t_0 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
        2000000.0)
     (sqrt
      (*
       (/ 1.0 2.0)
       (+
        1.0
        (/
         1.0
         (sqrt
          (fma
           (* (/ l_m Om_m) 4.0)
           (*
            (/ l_m Om_m)
            (+
             (+ 0.5 (* -0.5 (cos (+ kx kx))))
             (+ 0.5 (* -0.5 (cos (+ ky ky))))))
           1.0))))))
     (sqrt
      (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (* t_0 (hypot (sin kx) (sin ky))))))))))

C

Om_m = fabs(Om);
l_m = fabs(l);
double code(double l_m, double Om_m, double kx, double ky) {
	double t_0 = (2.0 * l_m) / Om_m;
	double tmp;
	if ((pow(t_0, 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 2000000.0) {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt(fma(((l_m / Om_m) * 4.0), ((l_m / Om_m) * ((0.5 + (-0.5 * cos((kx + kx)))) + (0.5 + (-0.5 * cos((ky + ky)))))), 1.0))))));
	} else {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (t_0 * hypot(sin(kx), sin(ky)))))));
	}
	return tmp;
}

Julia

Om_m = abs(Om)
l_m = abs(l)
function code(l_m, Om_m, kx, ky)
	t_0 = Float64(Float64(2.0 * l_m) / Om_m)
	tmp = 0.0
	if (Float64((t_0 ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 2000000.0)
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(fma(Float64(Float64(l_m / Om_m) * 4.0), Float64(Float64(l_m / Om_m) * Float64(Float64(0.5 + Float64(-0.5 * cos(Float64(kx + kx)))) + Float64(0.5 + Float64(-0.5 * cos(Float64(ky + ky)))))), 1.0))))));
	else
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / Float64(t_0 * hypot(sin(kx), sin(ky)))))));
	end
	return tmp
end

Wolfram

Om_m = N[Abs[Om], $MachinePrecision]
l_m = N[Abs[l], $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[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], 2000000.0], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(N[(N[(l$95$m / Om$95$m), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(N[(0.5 + N[(-0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(-0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[(t$95$0 * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 2e6

   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 rewrites 99.7%

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

   if 2e6 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 96.5%

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

   3. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

Reproduce

herbie shell --seed 2024223 
(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))))))))))