Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 18.6s

Alternatives: 7

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% 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

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. inv-pow 99.2%

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

   2. add-sqr-sqrt 99.2%

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

   3. unpow-prod-down 99.2%

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

5. Applied egg-rr 100.0%

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

   1. pow-sqr 100.0%

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

   2. associate-*l* 100.0%

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

   3. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 93.7% accurate, 1.4× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. inv-pow 99.2%

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

   2. add-sqr-sqrt 99.2%

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

   3. unpow-prod-down 99.2%

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

5. Applied egg-rr 100.0%

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

   1. pow-sqr 100.0%

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

   2. associate-*l* 100.0%

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

   3. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 93.8%

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

9. Final simplification 93.8%

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

Alternative 3: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

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

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 / (2.0 * l)))))))

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(2.0 * l)))))))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. hypot-1-def 99.2%

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

   3. sqrt-prod 99.2%

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

   4. sqrt-pow1 99.4%

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

   5. metadata-eval 99.4%

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

   6. div-inv 99.4%

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

   7. pow1 99.4%

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

   8. clear-num 99.4%

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

   9. unpow2 99.4%

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

   10. unpow2 99.4%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

   3. un-div-inv 99.3%

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

   4. associate-*r* 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

7. Applied egg-rr 99.3%

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

   1. expm1-def 99.3%

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

   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%

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

Alternative 4: 93.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. hypot-1-def 99.2%

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

   3. sqrt-prod 99.2%

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

   4. sqrt-pow1 99.4%

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

   5. metadata-eval 99.4%

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

   6. div-inv 99.4%

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

   7. pow1 99.4%

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

   8. clear-num 99.4%

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

   9. unpow2 99.4%

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

   10. unpow2 99.4%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

   3. un-div-inv 99.3%

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

   4. associate-*r* 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

7. Applied egg-rr 99.3%

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

   1. expm1-def 99.3%

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

   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 93.8%

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

    1. associate-/l* 93.8%

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

    2. associate-*r/ 93.8%

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

12. Simplified 93.8%

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

13. Final simplification 93.8%

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

Alternative 5: 74.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot \ell \leq 7 \cdot 10^{-57}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{\frac{Om}{ky}}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (* 2.0 l) 7e-57)
   1.0
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (/ (* 2.0 l) (/ Om ky))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((2.0 * l) <= 7e-57) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * l) / (Om / ky))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((2.0 * l) <= 7e-57) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 * l) / (Om / ky))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if (2.0 * l) <= 7e-57:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 * l) / (Om / ky))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(2.0 * l) <= 7e-57)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * l) / Float64(Om / ky))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((2.0 * l) <= 7e-57)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * l) / (Om / ky))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(2.0 * l), $MachinePrecision], 7e-57], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * l), $MachinePrecision] / N[(Om / ky), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 l) < 6.99999999999999983e-57

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. add-sqr-sqrt 98.9%

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

      2. hypot-1-def 98.9%

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

      3. sqrt-prod 98.9%

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

      4. sqrt-pow1 99.2%

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

      5. metadata-eval 99.2%

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

      6. div-inv 99.2%

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

      7. pow1 99.2%

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

      8. clear-num 99.2%

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

      9. unpow2 99.2%

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

      10. unpow2 99.2%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 99.4%

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

      2. expm1-udef 99.4%

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

      3. un-div-inv 99.4%

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

      4. associate-*r* 99.4%

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

      5. *-commutative 99.4%

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

      6. 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 kx around 0 94.7%

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

       1. associate-/l* 94.7%

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

       2. associate-*r/ 94.7%

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

   12. Simplified 94.7%

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

   13. Taylor expanded in l around 0 70.1%

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

   if 6.99999999999999983e-57 < (*.f64 2 l)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

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

      3. sqrt-prod 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

      6. div-inv 100.0%

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

      7. pow1 100.0%

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

      8. clear-num 100.0%

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

      9. unpow2 100.0%

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

      10. unpow2 100.0%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 98.9%

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

      2. expm1-udef 98.9%

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

      3. un-div-inv 98.9%

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

      4. associate-*r* 98.9%

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

      5. *-commutative 98.9%

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

      6. associate-*l* 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. expm1-def 98.9%

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

      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 91.4%

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

       1. associate-/l* 91.4%

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

       2. associate-*r/ 91.4%

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

   12. Simplified 91.4%

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

   13. Taylor expanded in ky around 0 83.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot \ell \leq 7 \cdot 10^{-57}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{\frac{Om}{ky}}\right)}}\\ \end{array} \]

Alternative 6: 67.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.9 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 2.9e-60) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.9e-60) {
		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 <= 2.9d-60) 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 <= 2.9e-60) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2.9e-60:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2.9e-60)
		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 <= 2.9e-60)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2.9e-60], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 2.8999999999999999e-60

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in Om around 0 56.2%

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

      1. unpow2 56.2%

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

      2. unpow2 56.2%

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

      3. hypot-def 56.4%

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

   6. Simplified 56.4%

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

   7. Taylor expanded in l around inf 63.5%

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

   if 2.8999999999999999e-60 < Om

   1. Initial program 98.8%

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

   3. Simplified 98.8%

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

      1. add-sqr-sqrt 98.8%

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

      2. hypot-1-def 98.8%

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

      3. sqrt-prod 98.8%

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

      4. sqrt-pow1 99.1%

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

      5. metadata-eval 99.1%

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

      6. div-inv 99.1%

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

      7. pow1 99.1%

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

      8. clear-num 99.1%

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

      9. unpow2 99.1%

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

      10. unpow2 99.1%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 99.6%

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

      3. un-div-inv 99.6%

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

      4. associate-*r* 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*l* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. expm1-def 99.6%

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

      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 92.9%

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

       1. associate-/l* 92.9%

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

       2. associate-*r/ 92.9%

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

   12. Simplified 92.9%

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

   13. Taylor expanded in l around 0 78.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2.9 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 62.5% 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 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. hypot-1-def 99.2%

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

   3. sqrt-prod 99.2%

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

   4. sqrt-pow1 99.4%

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

   5. metadata-eval 99.4%

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

   6. div-inv 99.4%

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

   7. pow1 99.4%

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

   8. clear-num 99.4%

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

   9. unpow2 99.4%

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

   10. unpow2 99.4%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

   3. un-div-inv 99.3%

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

   4. associate-*r* 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

7. Applied egg-rr 99.3%

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

   1. expm1-def 99.3%

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

   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 93.8%

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

    1. associate-/l* 93.8%

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

    2. associate-*r/ 93.8%

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

12. Simplified 93.8%

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

13. Taylor expanded in l around 0 63.3%

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

14. Final simplification 63.3%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023299 
(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))))))))))