Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 17.8s

Alternatives: 8

Speedup: 1.4×

• Unsound rule application detected in e-graph. Results may not be sound.

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 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. 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: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

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. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 100.0%

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

   5. unpow2 100.0%

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

   6. unpow2 100.0%

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

   7. +-commutative 100.0%

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

   8. unpow2 100.0%

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

   9. unpow2 100.0%

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

   10. hypot-def 100.0%

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

   11. *-commutative 100.0%

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

   12. associate-*l/ 100.0%

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

   13. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 93.6% accurate, 1.4× speedup

Math

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

FPCore

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

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * (sqrt(hypot(1.0, Float64(2.0 * Float64(Float64(sin(kx) * 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(kx) * 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[(2.0 * N[(N[(N[Sin[kx], $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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 ky around 0 94.5%

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

9. Final simplification 94.5%

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

Alternative 4: 93.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. add-log-exp 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. hypot-1-def 100.0%

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

   4. sqrt-prod 100.0%

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

   5. unpow2 100.0%

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

   6. sqrt-prod 57.8%

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

   7. add-sqr-sqrt 100.0%

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

   8. div-inv 100.0%

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

   9. clear-num 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in ky around 0 94.5%

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

   1. *-un-lft-identity 94.5%

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

   2. add-log-exp 94.5%

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

   3. associate-*l* 94.5%

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

8. Applied egg-rr 94.5%

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

   1. *-lft-identity 94.5%

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

   2. associate-*l/ 94.5%

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

   3. metadata-eval 94.5%

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

   4. associate-*l/ 94.5%

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

   5. *-commutative 94.5%

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

   6. associate-/l* 94.5%

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

10. Simplified 94.5%

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

11. Final simplification 94.5%

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

Alternative 5: 81.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{kx \cdot kx}}}}\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+85}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0
         (sqrt
          (+
           0.5
           (*
            0.5
            (/ 1.0 (+ 1.0 (* 2.0 (/ (* l l) (/ (* Om Om) (* kx kx)))))))))))
   (if (<= l -9e+153)
     (sqrt 0.5)
     (if (<= l -1.15e-35)
       t_0
       (if (<= l 1.15e-67)
         1.0
         (if (<= l 1.65e+40) t_0 (if (<= l 1.1e+85) 1.0 (sqrt 0.5))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (kx * kx)))))))));
	double tmp;
	if (l <= -9e+153) {
		tmp = sqrt(0.5);
	} else if (l <= -1.15e-35) {
		tmp = t_0;
	} else if (l <= 1.15e-67) {
		tmp = 1.0;
	} else if (l <= 1.65e+40) {
		tmp = t_0;
	} else if (l <= 1.1e+85) {
		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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((l * l) / ((om * om) / (kx * kx)))))))))
    if (l <= (-9d+153)) then
        tmp = sqrt(0.5d0)
    else if (l <= (-1.15d-35)) then
        tmp = t_0
    else if (l <= 1.15d-67) then
        tmp = 1.0d0
    else if (l <= 1.65d+40) then
        tmp = t_0
    else if (l <= 1.1d+85) 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 t_0 = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (kx * kx)))))))));
	double tmp;
	if (l <= -9e+153) {
		tmp = Math.sqrt(0.5);
	} else if (l <= -1.15e-35) {
		tmp = t_0;
	} else if (l <= 1.15e-67) {
		tmp = 1.0;
	} else if (l <= 1.65e+40) {
		tmp = t_0;
	} else if (l <= 1.1e+85) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (kx * kx)))))))))
	tmp = 0
	if l <= -9e+153:
		tmp = math.sqrt(0.5)
	elif l <= -1.15e-35:
		tmp = t_0
	elif l <= 1.15e-67:
		tmp = 1.0
	elif l <= 1.65e+40:
		tmp = t_0
	elif l <= 1.1e+85:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(l * l) / Float64(Float64(Om * Om) / Float64(kx * kx)))))))))
	tmp = 0.0
	if (l <= -9e+153)
		tmp = sqrt(0.5);
	elseif (l <= -1.15e-35)
		tmp = t_0;
	elseif (l <= 1.15e-67)
		tmp = 1.0;
	elseif (l <= 1.65e+40)
		tmp = t_0;
	elseif (l <= 1.1e+85)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (kx * kx)))))))));
	tmp = 0.0;
	if (l <= -9e+153)
		tmp = sqrt(0.5);
	elseif (l <= -1.15e-35)
		tmp = t_0;
	elseif (l <= 1.15e-67)
		tmp = 1.0;
	elseif (l <= 1.65e+40)
		tmp = t_0;
	elseif (l <= 1.1e+85)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -9e+153], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, -1.15e-35], t$95$0, If[LessEqual[l, 1.15e-67], 1.0, If[LessEqual[l, 1.65e+40], t$95$0, If[LessEqual[l, 1.1e+85], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.0000000000000002e153 or 1.1000000000000001e85 < 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. Taylor expanded in l around -inf 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l* 76.4%

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

      3. unpow2 76.4%

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

      4. unpow2 76.4%

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

      5. hypot-def 76.4%

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

   6. Simplified 76.4%

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

   7. Taylor expanded in l around inf 80.2%

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

   if -9.0000000000000002e153 < l < -1.1499999999999999e-35 or 1.15e-67 < l < 1.6499999999999999e40

   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 ky around 0 90.1%

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

      1. associate-/l* 91.7%

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

      2. associate-*r/ 91.7%

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

      3. unpow2 91.7%

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

      4. unpow2 91.7%

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

      5. times-frac 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in kx around 0 78.3%

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

      1. associate-/l* 81.5%

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

      2. unpow2 81.5%

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

      3. unpow2 81.5%

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

      4. unpow2 81.5%

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

   9. Simplified 81.5%

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

   if -1.1499999999999999e-35 < l < 1.15e-67 or 1.6499999999999999e40 < l < 1.1000000000000001e85

   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. add-log-exp 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

      4. sqrt-prod 100.0%

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

      5. unpow2 100.0%

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

      6. sqrt-prod 61.6%

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

      7. add-sqr-sqrt 100.0%

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

      8. div-inv 100.0%

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

      9. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in l around 0 89.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{kx \cdot kx}}}}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{kx \cdot kx}}}}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+85}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 82.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{kx \cdot kx}}}}\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{-92}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l -7.8e+149)
   (sqrt 0.5)
   (if (<= l -9e-37)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* l l) (/ (* Om Om) (* kx kx)))))))))
     (if (<= l 4.7e-92)
       1.0
       (if (<= l 1.8e+128)
         (sqrt
          (+
           0.5
           (*
            0.5
            (/ 1.0 (+ 1.0 (* 2.0 (/ (* l l) (/ (* Om Om) (* ky ky)))))))))
         (sqrt 0.5))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= -7.8e+149) {
		tmp = sqrt(0.5);
	} else if (l <= -9e-37) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (kx * kx)))))))));
	} else if (l <= 4.7e-92) {
		tmp = 1.0;
	} else if (l <= 1.8e+128) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	} 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 <= (-7.8d+149)) then
        tmp = sqrt(0.5d0)
    else if (l <= (-9d-37)) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((l * l) / ((om * om) / (kx * kx)))))))))
    else if (l <= 4.7d-92) then
        tmp = 1.0d0
    else if (l <= 1.8d+128) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((l * l) / ((om * om) / (ky * ky)))))))))
    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 <= -7.8e+149) {
		tmp = Math.sqrt(0.5);
	} else if (l <= -9e-37) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (kx * kx)))))))));
	} else if (l <= 4.7e-92) {
		tmp = 1.0;
	} else if (l <= 1.8e+128) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= -7.8e+149:
		tmp = math.sqrt(0.5)
	elif l <= -9e-37:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (kx * kx)))))))))
	elif l <= 4.7e-92:
		tmp = 1.0
	elif l <= 1.8e+128:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= -7.8e+149)
		tmp = sqrt(0.5);
	elseif (l <= -9e-37)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(l * l) / Float64(Float64(Om * Om) / Float64(kx * kx)))))))));
	elseif (l <= 4.7e-92)
		tmp = 1.0;
	elseif (l <= 1.8e+128)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(l * l) / Float64(Float64(Om * Om) / Float64(ky * ky)))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= -7.8e+149)
		tmp = sqrt(0.5);
	elseif (l <= -9e-37)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (kx * kx)))))))));
	elseif (l <= 4.7e-92)
		tmp = 1.0;
	elseif (l <= 1.8e+128)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky)))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, -7.8e+149], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, -9e-37], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 4.7e-92], 1.0, If[LessEqual[l, 1.8e+128], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -7.7999999999999998e149 or 1.80000000000000014e128 < 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. Taylor expanded in l around -inf 77.7%

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

      1. *-commutative 77.7%

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

      2. associate-*l* 77.7%

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

      3. unpow2 77.7%

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

      4. unpow2 77.7%

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

      5. hypot-def 77.7%

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

   6. Simplified 77.7%

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

   7. Taylor expanded in l around inf 81.1%

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

   if -7.7999999999999998e149 < l < -9.00000000000000081e-37

   1. Initial program 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-/l* 99.9%

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

      5. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in ky around 0 88.3%

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

      1. associate-/l* 88.3%

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

      2. associate-*r/ 88.3%

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

      3. unpow2 88.3%

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

      4. unpow2 88.3%

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

      5. times-frac 88.3%

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

   6. Simplified 88.3%

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

   7. Taylor expanded in kx around 0 77.9%

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

      1. associate-/l* 81.1%

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

      2. unpow2 81.1%

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

      3. unpow2 81.1%

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

      4. unpow2 81.1%

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

   9. Simplified 81.1%

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

   if -9.00000000000000081e-37 < l < 4.69999999999999993e-92

   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. add-log-exp 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

      4. sqrt-prod 100.0%

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

      5. unpow2 100.0%

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

      6. sqrt-prod 64.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. div-inv 100.0%

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

      9. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in l around 0 90.2%

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

   if 4.69999999999999993e-92 < l < 1.80000000000000014e128

   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 96.2%

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

      1. associate-*r/ 96.2%

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

      2. *-commutative 96.2%

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

      3. associate-*r* 96.2%

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

      4. unpow2 96.2%

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

      5. unpow2 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in ky around 0 76.6%

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

      1. associate-/l* 78.7%

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

      2. unpow2 78.7%

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

      3. unpow2 78.7%

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

      4. unpow2 78.7%

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

   9. Simplified 78.7%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}} \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq -9 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{kx \cdot kx}}}}\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{-92}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 78.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq -1.02 \cdot 10^{+80}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -1.55 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq -2.5 \cdot 10^{-86}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 10^{-37}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om -1.02e+80)
   1.0
   (if (<= Om -1.55e+65)
     (sqrt 0.5)
     (if (<= Om -2.5e-86) 1.0 (if (<= Om 1e-37) (sqrt 0.5) 1.0)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= -1.02e+80) {
		tmp = 1.0;
	} else if (Om <= -1.55e+65) {
		tmp = sqrt(0.5);
	} else if (Om <= -2.5e-86) {
		tmp = 1.0;
	} else if (Om <= 1e-37) {
		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 <= (-1.02d+80)) then
        tmp = 1.0d0
    else if (om <= (-1.55d+65)) then
        tmp = sqrt(0.5d0)
    else if (om <= (-2.5d-86)) then
        tmp = 1.0d0
    else if (om <= 1d-37) 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 <= -1.02e+80) {
		tmp = 1.0;
	} else if (Om <= -1.55e+65) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= -2.5e-86) {
		tmp = 1.0;
	} else if (Om <= 1e-37) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= -1.02e+80:
		tmp = 1.0
	elif Om <= -1.55e+65:
		tmp = math.sqrt(0.5)
	elif Om <= -2.5e-86:
		tmp = 1.0
	elif Om <= 1e-37:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= -1.02e+80)
		tmp = 1.0;
	elseif (Om <= -1.55e+65)
		tmp = sqrt(0.5);
	elseif (Om <= -2.5e-86)
		tmp = 1.0;
	elseif (Om <= 1e-37)
		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 <= -1.02e+80)
		tmp = 1.0;
	elseif (Om <= -1.55e+65)
		tmp = sqrt(0.5);
	elseif (Om <= -2.5e-86)
		tmp = 1.0;
	elseif (Om <= 1e-37)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, -1.02e+80], 1.0, If[LessEqual[Om, -1.55e+65], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, -2.5e-86], 1.0, If[LessEqual[Om, 1e-37], N[Sqrt[0.5], $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if Om < -1.02e80 or -1.54999999999999995e65 < Om < -2.4999999999999999e-86 or 1.00000000000000007e-37 < 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. add-log-exp 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

      4. sqrt-prod 100.0%

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

      5. unpow2 100.0%

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

      6. sqrt-prod 62.7%

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

      7. add-sqr-sqrt 100.0%

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

      8. div-inv 100.0%

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

      9. clear-num 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in l around 0 87.9%

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

   if -1.02e80 < Om < -1.54999999999999995e65 or -2.4999999999999999e-86 < Om < 1.00000000000000007e-37

   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 l around -inf 76.3%

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

      1. *-commutative 76.3%

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

      2. associate-*l* 76.3%

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

      3. unpow2 76.3%

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

      4. unpow2 76.3%

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

      5. hypot-def 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in l around inf 80.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -1.02 \cdot 10^{+80}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -1.55 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq -2.5 \cdot 10^{-86}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 10^{-37}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 56.0% 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 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 l around -inf 40.4%

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

   1. *-commutative 40.4%

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

   2. associate-*l* 40.4%

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

   3. unpow2 40.4%

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

   4. unpow2 40.4%

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

   5. hypot-def 40.4%

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

6. Simplified 40.4%

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

7. Taylor expanded in l around inf 50.5%

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

8. Final simplification 50.5%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023178 
(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))))))))))