Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.1%

Time: 14.2s

Alternatives: 7

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: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} t_0 := \frac{2 \cdot l\_m}{Om\_m}\\ \mathbf{if}\;{t\_0}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 5000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{l\_m \cdot \frac{l\_m \cdot 4}{Om\_m}}{Om\_m}, \left(0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{t\_0 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}\right)}\\ \end{array} \end{array} \]

FPCore

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 (<=
        (* (pow t_0 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
        5000000000.0)
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt
         (fma
          (/ (* l_m (/ (* l_m 4.0) Om_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

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 ((pow(t_0, 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 5000000000.0) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(((l_m * ((l_m * 4.0) / Om_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

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 (Float64((t_0 ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 5000000000.0)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(Float64(l_m * Float64(Float64(l_m * 4.0) / Om_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

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[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], 5000000000.0], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(N[(l$95$m * N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision] / 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] + 1.0), $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)))) < 5e9

   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 89.5%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   if 5e9 < (*.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 96.7%

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

   3. Taylor expanded in l around inf

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{2 \cdot \left(\frac{\ell}{Om} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}}\right)} \]
   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. *-lowering-*.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. /-lowering-/.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. *-lowering-*.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. accelerator-lowering-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. sin-lowering-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. sin-lowering-sin.f64 98.3

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

   5. Simplified 98.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 5000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \frac{\ell \cdot 4}{Om}}{Om}, \left(0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{2 \cdot \ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{l\_m \cdot \frac{l\_m \cdot 4}{Om\_m}}{Om\_m}, \left(0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      2e+28)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (fma
        (/ (* l_m (/ (* l_m 4.0) Om_m)) Om_m)
        (+ (+ 0.5 (* -0.5 (cos (+ kx kx)))) (+ 0.5 (* -0.5 (cos (+ ky ky)))))
        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 ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 2e+28) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(((l_m * ((l_m * 4.0) / Om_m)) / Om_m), ((0.5 + (-0.5 * cos((kx + kx)))) + (0.5 + (-0.5 * cos((ky + ky))))), 1.0)))));
	} else {
		tmp = 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(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 2e+28)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(Float64(l_m * Float64(Float64(l_m * 4.0) / Om_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(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[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $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], 2e+28], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(N[(l$95$m * N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision]), $MachinePrecision] / 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] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 1.99999999999999992e28

   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 89.0%

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

      1. associate-/l* N/A

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

      2. times-frac N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   if 1.99999999999999992e28 < (*.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 96.6%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 99.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \frac{\ell \cdot 4}{Om}}{Om}, \left(0.5 + -0.5 \cdot \cos \left(kx + kx\right)\right) + \left(0.5 + -0.5 \cdot \cos \left(ky + ky\right)\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{l\_m \cdot 4}{Om\_m}, \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right) \cdot \frac{l\_m}{Om\_m}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      4e+26)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (fma
        (/ (* l_m 4.0) Om_m)
        (* (fma -0.5 (cos (* ky -2.0)) 0.5) (/ l_m Om_m))
        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 ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 4e+26) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(((l_m * 4.0) / Om_m), (fma(-0.5, cos((ky * -2.0)), 0.5) * (l_m / Om_m)), 1.0)))));
	} else {
		tmp = 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(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 4e+26)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(Float64(l_m * 4.0) / Om_m), Float64(fma(-0.5, cos(Float64(ky * -2.0)), 0.5) * Float64(l_m / Om_m)), 1.0)))));
	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[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $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], 4e+26], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(N[(l$95$m * 4.0), $MachinePrecision] / Om$95$m), $MachinePrecision] * N[(N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 4.00000000000000019e26

   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 88.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. cos-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.3

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

   7. Simplified 88.3%

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

      1. associate-*l/ N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. frac-times N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

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

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

   9. Applied egg-rr 97.9%

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

   if 4.00000000000000019e26 < (*.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 96.6%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 99.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 4}{Om}, \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right) \cdot \frac{\ell}{Om}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, \frac{l\_m}{Om\_m} \cdot \left(\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right) \cdot \frac{l\_m}{Om\_m}\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      4e+26)
   (sqrt
    (+
     0.5
     (/
      0.5
      (fma
       2.0
       (* (/ l_m Om_m) (* (fma -0.5 (cos (* ky -2.0)) 0.5) (/ l_m Om_m)))
       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 ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 4e+26) {
		tmp = sqrt((0.5 + (0.5 / fma(2.0, ((l_m / Om_m) * (fma(-0.5, cos((ky * -2.0)), 0.5) * (l_m / Om_m))), 1.0))));
	} else {
		tmp = 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(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 4e+26)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / fma(2.0, Float64(Float64(l_m / Om_m) * Float64(fma(-0.5, cos(Float64(ky * -2.0)), 0.5) * Float64(l_m / Om_m))), 1.0))));
	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[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $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], 4e+26], N[Sqrt[N[(0.5 + N[(0.5 / N[(2.0 * N[(N[(l$95$m / Om$95$m), $MachinePrecision] * N[(N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(l$95$m / Om$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 4.00000000000000019e26

   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 88.9%

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

   5. Taylor expanded in kx around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft-neg-in N/A

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

      5. cos-neg N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.3

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

   7. Simplified 88.3%

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

   8. Taylor expanded in l around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 88.0

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

   10. Simplified 88.0%

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

       1. associate-*l* N/A

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

       2. times-frac N/A

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

       3. associate-*l/ N/A

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

       4. *-commutative N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. *-commutative N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

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

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

       13. /-lowering-/.f64 N/A

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

   12. Applied egg-rr 97.5%

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

   if 4.00000000000000019e26 < (*.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 96.6%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 99.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \left(\mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right) \cdot \frac{\ell}{Om}\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l_m) Om_m) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

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

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 = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l_m) / om_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
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 Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l_m) / Om_m), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

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

Julia

l_m = abs(l)
Om_m = abs(Om)
function code(l_m, Om_m, 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_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
code[l$95$m_, Om$95$m_, 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$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

Alternative 6: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 0.0002:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
(FPCore (l_m Om_m kx ky)
 :precision binary64
 (if (<=
      (*
       (pow (/ (* 2.0 l_m) Om_m) 2.0)
       (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      0.0002)
   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 ((pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 0.0002) {
		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 (((((2.0d0 * l_m) / om_m) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))) <= 0.0002d0) 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 ((Math.pow(((2.0 * l_m) / Om_m), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))) <= 0.0002) {
		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 (math.pow(((2.0 * l_m) / Om_m), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))) <= 0.0002:
		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 (Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.0002)
		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 (((((2.0 * l_m) / Om_m) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 0.0002)
		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[N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $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], 0.0002], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 2.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in ky around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 89.7%

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

   6. Taylor expanded in l around 0

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

   8. Simplified 99.3%

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

   if 2.0000000000000001e-4 < (*.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 96.8%

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

   3. Taylor expanded in l around inf

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

   5. Simplified 97.6%

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

Alternative 7: 62.6% accurate, 581.0× 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 98.4%

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

3. Taylor expanded in ky around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

5. Simplified 80.6%

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

6. Taylor expanded in l around 0

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

8. Simplified 61.2%

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

Reproduce

herbie shell --seed 2024198 
(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))))))))))