Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.6%

Time: 13.2s

Alternatives: 8

Speedup: 2.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot {\left(\frac{\ell}{Om} \cdot \sin ky\right)}^{2}}}} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (* 0.5 (/ 1.0 (sqrt (+ 1.0 (* 4.0 (pow (* (/ l Om) (sin ky)) 2.0)))))))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (4.0 * pow(((l / Om) * sin(ky)), 2.0))))))));
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + (4.0d0 * (((l / om) * sin(ky)) ** 2.0d0))))))))
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * (1.0 / Math.sqrt((1.0 + (4.0 * Math.pow(((l / Om) * Math.sin(ky)), 2.0))))))));
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + (4.0 * math.pow(((l / Om) * math.sin(ky)), 2.0))))))))

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64(4.0 * (Float64(Float64(l / Om) * sin(ky)) ^ 2.0))))))))
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (4.0 * (((l / Om) * sin(ky)) ^ 2.0))))))));
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(4.0 * N[Power[N[(N[(l / Om), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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. Taylor expanded in kx around 0 79.2%

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

   1. associate-/l* 79.2%

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

   2. unpow2 79.2%

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

   3. unpow2 79.2%

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

6. Simplified 79.2%

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

   1. add-sqr-sqrt 79.2%

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

   2. pow2 79.2%

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

   3. associate-/r/ 79.5%

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

   4. sqrt-prod 79.5%

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

   5. frac-times 86.5%

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

   6. sqrt-unprod 53.9%

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

   7. add-sqr-sqrt 87.2%

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

   8. unpow2 87.2%

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

   9. sqrt-prod 43.0%

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

   10. add-sqr-sqrt 92.4%

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

8. Applied egg-rr 92.4%

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

9. Final simplification 92.4%

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

Alternative 2: 99.6% accurate, 2.3× speedup

Math

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

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (/ l Om) (* (sin ky) 2.0)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((l / Om) * (sin(ky) * 2.0))))));
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(l / Om), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * 2.0), $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. Taylor expanded in kx around 0 79.2%

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

   1. associate-/l* 79.2%

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

   2. unpow2 79.2%

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

   3. unpow2 79.2%

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

6. Simplified 79.2%

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

   1. expm1-log1p-u 79.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

   2. expm1-udef 79.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

8. Applied egg-rr 92.4%

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

   1. expm1-def 92.4%

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

   2. expm1-log1p 92.4%

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

   3. *-commutative 92.4%

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

   4. *-commutative 92.4%

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

   5. associate-*l* 92.4%

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

10. Simplified 92.4%

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

    1. clear-num 92.4%

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

    2. un-div-inv 92.4%

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

    3. /-rgt-identity 92.4%

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

    4. associate-*r* 92.4%

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

    5. *-commutative 92.4%

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

    6. associate-*l* 92.4%

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

12. Applied egg-rr 92.4%

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

13. Final simplification 92.4%

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

Alternative 3: 66.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 4 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 5.8 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 4.4 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 4e-92)
   (sqrt 0.5)
   (if (<= Om 5.8e-73) 1.0 (if (<= Om 4.4e+32) (sqrt 0.5) 1.0))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 4e-92) {
		tmp = sqrt(0.5);
	} else if (Om <= 5.8e-73) {
		tmp = 1.0;
	} else if (Om <= 4.4e+32) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
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 <= 4d-92) then
        tmp = sqrt(0.5d0)
    else if (om <= 5.8d-73) then
        tmp = 1.0d0
    else if (om <= 4.4d+32) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 4e-92) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 5.8e-73) {
		tmp = 1.0;
	} else if (Om <= 4.4e+32) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 4e-92:
		tmp = math.sqrt(0.5)
	elif Om <= 5.8e-73:
		tmp = 1.0
	elif Om <= 4.4e+32:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 4e-92)
		tmp = sqrt(0.5);
	elseif (Om <= 5.8e-73)
		tmp = 1.0;
	elseif (Om <= 4.4e+32)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 4e-92)
		tmp = sqrt(0.5);
	elseif (Om <= 5.8e-73)
		tmp = 1.0;
	elseif (Om <= 4.4e+32)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 4e-92], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 5.8e-73], 1.0, If[LessEqual[Om, 4.4e+32], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if Om < 3.99999999999999995e-92 or 5.8e-73 < Om < 4.40000000000000002e32

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in Om around 0 63.5%

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

      1. unpow2 63.5%

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

      2. unpow2 63.5%

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

      3. hypot-def 64.5%

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

   6. Simplified 64.5%

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

   7. Taylor expanded in l around inf 70.6%

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

   if 3.99999999999999995e-92 < Om < 5.8e-73 or 4.40000000000000002e32 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 89.6%

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

      1. associate-/l* 87.9%

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

      2. unpow2 87.9%

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

      3. unpow2 87.9%

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

   6. Simplified 87.9%

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

      1. expm1-log1p-u 87.9%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

      2. expm1-udef 87.9%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

   8. Applied egg-rr 95.7%

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

      1. expm1-def 95.7%

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

      2. expm1-log1p 95.7%

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

      3. *-commutative 95.7%

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

      4. *-commutative 95.7%

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

      5. associate-*l* 95.7%

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

   10. Simplified 95.7%

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

       1. clear-num 95.7%

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

       2. un-div-inv 95.7%

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

       3. /-rgt-identity 95.7%

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

       4. associate-*r* 95.7%

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

       5. *-commutative 95.7%

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

       6. associate-*l* 95.7%

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

   12. Applied egg-rr 95.7%

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

   13. Taylor expanded in l around 0 84.3%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 4 \cdot 10^{-92}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 5.8 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 4.4 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 61.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.6 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky} \cdot \frac{1}{ky}} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 2.6e+131)
   (sqrt 0.5)
   (+ 1.0 (* (/ (* l l) (* (/ (* Om Om) ky) (/ 1.0 ky))) -0.5))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.6e+131) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0 + (((l * l) / (((Om * Om) / ky) * (1.0 / ky))) * -0.5);
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
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 <= 2.6d+131) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0 + (((l * l) / (((om * om) / ky) * (1.0d0 / ky))) * (-0.5d0))
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2.6e+131) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0 + (((l * l) / (((Om * Om) / ky) * (1.0 / ky))) * -0.5);
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2.6e+131:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0 + (((l * l) / (((Om * Om) / ky) * (1.0 / ky))) * -0.5)
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2.6e+131)
		tmp = sqrt(0.5);
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(l * l) / Float64(Float64(Float64(Om * Om) / ky) * Float64(1.0 / ky))) * -0.5));
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 2.6e+131)
		tmp = sqrt(0.5);
	else
		tmp = 1.0 + (((l * l) / (((Om * Om) / ky) * (1.0 / ky))) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2.6e+131], N[Sqrt[0.5], $MachinePrecision], N[(1.0 + N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[(Om * Om), $MachinePrecision] / ky), $MachinePrecision] * N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 2.6e131

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in Om around 0 60.6%

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

      1. unpow2 60.6%

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

      2. unpow2 60.6%

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

      3. hypot-def 61.5%

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

   6. Simplified 61.5%

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

   7. Taylor expanded in l around inf 68.1%

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

   if 2.6e131 < Om

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 89.7%

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

      1. associate-/l* 89.7%

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

      2. unpow2 89.7%

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

      3. unpow2 89.7%

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

   6. Simplified 89.7%

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

      1. expm1-log1p-u 89.7%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

      2. expm1-udef 89.7%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

   8. Applied egg-rr 100.0%

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

      1. expm1-def 99.9%

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

      2. expm1-log1p 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in ky around 0 58.6%

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

       1. *-commutative 58.6%

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

       2. associate-/l* 62.1%

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

       3. unpow2 62.1%

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

       4. unpow2 62.1%

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

       5. unpow2 62.1%

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

   13. Simplified 62.1%

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

       1. associate-/r* 89.7%

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

       2. div-inv 89.7%

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

   15. Applied egg-rr 89.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2.6 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky} \cdot \frac{1}{ky}} \cdot -0.5\\ \end{array} \]

Alternative 5: 39.0% accurate, 37.9× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{+121}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{1}{\frac{Om}{ky}} \cdot \frac{\ell \cdot \ell}{\frac{Om}{ky}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky} \cdot \frac{1}{ky}} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 2e+121)
   (+ 1.0 (* -0.5 (* (/ 1.0 (/ Om ky)) (/ (* l l) (/ Om ky)))))
   (+ 1.0 (* (/ (* l l) (* (/ (* Om Om) ky) (/ 1.0 ky))) -0.5))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2e+121) {
		tmp = 1.0 + (-0.5 * ((1.0 / (Om / ky)) * ((l * l) / (Om / ky))));
	} else {
		tmp = 1.0 + (((l * l) / (((Om * Om) / ky) * (1.0 / ky))) * -0.5);
	}
	return tmp;
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
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 <= 2d+121) then
        tmp = 1.0d0 + ((-0.5d0) * ((1.0d0 / (om / ky)) * ((l * l) / (om / ky))))
    else
        tmp = 1.0d0 + (((l * l) / (((om * om) / ky) * (1.0d0 / ky))) * (-0.5d0))
    end if
    code = tmp
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2e+121) {
		tmp = 1.0 + (-0.5 * ((1.0 / (Om / ky)) * ((l * l) / (Om / ky))));
	} else {
		tmp = 1.0 + (((l * l) / (((Om * Om) / ky) * (1.0 / ky))) * -0.5);
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2e+121:
		tmp = 1.0 + (-0.5 * ((1.0 / (Om / ky)) * ((l * l) / (Om / ky))))
	else:
		tmp = 1.0 + (((l * l) / (((Om * Om) / ky) * (1.0 / ky))) * -0.5)
	return tmp

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2e+121)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(1.0 / Float64(Om / ky)) * Float64(Float64(l * l) / Float64(Om / ky)))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(l * l) / Float64(Float64(Float64(Om * Om) / ky) * Float64(1.0 / ky))) * -0.5));
	end
	return tmp
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 2e+121)
		tmp = 1.0 + (-0.5 * ((1.0 / (Om / ky)) * ((l * l) / (Om / ky))));
	else
		tmp = 1.0 + (((l * l) / (((Om * Om) / ky) * (1.0 / ky))) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2e+121], N[(1.0 + N[(-0.5 * N[(N[(1.0 / N[(Om / ky), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(Om / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(l * l), $MachinePrecision] / N[(N[(N[(Om * Om), $MachinePrecision] / ky), $MachinePrecision] * N[(1.0 / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 2.00000000000000007e121

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in kx around 0 77.7%

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

      1. associate-/l* 77.7%

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

      2. unpow2 77.7%

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

      3. unpow2 77.7%

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

   6. Simplified 77.7%

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

      1. expm1-log1p-u 77.7%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

      2. expm1-udef 77.7%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

   8. Applied egg-rr 91.4%

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

      1. expm1-def 91.4%

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

      2. expm1-log1p 91.4%

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

      3. *-commutative 91.4%

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

      4. *-commutative 91.4%

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

      5. associate-*l* 91.4%

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

   10. Simplified 91.4%

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

   11. Taylor expanded in ky around 0 24.1%

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

       1. *-commutative 24.1%

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

       2. associate-/l* 24.1%

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

       3. unpow2 24.1%

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

       4. unpow2 24.1%

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

       5. unpow2 24.1%

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

   13. Simplified 24.1%

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

       1. *-un-lft-identity 24.1%

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

       2. add-sqr-sqrt 24.1%

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

       3. times-frac 24.1%

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

       4. times-frac 24.1%

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

       5. sqrt-prod 12.6%

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

       6. add-sqr-sqrt 24.8%

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

       7. times-frac 30.7%

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

       8. sqrt-prod 16.1%

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

       9. add-sqr-sqrt 33.2%

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

   15. Applied egg-rr 33.2%

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

   if 2.00000000000000007e121 < Om

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 90.0%

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

      1. associate-/l* 90.0%

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

      2. unpow2 90.0%

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

      3. unpow2 90.0%

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

   6. Simplified 90.0%

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

      1. expm1-log1p-u 90.0%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

      2. expm1-udef 90.0%

         \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

   8. Applied egg-rr 100.0%

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

      1. expm1-def 99.9%

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

      2. expm1-log1p 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*l* 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in ky around 0 60.0%

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

       1. *-commutative 60.0%

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

       2. associate-/l* 63.3%

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

       3. unpow2 63.3%

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

       4. unpow2 63.3%

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

       5. unpow2 63.3%

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

   13. Simplified 63.3%

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

       1. associate-/r* 90.0%

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

       2. div-inv 90.0%

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

   15. Applied egg-rr 90.0%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{+121}:\\ \;\;\;\;1 + -0.5 \cdot \left(\frac{1}{\frac{Om}{ky}} \cdot \frac{\ell \cdot \ell}{\frac{Om}{ky}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky} \cdot \frac{1}{ky}} \cdot -0.5\\ \end{array} \]

Alternative 6: 35.0% accurate, 48.1× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ 1 + -0.5 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{ky}{Om} \cdot \frac{ky}{Om}\right)\right)\right) \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (+ 1.0 (* -0.5 (* l (* l (* (/ ky Om) (/ ky Om)))))))

C

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

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0 + ((-0.5d0) * (l * (l * ((ky / om) * (ky / om)))))
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	return 1.0 + (-0.5 * (l * (l * ((ky / Om) * (ky / Om)))));
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return 1.0 + (-0.5 * (l * (l * ((ky / Om) * (ky / Om)))))

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	return Float64(1.0 + Float64(-0.5 * Float64(l * Float64(l * Float64(Float64(ky / Om) * Float64(ky / Om))))))
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp = code(l, Om, kx, ky)
	tmp = 1.0 + (-0.5 * (l * (l * ((ky / Om) * (ky / Om)))));
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := N[(1.0 + N[(-0.5 * N[(l * N[(l * N[(N[(ky / Om), $MachinePrecision] * N[(ky / Om), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in kx around 0 79.2%

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

   1. associate-/l* 79.2%

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

   2. unpow2 79.2%

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

   3. unpow2 79.2%

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

6. Simplified 79.2%

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

   1. expm1-log1p-u 79.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

   2. expm1-udef 79.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

8. Applied egg-rr 92.4%

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

   1. expm1-def 92.4%

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

   2. expm1-log1p 92.4%

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

   3. *-commutative 92.4%

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

   4. *-commutative 92.4%

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

   5. associate-*l* 92.4%

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

10. Simplified 92.4%

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

11. Taylor expanded in ky around 0 28.3%

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

    1. *-commutative 28.3%

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

    2. associate-/l* 28.7%

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

    3. unpow2 28.7%

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

    4. unpow2 28.7%

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

    5. unpow2 28.7%

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

13. Simplified 28.7%

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

    1. div-inv 28.3%

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

    2. clear-num 28.3%

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

    3. associate-*l* 31.1%

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

    4. times-frac 37.6%

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

15. Applied egg-rr 37.6%

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

16. Final simplification 37.6%

    \[\leadsto 1 + -0.5 \cdot \left(\ell \cdot \left(\ell \cdot \left(\frac{ky}{Om} \cdot \frac{ky}{Om}\right)\right)\right) \]

Alternative 7: 37.4% accurate, 48.1× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} t_0 := \frac{\ell}{Om} \cdot ky\\ 1 + -0.5 \cdot \left(t_0 \cdot t_0\right) \end{array} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (* (/ l Om) ky))) (+ 1.0 (* -0.5 (* t_0 t_0)))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double t_0 = (l / Om) * ky;
	return 1.0 + (-0.5 * (t_0 * t_0));
}

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: t_0
    t_0 = (l / om) * ky
    code = 1.0d0 + ((-0.5d0) * (t_0 * t_0))
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	double t_0 = (l / Om) * ky;
	return 1.0 + (-0.5 * (t_0 * t_0));
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	t_0 = (l / Om) * ky
	return 1.0 + (-0.5 * (t_0 * t_0))

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	t_0 = Float64(Float64(l / Om) * ky)
	return Float64(1.0 + Float64(-0.5 * Float64(t_0 * t_0)))
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp = code(l, Om, kx, ky)
	t_0 = (l / Om) * ky;
	tmp = 1.0 + (-0.5 * (t_0 * t_0));
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(N[(l / Om), $MachinePrecision] * ky), $MachinePrecision]}, N[(1.0 + N[(-0.5 * N[(t$95$0 * t$95$0), $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. Taylor expanded in kx around 0 79.2%

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

   1. associate-/l* 79.2%

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

   2. unpow2 79.2%

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

   3. unpow2 79.2%

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

6. Simplified 79.2%

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

   1. expm1-log1p-u 79.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

   2. expm1-udef 79.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

8. Applied egg-rr 92.4%

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

   1. expm1-def 92.4%

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

   2. expm1-log1p 92.4%

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

   3. *-commutative 92.4%

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

   4. *-commutative 92.4%

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

   5. associate-*l* 92.4%

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

10. Simplified 92.4%

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

11. Taylor expanded in ky around 0 28.3%

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

    1. *-commutative 28.3%

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

    2. associate-/l* 28.7%

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

    3. unpow2 28.7%

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

    4. unpow2 28.7%

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

    5. unpow2 28.7%

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

13. Simplified 28.7%

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

    1. add-sqr-sqrt 28.7%

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

    2. associate-/r/ 28.6%

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

    3. sqrt-prod 28.6%

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

    4. frac-times 28.6%

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

    5. sqrt-unprod 20.5%

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

    6. add-sqr-sqrt 29.1%

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

    7. sqrt-prod 12.8%

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

    8. add-sqr-sqrt 29.3%

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

    9. associate-/r/ 30.0%

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

    10. sqrt-prod 30.0%

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

    11. frac-times 33.0%

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

    12. sqrt-unprod 22.7%

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

    13. add-sqr-sqrt 33.8%

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

    14. sqrt-prod 16.4%

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

    15. add-sqr-sqrt 39.1%

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

15. Applied egg-rr 39.1%

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

16. Final simplification 39.1%

    \[\leadsto 1 + -0.5 \cdot \left(\left(\frac{\ell}{Om} \cdot ky\right) \cdot \left(\frac{\ell}{Om} \cdot ky\right)\right) \]

Alternative 8: 38.2% accurate, 48.1× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ 1 + -0.5 \cdot \frac{ky \cdot \frac{\ell \cdot \ell}{Om}}{\frac{Om}{ky}} \end{array} \]

FPCore

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
(FPCore (l Om kx ky)
 :precision binary64
 (+ 1.0 (* -0.5 (/ (* ky (/ (* l l) Om)) (/ Om ky)))))

C

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

Fortran

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0 + ((-0.5d0) * ((ky * ((l * l) / om)) / (om / ky)))
end function

Java

kx = Math.abs(kx);
ky = Math.abs(ky);
assert kx < ky;
public static double code(double l, double Om, double kx, double ky) {
	return 1.0 + (-0.5 * ((ky * ((l * l) / Om)) / (Om / ky)));
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return 1.0 + (-0.5 * ((ky * ((l * l) / Om)) / (Om / ky)))

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	return Float64(1.0 + Float64(-0.5 * Float64(Float64(ky * Float64(Float64(l * l) / Om)) / Float64(Om / ky))))
end

MATLAB

kx = abs(kx)
ky = abs(ky)
kx, ky = num2cell(sort([kx, ky])){:}
function tmp = code(l, Om, kx, ky)
	tmp = 1.0 + (-0.5 * ((ky * ((l * l) / Om)) / (Om / ky)));
end

Wolfram

NOTE: kx should be positive before calling this function
NOTE: ky should be positive before calling this function
NOTE: kx and ky should be sorted in increasing order before calling this function.
code[l_, Om_, kx_, ky_] := N[(1.0 + N[(-0.5 * N[(N[(ky * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / N[(Om / ky), $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. Taylor expanded in kx around 0 79.2%

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

   1. associate-/l* 79.2%

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

   2. unpow2 79.2%

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

   3. unpow2 79.2%

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

6. Simplified 79.2%

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

   1. expm1-log1p-u 79.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)\right)}}} \]

   2. expm1-udef 79.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{{\sin ky}^{2}}}}\right)} - 1}}} \]

8. Applied egg-rr 92.4%

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

   1. expm1-def 92.4%

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

   2. expm1-log1p 92.4%

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

   3. *-commutative 92.4%

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

   4. *-commutative 92.4%

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

   5. associate-*l* 92.4%

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

10. Simplified 92.4%

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

11. Taylor expanded in ky around 0 28.3%

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

    1. *-commutative 28.3%

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

    2. associate-/l* 28.7%

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

    3. unpow2 28.7%

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

    4. unpow2 28.7%

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

    5. unpow2 28.7%

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

13. Simplified 28.7%

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

    1. *-un-lft-identity 28.7%

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

    2. add-sqr-sqrt 28.7%

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

    3. times-frac 28.7%

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

    4. times-frac 28.7%

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

    5. sqrt-prod 14.2%

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

    6. add-sqr-sqrt 29.3%

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

    7. times-frac 36.5%

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

    8. sqrt-prod 17.8%

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

    9. add-sqr-sqrt 38.7%

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

15. Applied egg-rr 38.7%

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

    1. associate-*l/ 38.7%

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

    2. *-lft-identity 38.7%

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

    3. unpow2 38.7%

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

    4. associate-/r/ 38.6%

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

    5. unpow2 38.6%

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

17. Simplified 38.6%

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

18. Final simplification 38.6%

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

Reproduce

herbie shell --seed 2023264 
(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))))))))))