Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 13.7s

Alternatives: 7

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, \left(\ell \cdot \frac{2}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}^{3}}}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (*
    0.5
    (/
     1.0
     (cbrt
      (pow
       (hypot 1.0 (* (* l (/ 2.0 Om)) (hypot (sin kx) (sin ky))))
       3.0)))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * (1.0 / cbrt(pow(hypot(1.0, ((l * (2.0 / Om)) * hypot(sin(kx), sin(ky)))), 3.0))))));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * (1.0 / Math.cbrt(Math.pow(Math.hypot(1.0, ((l * (2.0 / Om)) * Math.hypot(Math.sin(kx), Math.sin(ky)))), 3.0))))));
}

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / cbrt((hypot(1.0, Float64(Float64(l * Float64(2.0 / Om)) * hypot(sin(kx), sin(ky)))) ^ 3.0))))))
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Power[N[Power[N[Sqrt[1.0 ^ 2 + N[(N[(l * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. add-cbrt-cube 97.3%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)} \cdot \sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}}}} \]

   2. pow3 97.3%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt[3]{\color{blue}{{\left(\sqrt{1 + {\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}\right)}^{3}}}}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, \left(\ell \cdot \frac{2}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}^{3}}}}} \]

6. Final simplification 100.0%

   \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt[3]{{\left(\mathsf{hypot}\left(1, \left(\ell \cdot \frac{2}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}^{3}}}} \]

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (*
    0.5
    (/ 1.0 (hypot 1.0 (* (* l (/ 2.0 Om)) (hypot (sin kx) (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)) * hypot(sin(kx), 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.hypot(Math.sin(kx), 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.hypot(math.sin(kx), 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 * Float64(2.0 / Om)) * hypot(sin(kx), 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)) * hypot(sin(kx), 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 * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. add-sqr-sqrt 97.3%

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

   2. hypot-1-def 97.3%

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

   3. sqrt-prod 97.3%

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

   4. unpow2 97.3%

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

   5. sqrt-prod 57.0%

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

   6. add-sqr-sqrt 97.7%

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

   7. associate-/r/ 97.7%

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

   8. *-commutative 97.7%

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

   9. unpow2 97.7%

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

   10. unpow2 97.7%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 94.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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, ((2.0 * (l * Math.sin(ky))) / Om))))));
}

Python

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

Julia

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

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 * (l * sin(ky))) / Om))))));
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[(2.0 * N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. add-sqr-sqrt 97.3%

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

   2. hypot-1-def 97.3%

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

   3. sqrt-prod 97.3%

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

   4. unpow2 97.3%

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

   5. sqrt-prod 57.0%

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

   6. add-sqr-sqrt 97.7%

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

   7. associate-/r/ 97.7%

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

   8. *-commutative 97.7%

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

   9. unpow2 97.7%

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

   10. unpow2 97.7%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 95.5%

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

   1. associate-*r/ 95.5%

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

8. Simplified 95.5%

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

9. Final simplification 95.5%

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

Alternative 4: 74.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-80}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 5e-80)
   1.0
   (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (/ (* 2.0 (* l ky)) Om))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5e-80) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 * (l * ky)) / Om))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5e-80) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, ((2.0 * (l * ky)) / Om))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 5e-80:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, ((2.0 * (l * ky)) / Om))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 5e-80)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(Float64(2.0 * Float64(l * ky)) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 5e-80)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 * (l * ky)) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 5e-80], 1.0, N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[(l * ky), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5e-80

   1. Initial program 98.3%

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

   3. Simplified 98.3%

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

      1. add-sqr-sqrt 98.3%

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

      2. hypot-1-def 98.3%

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

      3. sqrt-prod 98.3%

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

      4. unpow2 98.3%

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

      5. sqrt-prod 62.8%

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

      6. add-sqr-sqrt 98.9%

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

      7. associate-/r/ 98.9%

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

      8. *-commutative 98.9%

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

      9. unpow2 98.9%

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

      10. unpow2 98.9%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 97.4%

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

      1. associate-*r/ 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in l around 0 72.5%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{1}} \]

   if 5e-80 < l

   1. Initial program 94.7%

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

   3. Simplified 94.7%

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

      1. add-sqr-sqrt 94.7%

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

      2. hypot-1-def 94.7%

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

      3. sqrt-prod 94.7%

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

      4. unpow2 94.7%

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

      5. sqrt-prod 43.4%

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

      6. add-sqr-sqrt 95.0%

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

      7. associate-/r/ 95.0%

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

      8. *-commutative 95.0%

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

      9. unpow2 95.0%

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

      10. unpow2 95.0%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 91.1%

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

      1. associate-*r/ 91.1%

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

   8. Simplified 91.1%

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

   9. Taylor expanded in ky around 0 83.7%

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

       1. associate-*r/ 83.7%

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

       2. *-commutative 83.7%

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

   11. Simplified 83.7%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-80}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}}\\ \end{array} \]

Alternative 5: 69.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 9 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 8.4e-166)
   (sqrt 0.5)
   (if (<= Om 9e+97)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* (* l l) (* ky ky)) (* Om Om))))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8.4e-166) {
		tmp = sqrt(0.5);
	} else if (Om <= 9e+97) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * (((l * l) * (ky * ky)) / (Om * Om))))))));
	} 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 <= 8.4d-166) then
        tmp = sqrt(0.5d0)
    else if (om <= 9d+97) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * (((l * l) * (ky * ky)) / (om * om))))))))
    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 <= 8.4e-166) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 9e+97) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * (((l * l) * (ky * ky)) / (Om * Om))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 8.4e-166:
		tmp = math.sqrt(0.5)
	elif Om <= 9e+97:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * (((l * l) * (ky * ky)) / (Om * Om))))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 8.4e-166)
		tmp = sqrt(0.5);
	elseif (Om <= 9e+97)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(Float64(l * l) * Float64(ky * ky)) / Float64(Om * Om))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 8.4e-166)
		tmp = sqrt(0.5);
	elseif (Om <= 9e+97)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * (((l * l) * (ky * ky)) / (Om * Om))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 8.4e-166], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 9e+97], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 8.3999999999999998e-166

   1. Initial program 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in Om around 0 49.7%

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

      1. unpow2 49.7%

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

      2. unpow2 49.7%

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

      3. hypot-def 53.3%

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

   6. Simplified 53.3%

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

   7. Taylor expanded in l around inf 61.2%

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

   if 8.3999999999999998e-166 < Om < 8.99999999999999952e97

   1. Initial program 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in kx around 0 87.7%

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

      1. *-commutative 87.7%

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

      2. associate-/l* 87.7%

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

      3. associate-*l/ 87.7%

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

      4. unpow2 87.7%

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

      5. unpow2 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in ky around 0 78.2%

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

      1. *-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. unpow2 78.2%

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

      4. unpow2 78.2%

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

   9. Simplified 78.2%

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

   if 8.99999999999999952e97 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

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

      3. sqrt-prod 100.0%

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

      4. unpow2 100.0%

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

      5. sqrt-prod 72.2%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/r/ 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. unpow2 100.0%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in l around 0 100.0%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 8.4 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 9 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 71.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 50000000.0) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 50000000.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 50000000.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 50000000.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 50000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 50000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 50000000.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5e7

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

      1. add-sqr-sqrt 98.5%

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

      2. hypot-1-def 98.5%

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

      3. sqrt-prod 98.5%

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

      4. unpow2 98.5%

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

      5. sqrt-prod 60.8%

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

      6. add-sqr-sqrt 99.0%

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

      7. associate-/r/ 99.0%

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

      8. *-commutative 99.0%

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

      9. unpow2 99.0%

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

      10. unpow2 99.0%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 95.9%

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

      1. associate-*r/ 95.9%

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

   8. Simplified 95.9%

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

   9. Taylor expanded in l around 0 71.5%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{1}} \]

   if 5e7 < l

   1. Initial program 93.0%

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

   3. Simplified 93.0%

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

   4. Taylor expanded in Om around 0 69.5%

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

      1. unpow2 69.5%

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

      2. unpow2 69.5%

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

      3. hypot-def 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in l around inf 80.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 50000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 56.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (sqrt 0.5))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

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(0.5d0)
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(0.5);
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

function code(l, Om, kx, ky)
	return sqrt(0.5)
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(0.5);
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in Om around 0 42.8%

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

   1. unpow2 42.8%

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

   2. unpow2 42.8%

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

   3. hypot-def 45.2%

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

6. Simplified 45.2%

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

7. Taylor expanded in l around inf 54.7%

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

8. Final simplification 54.7%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023271 
(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))))))))))