Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 100.0%

Time: 15.1s

Alternatives: 5

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: 98.2% 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, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-sqr-sqrt 97.3%

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

   2. hypot-1-def 97.3%

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

   3. sqrt-prod 97.3%

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

   4. sqrt-pow1 97.9%

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

   5. metadata-eval 97.9%

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

   6. div-inv 97.9%

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

   7. pow1 97.9%

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

   8. clear-num 97.9%

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

   9. unpow2 97.9%

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

   10. unpow2 97.9%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*l* 99.2%

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

7. Applied egg-rr 99.2%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. *-commutative 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 93.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-sqr-sqrt 97.3%

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

   2. hypot-1-def 97.3%

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

   3. sqrt-prod 97.3%

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

   4. sqrt-pow1 97.9%

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

   5. metadata-eval 97.9%

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

   6. div-inv 97.9%

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

   7. pow1 97.9%

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

   8. clear-num 97.9%

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

   9. unpow2 97.9%

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

   10. unpow2 97.9%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*l* 99.2%

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

7. Applied egg-rr 99.2%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. *-commutative 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in ky around 0 93.6%

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

11. Final simplification 93.6%

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

Alternative 3: 94.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-sqr-sqrt 97.3%

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

   2. hypot-1-def 97.3%

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

   3. sqrt-prod 97.3%

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

   4. sqrt-pow1 97.9%

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

   5. metadata-eval 97.9%

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

   6. div-inv 97.9%

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

   7. pow1 97.9%

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

   8. clear-num 97.9%

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

   9. unpow2 97.9%

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

   10. unpow2 97.9%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*l* 99.2%

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

7. Applied egg-rr 99.2%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. *-commutative 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in kx around 0 93.0%

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

11. Final simplification 93.0%

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

Alternative 4: 70.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{+23}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 9e+23) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 9e+23) {
		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 <= 9d+23) 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 <= 9e+23) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 9e+23:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 9e+23)
		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 <= 9e+23)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 9e+23], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.99999999999999958e23

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

      1. add-sqr-sqrt 98.4%

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

      2. hypot-1-def 98.4%

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

      3. sqrt-prod 98.4%

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

      4. sqrt-pow1 99.1%

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

      5. metadata-eval 99.1%

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

      6. div-inv 99.1%

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

      7. pow1 99.1%

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

      8. clear-num 99.1%

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

      9. unpow2 99.1%

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

      10. unpow2 99.1%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 99.4%

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

      2. expm1-udef 99.4%

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

      3. associate-*l/ 99.4%

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

      4. metadata-eval 99.4%

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

      5. associate-*r* 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-*l* 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. expm1-def 99.4%

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

      2. expm1-log1p 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in Om around inf 70.0%

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

   if 8.99999999999999958e23 < l

   1. Initial program 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in Om around 0 76.9%

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

      1. unpow2 76.9%

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

      2. unpow2 76.9%

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

      3. hypot-def 82.9%

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

   6. Simplified 82.9%

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

   7. Taylor expanded in l around inf 85.0%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{+23}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 62.6% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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%

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

   1. add-sqr-sqrt 97.3%

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

   2. hypot-1-def 97.3%

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

   3. sqrt-prod 97.3%

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

   4. sqrt-pow1 97.9%

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

   5. metadata-eval 97.9%

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

   6. div-inv 97.9%

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

   7. pow1 97.9%

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

   8. clear-num 97.9%

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

   9. unpow2 97.9%

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

   10. unpow2 97.9%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

   3. associate-*l/ 99.2%

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

   4. metadata-eval 99.2%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*l* 99.2%

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

7. Applied egg-rr 99.2%

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

   1. expm1-def 99.2%

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

   2. expm1-log1p 100.0%

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

   3. associate-*l/ 100.0%

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

   4. associate-*l* 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-*r* 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. *-commutative 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in Om around inf 60.5%

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

11. Final simplification 60.5%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023334 
(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))))))))))