Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 18.3s

Alternatives: 9

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

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

Python

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

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

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

Python

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

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

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

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (*
    0.5
    (pow
     (sqrt (hypot 1.0 (* (hypot (sin ky) (sin kx)) (* l (/ 2.0 Om)))))
     -2.0)))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * pow(sqrt(hypot(1.0, (hypot(sin(ky), sin(kx)) * (l * (2.0 / Om))))), -2.0))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * Math.pow(Math.sqrt(Math.hypot(1.0, (Math.hypot(Math.sin(ky), Math.sin(kx)) * (l * (2.0 / Om))))), -2.0))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * math.pow(math.sqrt(math.hypot(1.0, (math.hypot(math.sin(ky), math.sin(kx)) * (l * (2.0 / Om))))), -2.0))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * (sqrt(hypot(1.0, Float64(hypot(sin(ky), sin(kx)) * Float64(l * Float64(2.0 / Om))))) ^ -2.0))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (sqrt(hypot(1.0, (hypot(sin(ky), sin(kx)) * (l * (2.0 / Om))))) ^ -2.0))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[Sqrt[N[Sqrt[1.0 ^ 2 + N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(l * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision], -2.0], $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. Step-by-step derivation

   1. distribute-rgt-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-/l* 98.4%

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

   5. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. inv-pow 98.4%

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

   2. add-sqr-sqrt 98.4%

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

   3. unpow-prod-down 98.4%

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

5. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
6. Step-by-step derivation

   1. pow-sqr 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

   \[\leadsto \sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}} \]

Alternative 2: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\sin ky \cdot \ell\right)}{Om}\right)}\right)}^{-2}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+ 0.5 (* 0.5 (pow (sqrt (hypot 1.0 (/ (* 2.0 (* (sin ky) l)) Om))) -2.0)))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * pow(sqrt(hypot(1.0, ((2.0 * (sin(ky) * l)) / Om))), -2.0))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * Math.pow(Math.sqrt(Math.hypot(1.0, ((2.0 * (Math.sin(ky) * l)) / Om))), -2.0))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * math.pow(math.sqrt(math.hypot(1.0, ((2.0 * (math.sin(ky) * l)) / Om))), -2.0))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * (sqrt(hypot(1.0, Float64(Float64(2.0 * Float64(sin(ky) * l)) / Om))) ^ -2.0))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (sqrt(hypot(1.0, ((2.0 * (sin(ky) * l)) / Om))) ^ -2.0))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[Sqrt[N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[(N[Sin[ky], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision], -2.0], $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. Step-by-step derivation

   1. distribute-rgt-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-/l* 98.4%

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

   5. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. inv-pow 98.4%

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

   2. add-sqr-sqrt 98.4%

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

   3. unpow-prod-down 98.4%

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

5. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
6. Step-by-step derivation

   1. pow-sqr 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 93.9%

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

   1. associate-*r/ 93.9%

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

10. Simplified 93.9%

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

11. Final simplification 93.9%

    \[\leadsto \sqrt{0.5 + 0.5 \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\sin ky \cdot \ell\right)}{Om}\right)}\right)}^{-2}} \]

Alternative 3: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+ 0.5 (/ 0.5 (hypot 1.0 (* (hypot (sin ky) (sin kx)) (* 2.0 (/ l Om))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 / hypot(1.0, (hypot(sin(ky), sin(kx)) * (2.0 * (l / Om)))))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.hypot(Math.sin(ky), Math.sin(kx)) * (2.0 * (l / Om)))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.hypot(math.sin(ky), math.sin(kx)) * (2.0 * (l / Om)))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(hypot(sin(ky), sin(kx)) * Float64(2.0 * Float64(l / Om)))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 / hypot(1.0, (hypot(sin(ky), sin(kx)) * (2.0 * (l / Om)))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $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. Step-by-step derivation

   1. distribute-rgt-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-/l* 98.4%

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

   5. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. inv-pow 98.4%

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

   2. add-sqr-sqrt 98.4%

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

   3. unpow-prod-down 98.4%

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

5. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
6. Step-by-step derivation

   1. pow-sqr 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)\right)} \cdot 0.5} \]

   2. expm1-udef 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)} - 1\right)} \cdot 0.5} \]

   3. sqrt-pow2 100.0%

      \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\left(\frac{-2}{2}\right)}}\right)} - 1\right) \cdot 0.5} \]

   4. metadata-eval 100.0%

      \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\color{blue}{-1}}\right)} - 1\right) \cdot 0.5} \]

9. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)} - 1\right)} \cdot 0.5} \]
10. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)\right)} \cdot 0.5} \]

    2. expm1-log1p 100.0%

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

    3. unpow-1 100.0%

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

    4. associate-*r/ 100.0%

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

    5. associate-/l* 100.0%

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

11. Simplified 100.0%

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

    1. expm1-log1p-u 100.0%

       \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)} \cdot 0.5\right)\right)}} \]

    2. expm1-udef 100.0%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)} \cdot 0.5\right)} - 1}} \]

    3. associate-*l/ 100.0%

       \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \color{blue}{\frac{1 \cdot 0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)}}\right)} - 1} \]

    4. metadata-eval 100.0%

       \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)}\right)} - 1} \]

    5. associate-/r/ 100.0%

       \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot 2\right)}\right)}\right)} - 1} \]

13. Applied egg-rr 100.0%

    \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)\right)}\right)} - 1}} \]
14. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)\right)}\right)\right)}} \]

    2. expm1-log1p 100.0%

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

    3. *-commutative 100.0%

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

15. Simplified 100.0%

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

16. Final simplification 100.0%

    \[\leadsto \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}} \]

Alternative 4: 90.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2}{Om} \cdot \left(ky \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx}{\frac{0.5 \cdot Om}{\ell}}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= kx 2.5e-141)
   (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (* (/ 2.0 Om) (* ky l)))))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (/ (sin kx) (/ (* 0.5 Om) l))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 2.5e-141) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 / Om) * (ky * l)))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) / ((0.5 * Om) / l))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 2.5e-141) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, ((2.0 / Om) * (ky * l)))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) / ((0.5 * Om) / l))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if kx <= 2.5e-141:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, ((2.0 / Om) * (ky * l)))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.sin(kx) / ((0.5 * Om) / l))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (kx <= 2.5e-141)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(Float64(2.0 / Om) * Float64(ky * l)))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) / Float64(Float64(0.5 * Om) / l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (kx <= 2.5e-141)
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 / Om) * (ky * l)))))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) / ((0.5 * Om) / l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[kx, 2.5e-141], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 / Om), $MachinePrecision] * N[(ky * l), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] / N[(N[(0.5 * Om), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 2.5e-141

   1. Initial program 97.4%

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

      1. distribute-rgt-in 97.4%

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

      2. metadata-eval 97.4%

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

      3. metadata-eval 97.4%

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

      4. associate-/l* 97.4%

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

      5. metadata-eval 97.4%

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

   3. Simplified 97.4%

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

      1. inv-pow 97.4%

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

      2. add-sqr-sqrt 97.4%

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

      3. unpow-prod-down 97.4%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
   6. Step-by-step derivation

      1. pow-sqr 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 96.1%

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

      1. associate-*r/ 96.1%

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

   10. Simplified 96.1%

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

   11. Taylor expanded in ky around 0 88.0%

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

       1. *-un-lft-identity 88.0%

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

       2. sqrt-pow2 88.0%

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

       3. associate-/l* 88.0%

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

       4. *-commutative 88.0%

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

       5. metadata-eval 88.0%

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

   13. Applied egg-rr 88.0%

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

       1. *-lft-identity 88.0%

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

       2. unpow-1 88.0%

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

       3. associate-/r/ 88.0%

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

       4. *-commutative 88.0%

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

   15. Simplified 88.0%

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

   if 2.5e-141 < kx

   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. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. inv-pow 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. unpow-prod-down 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
   6. Step-by-step derivation

      1. pow-sqr 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)\right)} \cdot 0.5} \]

      2. expm1-udef 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)} - 1\right)} \cdot 0.5} \]

      3. sqrt-pow2 100.0%

         \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\left(\frac{-2}{2}\right)}}\right)} - 1\right) \cdot 0.5} \]

      4. metadata-eval 100.0%

         \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\color{blue}{-1}}\right)} - 1\right) \cdot 0.5} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)} - 1\right)} \cdot 0.5} \]
   10. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)\right)} \cdot 0.5} \]

       2. expm1-log1p 100.0%

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

       3. unpow-1 100.0%

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

       4. associate-*r/ 100.0%

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

       5. associate-/l* 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in ky around 0 96.5%

       \[\leadsto \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sin kx} \cdot \frac{\ell}{\frac{Om}{2}}\right)} \cdot 0.5} \]
   13. Step-by-step derivation

       1. *-un-lft-identity 96.5%

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

       2. associate-*l/ 96.5%

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

       3. metadata-eval 96.5%

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

       4. associate-*r/ 96.5%

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

       5. div-inv 96.5%

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

       6. metadata-eval 96.5%

          \[\leadsto \sqrt{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx \cdot \ell}{Om \cdot \color{blue}{0.5}}\right)}\right)} \]

   14. Applied egg-rr 96.5%

       \[\leadsto \sqrt{\color{blue}{1 \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx \cdot \ell}{Om \cdot 0.5}\right)}\right)}} \]
   15. Step-by-step derivation

       1. *-lft-identity 96.5%

          \[\leadsto \sqrt{\color{blue}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx \cdot \ell}{Om \cdot 0.5}\right)}}} \]

       2. associate-/l* 96.5%

          \[\leadsto \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \color{blue}{\frac{\sin kx}{\frac{Om \cdot 0.5}{\ell}}}\right)}} \]

   16. Simplified 96.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2}{Om} \cdot \left(ky \cdot \ell\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\sin kx}{\frac{0.5 \cdot Om}{\ell}}\right)}}\\ \end{array} \]

Alternative 5: 93.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\ell \cdot 2}{\frac{Om}{\sin ky}}\right)}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (/ (* l 2.0) (/ Om (sin ky)))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((l * 2.0) / (Om / sin(ky))))))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, ((l * 2.0) / (Om / Math.sin(ky))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, ((l * 2.0) / (Om / math.sin(ky))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(Float64(l * 2.0) / Float64(Om / sin(ky))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((l * 2.0) / (Om / sin(ky))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(l * 2.0), $MachinePrecision] / N[(Om / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $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. Step-by-step derivation

   1. distribute-rgt-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-/l* 98.4%

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

   5. metadata-eval 98.4%

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

3. Simplified 98.4%

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

   1. inv-pow 98.4%

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

   2. add-sqr-sqrt 98.4%

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

   3. unpow-prod-down 98.4%

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

5. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
6. Step-by-step derivation

   1. pow-sqr 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)\right)} \cdot 0.5} \]

   2. expm1-udef 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)} - 1\right)} \cdot 0.5} \]

   3. sqrt-pow2 100.0%

      \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\left(\frac{-2}{2}\right)}}\right)} - 1\right) \cdot 0.5} \]

   4. metadata-eval 100.0%

      \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\color{blue}{-1}}\right)} - 1\right) \cdot 0.5} \]

9. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)} - 1\right)} \cdot 0.5} \]
10. Step-by-step derivation

    1. expm1-def 100.0%

       \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)\right)} \cdot 0.5} \]

    2. expm1-log1p 100.0%

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

    3. unpow-1 100.0%

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

    4. associate-*r/ 100.0%

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

    5. associate-/l* 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in kx around 0 93.9%

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

    1. associate-*r/ 93.9%

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

    2. associate-*l* 93.9%

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

    3. associate-/l* 93.9%

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

14. Simplified 93.9%

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

15. Final simplification 93.9%

    \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{\ell \cdot 2}{\frac{Om}{\sin ky}}\right)}} \]

Alternative 6: 73.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2}{Om} \cdot \left(ky \cdot \ell\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.75e-50)
   1.0
   (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (* (/ 2.0 Om) (* ky l)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.75e-50) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 / Om) * (ky * l)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.75e-50) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, ((2.0 / Om) * (ky * l)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.75e-50:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, ((2.0 / Om) * (ky * l)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.75e-50)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(Float64(2.0 / Om) * Float64(ky * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.75e-50)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 / Om) * (ky * l)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.75e-50], 1.0, N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 / Om), $MachinePrecision] * N[(ky * l), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.74999999999999998e-50

   1. Initial program 97.7%

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

      1. distribute-rgt-in 97.7%

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

      2. metadata-eval 97.7%

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

      3. metadata-eval 97.7%

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

      4. associate-/l* 97.7%

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

      5. metadata-eval 97.7%

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

   3. Simplified 97.7%

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

      1. inv-pow 97.7%

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

      2. add-sqr-sqrt 97.7%

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

      3. unpow-prod-down 97.7%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
   6. Step-by-step derivation

      1. pow-sqr 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)\right)} \cdot 0.5} \]

      2. expm1-udef 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)} - 1\right)} \cdot 0.5} \]

      3. sqrt-pow2 100.0%

         \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\left(\frac{-2}{2}\right)}}\right)} - 1\right) \cdot 0.5} \]

      4. metadata-eval 100.0%

         \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\color{blue}{-1}}\right)} - 1\right) \cdot 0.5} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)} - 1\right)} \cdot 0.5} \]
   10. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)\right)} \cdot 0.5} \]

       2. expm1-log1p 100.0%

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

       3. unpow-1 100.0%

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

       4. associate-*r/ 100.0%

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

       5. associate-/l* 100.0%

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

   11. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)} \cdot 0.5\right)\right)}} \]

       2. expm1-udef 100.0%

          \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)} \cdot 0.5\right)} - 1}} \]

       3. associate-*l/ 100.0%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \color{blue}{\frac{1 \cdot 0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)}}\right)} - 1} \]

       4. metadata-eval 100.0%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)}\right)} - 1} \]

       5. associate-/r/ 100.0%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot 2\right)}\right)}\right)} - 1} \]

   13. Applied egg-rr 100.0%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)\right)}\right)} - 1}} \]
   14. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)\right)}\right)\right)}} \]

       2. expm1-log1p 100.0%

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

       3. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   16. Taylor expanded in l around 0 73.1%

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

   if 1.74999999999999998e-50 < l

   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. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. inv-pow 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. unpow-prod-down 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
   6. Step-by-step derivation

      1. pow-sqr 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 92.8%

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

      1. associate-*r/ 92.8%

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

   10. Simplified 92.8%

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

   11. Taylor expanded in ky around 0 85.6%

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

       1. *-un-lft-identity 85.6%

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

       2. sqrt-pow2 85.6%

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

       3. associate-/l* 85.6%

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

       4. *-commutative 85.6%

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

       5. metadata-eval 85.6%

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

   13. Applied egg-rr 85.6%

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

       1. *-lft-identity 85.6%

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

       2. unpow-1 85.6%

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

       3. associate-/r/ 85.6%

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

       4. *-commutative 85.6%

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

   15. Simplified 85.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.75 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2}{Om} \cdot \left(ky \cdot \ell\right)\right)}}\\ \end{array} \]

Alternative 7: 68.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 10^{-59}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 5000000000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1e-59)
   (sqrt 0.5)
   (if (<= Om 2e-50) 1.0 (if (<= Om 5000000000000.0) (sqrt 0.5) 1.0))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1e-59) {
		tmp = sqrt(0.5);
	} else if (Om <= 2e-50) {
		tmp = 1.0;
	} else if (Om <= 5000000000000.0) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	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 (om <= 1d-59) then
        tmp = sqrt(0.5d0)
    else if (om <= 2d-50) then
        tmp = 1.0d0
    else if (om <= 5000000000000.0d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1e-59) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 2e-50) {
		tmp = 1.0;
	} else if (Om <= 5000000000000.0) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1e-59:
		tmp = math.sqrt(0.5)
	elif Om <= 2e-50:
		tmp = 1.0
	elif Om <= 5000000000000.0:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1e-59)
		tmp = sqrt(0.5);
	elseif (Om <= 2e-50)
		tmp = 1.0;
	elseif (Om <= 5000000000000.0)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1e-59)
		tmp = sqrt(0.5);
	elseif (Om <= 2e-50)
		tmp = 1.0;
	elseif (Om <= 5000000000000.0)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1e-59], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 2e-50], 1.0, If[LessEqual[Om, 5000000000000.0], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if Om < 1e-59 or 2.00000000000000002e-50 < Om < 5e12

   1. Initial program 98.0%

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

      1. distribute-rgt-in 98.0%

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

      2. metadata-eval 98.0%

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

      3. metadata-eval 98.0%

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

      4. associate-/l* 98.0%

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

      5. metadata-eval 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in Om around 0 51.6%

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

      1. *-commutative 51.6%

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

      2. associate-*r* 51.6%

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

      3. unpow2 51.6%

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

      4. unpow2 51.6%

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

      5. hypot-def 53.3%

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

      6. associate-*l/ 53.3%

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

      7. associate-*r/ 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in l around inf 61.1%

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

   if 1e-59 < Om < 2.00000000000000002e-50 or 5e12 < 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. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. inv-pow 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. unpow-prod-down 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left({\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1} \cdot {\left(\sqrt{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}\right)}^{-1}\right)} \cdot 0.5} \]
   6. Step-by-step derivation

      1. pow-sqr 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)\right)} \cdot 0.5} \]

      2. expm1-udef 100.0%

         \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\sqrt{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}\right)}^{-2}\right)} - 1\right)} \cdot 0.5} \]

      3. sqrt-pow2 100.0%

         \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\left(\frac{-2}{2}\right)}}\right)} - 1\right) \cdot 0.5} \]

      4. metadata-eval 100.0%

         \[\leadsto \sqrt{0.5 + \left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{\color{blue}{-1}}\right)} - 1\right) \cdot 0.5} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)} - 1\right)} \cdot 0.5} \]
   10. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \sqrt{0.5 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)\right)}^{-1}\right)\right)} \cdot 0.5} \]

       2. expm1-log1p 100.0%

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

       3. unpow-1 100.0%

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

       4. associate-*r/ 100.0%

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

       5. associate-/l* 100.0%

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

   11. Simplified 100.0%

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

       1. expm1-log1p-u 100.0%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)} \cdot 0.5\right)\right)}} \]

       2. expm1-udef 100.0%

          \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{1}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)} \cdot 0.5\right)} - 1}} \]

       3. associate-*l/ 100.0%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \color{blue}{\frac{1 \cdot 0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)}}\right)} - 1} \]

       4. metadata-eval 100.0%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right)}\right)} - 1} \]

       5. associate-/r/ 100.0%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \color{blue}{\left(\frac{\ell}{Om} \cdot 2\right)}\right)}\right)} - 1} \]

   13. Applied egg-rr 100.0%

       \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)\right)}\right)} - 1}} \]
   14. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \mathsf{hypot}\left(\sin ky, \sin kx\right) \cdot \left(\frac{\ell}{Om} \cdot 2\right)\right)}\right)\right)}} \]

       2. expm1-log1p 100.0%

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

       3. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   16. Taylor expanded in l around 0 87.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 10^{-59}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 2 \cdot 10^{-50}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 5000000000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 61.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 8 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 8e+76)
   (sqrt 0.5)
   (+ 1.0 (* -0.5 (/ (* l l) (/ (* Om Om) (* ky ky)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8e+76) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * l) / ((Om * Om) / (ky * ky))));
	}
	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 (om <= 8d+76) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0 + ((-0.5d0) * ((l * l) / ((om * om) / (ky * ky))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8e+76) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * l) / ((Om * Om) / (ky * ky))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 8e+76:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0 + (-0.5 * ((l * l) / ((Om * Om) / (ky * ky))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 8e+76)
		tmp = sqrt(0.5);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(l * l) / Float64(Float64(Om * Om) / Float64(ky * ky)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 8e+76)
		tmp = sqrt(0.5);
	else
		tmp = 1.0 + (-0.5 * ((l * l) / ((Om * Om) / (ky * ky))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 8e+76], N[Sqrt[0.5], $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 8.0000000000000004e76

   1. Initial program 98.1%

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

      1. distribute-rgt-in 98.1%

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

      2. metadata-eval 98.1%

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

      3. metadata-eval 98.1%

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

      4. associate-/l* 98.1%

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

      5. metadata-eval 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in Om around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. associate-*r* 51.4%

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

      3. unpow2 51.4%

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

      4. unpow2 51.4%

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

      5. hypot-def 53.0%

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

      6. associate-*l/ 53.0%

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

      7. associate-*r/ 53.0%

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

   6. Simplified 53.0%

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

   7. Taylor expanded in l around inf 60.8%

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

   if 8.0000000000000004e76 < 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. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 82.9%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
   5. Step-by-step derivation

      1. associate-*r/ 82.9%

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

      2. *-commutative 82.9%

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

      3. associate-*r* 82.9%

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

      4. unpow2 82.9%

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

      5. unpow2 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in l around 0 78.9%

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

      1. associate-*r/ 78.9%

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

      2. associate-*r* 78.9%

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

      3. unpow2 78.9%

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

      4. unpow2 78.9%

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

   9. Simplified 78.9%

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

   10. Taylor expanded in ky around 0 60.6%

       \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{\ell}^{2} \cdot {ky}^{2}}{{Om}^{2}}} \]
   11. Step-by-step derivation

       1. associate-/l* 60.6%

          \[\leadsto 1 + -0.5 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{ky}^{2}}}} \]

       2. unpow2 60.6%

          \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{{Om}^{2}}{{ky}^{2}}} \]

       3. unpow2 60.6%

          \[\leadsto 1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{\color{blue}{Om \cdot Om}}{{ky}^{2}}} \]

       4. unpow2 60.6%

          \[\leadsto 1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{\color{blue}{ky \cdot ky}}} \]

   12. Simplified 60.6%

       \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 8 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}\\ \end{array} \]

Alternative 9: 33.1% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (+ 1.0 (* -0.5 (/ (* l l) (/ (* Om Om) (* ky ky))))))

C

double code(double l, double Om, double kx, double ky) {
	return 1.0 + (-0.5 * ((l * l) / ((Om * Om) / (ky * ky))));
}

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 + ((-0.5d0) * ((l * l) / ((om * om) / (ky * ky))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0 + (-0.5 * ((l * l) / ((Om * Om) / (ky * ky))));
}

Python

def code(l, Om, kx, ky):
	return 1.0 + (-0.5 * ((l * l) / ((Om * Om) / (ky * ky))))

Julia

function code(l, Om, kx, ky)
	return Float64(1.0 + Float64(-0.5 * Float64(Float64(l * l) / Float64(Float64(Om * Om) / Float64(ky * ky)))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = 1.0 + (-0.5 * ((l * l) / ((Om * Om) / (ky * ky))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[(1.0 + N[(-0.5 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(ky * ky), $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. Step-by-step derivation

   1. distribute-rgt-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-/l* 98.4%

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

   5. metadata-eval 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in kx around 0 79.7%

   \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]
5. Step-by-step derivation

   1. associate-*r/ 79.7%

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

   2. *-commutative 79.7%

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

   3. associate-*r* 79.7%

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

   4. unpow2 79.7%

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

   5. unpow2 79.7%

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

6. Simplified 79.7%

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

7. Taylor expanded in l around 0 50.0%

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

   1. associate-*r/ 50.0%

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

   2. associate-*r* 50.0%

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

   3. unpow2 50.0%

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

   4. unpow2 50.0%

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

9. Simplified 50.0%

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

10. Taylor expanded in ky around 0 37.6%

    \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{\ell}^{2} \cdot {ky}^{2}}{{Om}^{2}}} \]
11. Step-by-step derivation

    1. associate-/l* 36.8%

       \[\leadsto 1 + -0.5 \cdot \color{blue}{\frac{{\ell}^{2}}{\frac{{Om}^{2}}{{ky}^{2}}}} \]

    2. unpow2 36.8%

       \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{\ell \cdot \ell}}{\frac{{Om}^{2}}{{ky}^{2}}} \]

    3. unpow2 36.8%

       \[\leadsto 1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{\color{blue}{Om \cdot Om}}{{ky}^{2}}} \]

    4. unpow2 36.8%

       \[\leadsto 1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{\color{blue}{ky \cdot ky}}} \]

12. Simplified 36.8%

    \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}} \]

13. Final simplification 36.8%

    \[\leadsto 1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}} \]

Reproduce

herbie shell --seed 2023192 
(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))))))))))