Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 98.0%

Time: 14.9s

Alternatives: 7

Speedup: N/A×

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: 98.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}}}\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))
        (/ (* l (* (/ l Om) 4.0)) Om)))))))))

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(sin(kx), 2.0) + pow(sin(ky), 2.0)) * ((l * ((l / Om) * 4.0)) / Om))))))));
}

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 + (((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)) * ((l * ((l / om) * 4.0d0)) / om))))))))
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(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)) * ((l * ((l / Om) * 4.0)) / Om))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + ((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)) * ((l * ((l / Om) * 4.0)) / Om))))))))

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((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * Float64(Float64(l * Float64(Float64(l / Om) * 4.0)) / Om))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * ((l * ((l / Om) * 4.0)) / Om))))))));
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[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(N[(l / Om), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot \frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}}}\right)} \]
6. Add Preprocessing

Alternative 2: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{1 + \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}, \left(0.5 + -0.5\right) + ky \cdot ky, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (/
       1.0
       (sqrt
        (+
         1.0
         (*
          (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))
          (pow (/ (* 2.0 l) Om) 2.0)))))
      2e-6)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (fma (/ (* l (* (/ l Om) 4.0)) Om) (+ (+ 0.5 -0.5) (* ky ky)) 1.0)))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((1.0 / sqrt((1.0 + ((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * pow(((2.0 * l) / Om), 2.0))))) <= 2e-6) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(((l * ((l / Om) * 4.0)) / Om), ((0.5 + -0.5) + (ky * ky)), 1.0)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (Float64(Float64(2.0 * l) / Om) ^ 2.0))))) <= 2e-6)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(Float64(l * Float64(Float64(l / Om) * 4.0)) / Om), Float64(Float64(0.5 + -0.5) + Float64(ky * ky)), 1.0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-6], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(N[(l * N[(N[(l / Om), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(0.5 + -0.5), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.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))))))) < 1.99999999999999991e-6

   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 73.3%

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

   5. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.0

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

   7. Simplified 86.0%

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

   8. Taylor expanded in kx around 0

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

   10. Simplified 77.3%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

       3. associate-*r/ N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 83.2

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

   12. Applied egg-rr 83.2%

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

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.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 99.2%

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

   3. Taylor expanded in l around 0

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

   5. Simplified 98.7%

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

      1. metadata-eval 98.7

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

   7. Applied egg-rr 98.7%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{1 + \left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}}} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}, \left(0.5 + -0.5\right) + ky \cdot ky, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)) (pow (/ (* 2.0 l) Om) 2.0))
      2e-9)
   1.0
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)) * pow(((2.0 * l) / Om), 2.0)) <= 2e-9) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if ((((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)) * (((2.0d0 * l) / om) ** 2.0d0)) <= 2d-9) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)) * Math.pow(((2.0 * l) / Om), 2.0)) <= 2e-9) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)) * math.pow(((2.0 * l) / Om), 2.0)) <= 2e-9:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (Float64(Float64(2.0 * l) / Om) ^ 2.0)) <= 2e-9)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)) * (((2.0 * l) / Om) ^ 2.0)) <= 2e-9)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-9], 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)))) < 2.00000000000000012e-9

   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

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

   5. Simplified 99.4%

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

      1. metadata-eval 99.4

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

   7. Applied egg-rr 99.4%

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

   if 2.00000000000000012e-9 < (*.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 99.2%

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

   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 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\sin kx}^{2} + {\sin ky}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell}{Om} \cdot 4\\ \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(t\_0, \frac{\ell}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right) + \cos \left(kx \cdot -2\right), 1\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot t\_0}{Om}, \left(0.5 + -0.5\right) + ky \cdot ky, 1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (* (/ l Om) 4.0)))
   (if (<= (/ (* 2.0 l) Om) 5e+15)
     (sqrt
      (*
       0.5
       (+
        1.0
        (/
         1.0
         (sqrt
          (fma
           t_0
           (* (/ l Om) (fma -0.5 (+ (cos (* ky -2.0)) (cos (* kx -2.0))) 1.0))
           1.0))))))
     (sqrt
      (+
       0.5
       (/
        0.5
        (sqrt (fma (/ (* l t_0) Om) (+ (+ 0.5 -0.5) (* ky ky)) 1.0))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = (l / Om) * 4.0;
	double tmp;
	if (((2.0 * l) / Om) <= 5e+15) {
		tmp = sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(t_0, ((l / Om) * fma(-0.5, (cos((ky * -2.0)) + cos((kx * -2.0))), 1.0)), 1.0))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(((l * t_0) / Om), ((0.5 + -0.5) + (ky * ky)), 1.0)))));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	t_0 = Float64(Float64(l / Om) * 4.0)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 5e+15)
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(t_0, Float64(Float64(l / Om) * fma(-0.5, Float64(cos(Float64(ky * -2.0)) + cos(Float64(kx * -2.0))), 1.0)), 1.0))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(Float64(l * t_0) / Om), Float64(Float64(0.5 + -0.5) + Float64(ky * ky)), 1.0)))));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(N[(l / Om), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 5e+15], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(t$95$0 * N[(N[(l / Om), $MachinePrecision] * N[(-0.5 * N[(N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(kx * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(N[(l * t$95$0), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(0.5 + -0.5), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 5e15

   1. Initial program 99.5%

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

   4. Applied egg-rr 94.5%

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

      1. metadata-eval 94.5

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

   6. Applied egg-rr 94.5%

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

   7. Taylor expanded in kx around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

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

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

      14. cos-neg N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 94.5

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

   9. Simplified 94.5%

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

   if 5e15 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 100.0%

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

   4. Applied egg-rr 73.5%

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

   5. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.8

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

   7. Simplified 85.8%

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

   8. Taylor expanded in kx around 0

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

   10. Simplified 76.1%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

       3. associate-*r/ N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 83.5

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

   12. Applied egg-rr 83.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{\ell}{Om} \cdot 4, \frac{\ell}{Om} \cdot \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right) + \cos \left(kx \cdot -2\right), 1\right), 1\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}, \left(0.5 + -0.5\right) + ky \cdot ky, 1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \frac{\ell \cdot 4}{Om}}{Om}, \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}, \left(0.5 + -0.5\right) + ky \cdot ky, 1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 5e+15)
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (fma
        (/ (* l (/ (* l 4.0) Om)) Om)
        (fma -0.5 (cos (* ky -2.0)) 0.5)
        1.0)))))
   (sqrt
    (+
     0.5
     (/
      0.5
      (sqrt
       (fma (/ (* l (* (/ l Om) 4.0)) Om) (+ (+ 0.5 -0.5) (* ky ky)) 1.0)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (((2.0 * l) / Om) <= 5e+15) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(((l * ((l * 4.0) / Om)) / Om), fma(-0.5, cos((ky * -2.0)), 0.5), 1.0)))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(((l * ((l / Om) * 4.0)) / Om), ((0.5 + -0.5) + (ky * ky)), 1.0)))));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 5e+15)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(Float64(l * Float64(Float64(l * 4.0) / Om)) / Om), fma(-0.5, cos(Float64(ky * -2.0)), 0.5), 1.0)))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(Float64(Float64(l * Float64(Float64(l / Om) * 4.0)) / Om), Float64(Float64(0.5 + -0.5) + Float64(ky * ky)), 1.0)))));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 5e+15], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(N[(l * N[(N[(l * 4.0), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(-0.5 * N[Cos[N[(ky * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(N[(N[(l * N[(N[(l / Om), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(0.5 + -0.5), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 5e15

   1. Initial program 99.5%

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

   4. Applied egg-rr 80.5%

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

   5. 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. lower-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. lower-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. lower-*.f64 75.5

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

   7. Simplified 75.5%

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

      1. associate-*r* N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 86.6

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 86.6

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

   9. Applied egg-rr 86.6%

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

   if 5e15 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 100.0%

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

   4. Applied egg-rr 73.5%

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

   5. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.8

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

   7. Simplified 85.8%

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

   8. Taylor expanded in kx around 0

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

   10. Simplified 76.1%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

       3. associate-*r/ N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 83.5

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

   12. Applied egg-rr 83.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \frac{\ell \cdot 4}{Om}}{Om}, \mathsf{fma}\left(-0.5, \cos \left(ky \cdot -2\right), 0.5\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}, \left(0.5 + -0.5\right) + ky \cdot ky, 1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}\\ \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 50000000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.5, \cos \left(-ky\right), 0.5\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(t\_0, \left(0.5 + -0.5\right) + ky \cdot ky, 1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (/ (* l (* (/ l Om) 4.0)) Om)))
   (if (<= (/ (* 2.0 l) Om) 50000000000000.0)
     (sqrt (+ 0.5 (/ 0.5 (sqrt (fma t_0 (fma -0.5 (cos (- ky)) 0.5) 1.0)))))
     (sqrt (+ 0.5 (/ 0.5 (sqrt (fma t_0 (+ (+ 0.5 -0.5) (* ky ky)) 1.0))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = (l * ((l / Om) * 4.0)) / Om;
	double tmp;
	if (((2.0 * l) / Om) <= 50000000000000.0) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(t_0, fma(-0.5, cos(-ky), 0.5), 1.0)))));
	} else {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(t_0, ((0.5 + -0.5) + (ky * ky)), 1.0)))));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	t_0 = Float64(Float64(l * Float64(Float64(l / Om) * 4.0)) / Om)
	tmp = 0.0
	if (Float64(Float64(2.0 * l) / Om) <= 50000000000000.0)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(t_0, fma(-0.5, cos(Float64(-ky)), 0.5), 1.0)))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(t_0, Float64(Float64(0.5 + -0.5) + Float64(ky * ky)), 1.0)))));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(N[(l * N[(N[(l / Om), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 50000000000000.0], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(t$95$0 * N[(-0.5 * N[Cos[(-ky)], $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(t$95$0 * N[(N[(0.5 + -0.5), $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 5e13

   1. Initial program 99.5%

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

   4. Applied egg-rr 80.5%

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

   5. 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. lower-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. lower-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. lower-*.f64 75.5

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

   7. Simplified 75.5%

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

   8. Taylor expanded in ky around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.3

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

   10. Simplified 75.3%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

       3. associate-*r/ N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 86.4

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

   12. Applied egg-rr 86.4%

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

   if 5e13 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 100.0%

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

   4. Applied egg-rr 73.5%

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

   5. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.8

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

   7. Simplified 85.8%

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

   8. Taylor expanded in kx around 0

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

   10. Simplified 76.1%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

       3. associate-*r/ N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 83.5

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

   12. Applied egg-rr 83.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 \cdot \ell}{Om} \leq 50000000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}, \mathsf{fma}\left(-0.5, \cos \left(-ky\right), 0.5\right), 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot \left(\frac{\ell}{Om} \cdot 4\right)}{Om}, \left(0.5 + -0.5\right) + ky \cdot ky, 1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.2% accurate, 581.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in l around 0

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

5. Simplified 59.3%

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

   1. metadata-eval 59.3

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

7. Applied egg-rr 59.3%

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

Reproduce

herbie shell --seed 2024214 
(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))))))))))