Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.6% → 99.6%

Time: 13.7s

Alternatives: 5

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: 98.6% 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, 2.3× speedup

Math

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

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 (* (sin ky) (* 2.0 (/ l Om))))))))

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

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

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, (math.sin(ky) * (2.0 * (l / Om)))))))

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(sin(ky) * Float64(2.0 * Float64(l / Om)))))))
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, (sin(ky) * (2.0 * (l / 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[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[ky], $MachinePrecision] * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.0%

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

   1. distribute-rgt-in 98.0%

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

   2. metadata-eval 98.0%

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

   3. metadata-eval 98.0%

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

   4. associate-/l* 98.0%

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

   5. metadata-eval 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in kx around 0 79.4%

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

   1. associate-/l* 79.8%

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

   2. associate-/r/ 80.1%

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

   3. unpow2 80.1%

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

   4. unpow2 80.1%

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

   5. times-frac 90.1%

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

6. Simplified 90.1%

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

   1. add-sqr-sqrt 90.1%

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

   2. hypot-1-def 90.1%

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

   3. sqrt-prod 90.1%

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

   4. metadata-eval 90.1%

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

   5. sqrt-prod 90.1%

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

   6. sqrt-prod 51.5%

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

   7. add-sqr-sqrt 90.7%

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

   8. unpow2 90.7%

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

   9. sqrt-prod 49.0%

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

   10. add-sqr-sqrt 93.7%

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

8. Applied egg-rr 93.7%

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

   1. expm1-log1p-u 93.2%

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

   2. expm1-udef 93.2%

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

   3. associate-*l/ 93.2%

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

   4. metadata-eval 93.2%

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

   5. associate-*r* 93.2%

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

10. Applied egg-rr 93.2%

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

    1. expm1-def 93.2%

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

    2. expm1-log1p 93.7%

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

    3. *-commutative 93.7%

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

12. Simplified 93.7%

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

13. Final simplification 93.7%

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

Alternative 2: 78.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 3.9 \cdot 10^{-69} \lor \neg \left(Om \leq 8 \cdot 10^{-62}\right) \land Om \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(ky \cdot ky\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 (or (<= Om 3.9e-69) (and (not (<= Om 8e-62)) (<= Om 1.7e-12)))
   (sqrt
    (+
     0.5
     (*
      0.5
      (/ 1.0 (sqrt (+ 1.0 (* 4.0 (* (* (/ l Om) (/ l Om)) (* ky ky)))))))))
   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 <= 3.9e-69) || (!(Om <= 8e-62) && (Om <= 1.7e-12))) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (4.0 * (((l / Om) * (l / Om)) * (ky * ky)))))))));
	} 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 <= 3.9d-69) .or. (.not. (om <= 8d-62)) .and. (om <= 1.7d-12)) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + (4.0d0 * (((l / om) * (l / om)) * (ky * ky)))))))))
    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 <= 3.9e-69) || (!(Om <= 8e-62) && (Om <= 1.7e-12))) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.sqrt((1.0 + (4.0 * (((l / Om) * (l / Om)) * (ky * ky)))))))));
	} 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 <= 3.9e-69) or (not (Om <= 8e-62) and (Om <= 1.7e-12)):
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + (4.0 * (((l / Om) * (l / Om)) * (ky * ky)))))))))
	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 <= 3.9e-69) || (!(Om <= 8e-62) && (Om <= 1.7e-12)))
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64(4.0 * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(ky * ky)))))))));
	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 <= 3.9e-69) || (~((Om <= 8e-62)) && (Om <= 1.7e-12)))
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (4.0 * (((l / Om) * (l / Om)) * (ky * ky)))))))));
	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[Or[LessEqual[Om, 3.9e-69], And[N[Not[LessEqual[Om, 8e-62]], $MachinePrecision], LessEqual[Om, 1.7e-12]]], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(4.0 * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 3.89999999999999981e-69 or 8.0000000000000003e-62 < Om < 1.7e-12

   1. Initial program 97.1%

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

      1. distribute-rgt-in 97.1%

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

      2. metadata-eval 97.1%

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

      3. metadata-eval 97.1%

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

      4. associate-/l* 97.1%

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

      5. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in kx around 0 75.9%

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

      1. associate-/l* 77.0%

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

      2. associate-/r/ 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. times-frac 87.9%

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

   6. Simplified 87.9%

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

   7. Taylor expanded in ky around 0 77.2%

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

      1. unpow2 77.2%

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

   9. Simplified 77.2%

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

   if 3.89999999999999981e-69 < Om < 8.0000000000000003e-62 or 1.7e-12 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 86.8%

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

      1. associate-/l* 85.6%

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

      2. associate-/r/ 86.8%

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

      3. unpow2 86.8%

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

      4. unpow2 86.8%

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

      5. times-frac 94.6%

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

   6. Simplified 94.6%

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

      1. add-sqr-sqrt 94.6%

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

      2. hypot-1-def 94.6%

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

      3. sqrt-prod 94.6%

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

      4. metadata-eval 94.6%

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

      5. sqrt-prod 94.6%

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

      6. sqrt-prod 60.0%

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

      7. add-sqr-sqrt 95.1%

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

      8. unpow2 95.1%

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

      9. sqrt-prod 57.8%

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

      10. add-sqr-sqrt 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. expm1-log1p-u 95.0%

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

      2. expm1-udef 95.0%

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

      3. associate-*l/ 95.0%

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

      4. metadata-eval 95.0%

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

      5. associate-*r* 95.0%

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

   10. Applied egg-rr 95.0%

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

       1. expm1-def 95.0%

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

       2. expm1-log1p 95.1%

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

       3. *-commutative 95.1%

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

   12. Simplified 95.1%

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

   13. Taylor expanded in ky around 0 87.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 3.9 \cdot 10^{-69} \lor \neg \left(Om \leq 8 \cdot 10^{-62}\right) \land Om \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(ky \cdot ky\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 67.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.5 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 2.5e-69)
   (sqrt 0.5)
   (if (<= Om 5e-62) 1.0 (if (<= Om 7e-12) (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 <= 2.5e-69) {
		tmp = sqrt(0.5);
	} else if (Om <= 5e-62) {
		tmp = 1.0;
	} else if (Om <= 7e-12) {
		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 <= 2.5d-69) then
        tmp = sqrt(0.5d0)
    else if (om <= 5d-62) then
        tmp = 1.0d0
    else if (om <= 7d-12) 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 <= 2.5e-69) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 5e-62) {
		tmp = 1.0;
	} else if (Om <= 7e-12) {
		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 <= 2.5e-69:
		tmp = math.sqrt(0.5)
	elif Om <= 5e-62:
		tmp = 1.0
	elif Om <= 7e-12:
		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 <= 2.5e-69)
		tmp = sqrt(0.5);
	elseif (Om <= 5e-62)
		tmp = 1.0;
	elseif (Om <= 7e-12)
		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 <= 2.5e-69)
		tmp = sqrt(0.5);
	elseif (Om <= 5e-62)
		tmp = 1.0;
	elseif (Om <= 7e-12)
		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, 2.5e-69], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 5e-62], 1.0, If[LessEqual[Om, 7e-12], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if Om < 2.50000000000000017e-69 or 5.0000000000000002e-62 < Om < 7.0000000000000001e-12

   1. Initial program 97.1%

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

      1. distribute-rgt-in 97.1%

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

      2. metadata-eval 97.1%

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

      3. metadata-eval 97.1%

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

      4. associate-/l* 97.1%

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

      5. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in Om around 0 57.4%

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

      1. associate-*r* 57.4%

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

      2. *-commutative 57.4%

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

      3. associate-*r* 57.4%

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

      4. unpow2 57.4%

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

      5. unpow2 57.4%

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

      6. hypot-def 59.8%

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

   6. Simplified 59.8%

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

   7. Taylor expanded in l around inf 66.0%

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

   if 2.50000000000000017e-69 < Om < 5.0000000000000002e-62 or 7.0000000000000001e-12 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 86.8%

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

      1. associate-/l* 85.6%

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

      2. associate-/r/ 86.8%

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

      3. unpow2 86.8%

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

      4. unpow2 86.8%

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

      5. times-frac 94.6%

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

   6. Simplified 94.6%

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

      1. add-sqr-sqrt 94.6%

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

      2. hypot-1-def 94.6%

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

      3. sqrt-prod 94.6%

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

      4. metadata-eval 94.6%

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

      5. sqrt-prod 94.6%

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

      6. sqrt-prod 60.0%

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

      7. add-sqr-sqrt 95.1%

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

      8. unpow2 95.1%

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

      9. sqrt-prod 57.8%

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

      10. add-sqr-sqrt 95.1%

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

   8. Applied egg-rr 95.1%

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

      1. expm1-log1p-u 95.0%

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

      2. expm1-udef 95.0%

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

      3. associate-*l/ 95.0%

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

      4. metadata-eval 95.0%

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

      5. associate-*r* 95.0%

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

   10. Applied egg-rr 95.0%

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

       1. expm1-def 95.0%

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

       2. expm1-log1p 95.1%

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

       3. *-commutative 95.1%

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

   12. Simplified 95.1%

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

   13. Taylor expanded in ky around 0 87.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2.5 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 5 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 60.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.7 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(\left(ky \cdot \frac{ky}{Om}\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= Om 1.7e+84)
   (sqrt 0.5)
   (+ 1.0 (* -0.5 (* (* ky (/ ky Om)) (* l (/ l Om)))))))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.7e+84) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((ky * (ky / Om)) * (l * (l / Om))));
	}
	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 <= 1.7d+84) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0 + ((-0.5d0) * ((ky * (ky / om)) * (l * (l / om))))
    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 <= 1.7e+84) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((ky * (ky / Om)) * (l * (l / Om))));
	}
	return tmp;
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.7e+84:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0 + (-0.5 * ((ky * (ky / Om)) * (l * (l / Om))))
	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 <= 1.7e+84)
		tmp = sqrt(0.5);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(ky * Float64(ky / Om)) * Float64(l * Float64(l / Om)))));
	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 <= 1.7e+84)
		tmp = sqrt(0.5);
	else
		tmp = 1.0 + (-0.5 * ((ky * (ky / Om)) * (l * (l / Om))));
	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, 1.7e+84], N[Sqrt[0.5], $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(ky * N[(ky / Om), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 1.6999999999999999e84

   1. Initial program 97.5%

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

      1. distribute-rgt-in 97.5%

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

      2. metadata-eval 97.5%

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

      3. metadata-eval 97.5%

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

      4. associate-/l* 97.5%

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

      5. metadata-eval 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in Om around 0 54.0%

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

      1. associate-*r* 54.0%

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

      2. *-commutative 54.0%

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

      3. associate-*r* 54.0%

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

      4. unpow2 54.0%

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

      5. unpow2 54.0%

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

      6. hypot-def 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in l around inf 62.7%

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

   if 1.6999999999999999e84 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 85.5%

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

      1. associate-/l* 83.6%

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

      2. associate-/r/ 85.5%

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

      3. unpow2 85.5%

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

      4. unpow2 85.5%

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

      5. times-frac 97.1%

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

   6. Simplified 97.1%

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

      1. add-sqr-sqrt 97.1%

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

      2. hypot-1-def 97.1%

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

      3. sqrt-prod 97.1%

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

      4. metadata-eval 97.1%

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

      5. sqrt-prod 97.1%

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

      6. sqrt-prod 63.6%

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

      7. add-sqr-sqrt 97.1%

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

      8. unpow2 97.1%

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

      9. sqrt-prod 56.7%

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

      10. add-sqr-sqrt 97.1%

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

   8. Applied egg-rr 97.1%

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

   9. Taylor expanded in ky around 0 63.7%

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

       1. unpow2 63.7%

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

       2. unpow2 63.7%

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

       3. unpow2 63.7%

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

   11. Simplified 63.7%

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

       1. expm1-log1p-u 63.7%

          \[\leadsto 1 + -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)}{Om \cdot Om}\right)\right)} \]

       2. expm1-udef 63.7%

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

       3. times-frac 67.4%

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

       4. associate-/l* 75.0%

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

   13. Applied egg-rr 75.0%

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

       1. expm1-def 75.0%

          \[\leadsto 1 + -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{Om} \cdot \frac{ky}{\frac{Om}{ky}}\right)\right)} \]

       2. expm1-log1p 75.0%

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

       3. associate-*r/ 82.7%

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

       4. *-commutative 82.7%

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

       5. associate-/r/ 82.7%

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

   15. Simplified 82.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.7 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(\left(ky \cdot \frac{ky}{Om}\right) \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\\ \end{array} \]

Alternative 5: 32.8% accurate, 48.1× speedup

Math

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

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 (* (* ky (/ ky Om)) (* l (/ l 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 * ((ky * (ky / Om)) * (l * (l / 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) * ((ky * (ky / om)) * (l * (l / 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 * ((ky * (ky / Om)) * (l * (l / Om))));
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return 1.0 + (-0.5 * ((ky * (ky / Om)) * (l * (l / 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(Float64(ky * Float64(ky / Om)) * Float64(l * Float64(l / 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 * ((ky * (ky / Om)) * (l * (l / 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[(N[(ky * N[(ky / Om), $MachinePrecision]), $MachinePrecision] * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

   1. distribute-rgt-in 98.0%

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

   2. metadata-eval 98.0%

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

   3. metadata-eval 98.0%

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

   4. associate-/l* 98.0%

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

   5. metadata-eval 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in kx around 0 79.4%

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

   1. associate-/l* 79.8%

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

   2. associate-/r/ 80.1%

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

   3. unpow2 80.1%

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

   4. unpow2 80.1%

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

   5. times-frac 90.1%

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

6. Simplified 90.1%

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

   1. add-sqr-sqrt 90.1%

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

   2. hypot-1-def 90.1%

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

   3. sqrt-prod 90.1%

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

   4. metadata-eval 90.1%

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

   5. sqrt-prod 90.1%

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

   6. sqrt-prod 51.5%

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

   7. add-sqr-sqrt 90.7%

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

   8. unpow2 90.7%

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

   9. sqrt-prod 49.0%

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

   10. add-sqr-sqrt 93.7%

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

8. Applied egg-rr 93.7%

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

9. Taylor expanded in ky around 0 33.7%

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

    1. unpow2 33.7%

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

    2. unpow2 33.7%

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

    3. unpow2 33.7%

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

11. Simplified 33.7%

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

    1. expm1-log1p-u 33.7%

       \[\leadsto 1 + -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)}{Om \cdot Om}\right)\right)} \]

    2. expm1-udef 33.7%

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

    3. times-frac 38.8%

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

    4. associate-/l* 40.4%

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

13. Applied egg-rr 40.4%

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

    1. expm1-def 40.4%

       \[\leadsto 1 + -0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \ell}{Om} \cdot \frac{ky}{\frac{Om}{ky}}\right)\right)} \]

    2. expm1-log1p 40.4%

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

    3. associate-*r/ 43.3%

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

    4. *-commutative 43.3%

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

    5. associate-/r/ 43.3%

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

15. Simplified 43.3%

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

16. Final simplification 43.3%

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

Reproduce

herbie shell --seed 2023258 
(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))))))))))