Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.6%

Time: 20.2s

Alternatives: 3

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, 2.3× speedup

Math

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

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in kx around 0 74.6%

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

   1. associate-*r/ 74.6%

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

   2. unpow2 74.6%

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

   3. times-frac 82.3%

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

   4. unpow2 82.3%

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

   5. associate-*l* 84.2%

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

6. Simplified 84.2%

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

   1. *-commutative 84.2%

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

   2. clear-num 84.2%

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

   3. un-div-inv 84.2%

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

   4. associate-*r* 82.3%

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

   5. pow2 82.3%

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

   6. pow-prod-down 89.2%

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

   7. div-inv 89.2%

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

   8. metadata-eval 89.2%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \frac{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om}}{Om \cdot \color{blue}{0.25}}}} \cdot 0.5} \]

8. Applied egg-rr 89.2%

   \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\frac{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om}}{Om \cdot 0.25}}}} \cdot 0.5} \]
9. Step-by-step derivation

   1. associate-/r* 82.5%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)}}}} \cdot 0.5} \]

10. Simplified 82.5%

    \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)}}}} \cdot 0.5} \]
11. Step-by-step derivation

    1. remove-double-neg 82.5%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{-\left(-\sqrt{1 + \frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)}}\right)}} \cdot 0.5} \]

    2. neg-sub0 82.5%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{0 - \left(-\sqrt{1 + \frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)}}\right)}} \cdot 0.5} \]

    3. +-commutative 82.5%

       \[\leadsto \sqrt{0.5 + \frac{1}{0 - \left(-\sqrt{\color{blue}{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)} + 1}}\right)} \cdot 0.5} \]

12. Applied egg-rr 93.9%

    \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{0 - \left(-\sqrt{{\left(\frac{-2}{\frac{\frac{Om}{\ell}}{\sin ky}}\right)}^{2} + 1}\right)}} \cdot 0.5} \]
13. Step-by-step derivation

    1. neg-sub0 93.9%

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

    2. remove-double-neg 93.9%

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

    3. +-commutative 93.9%

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

    4. unpow2 93.9%

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

    5. hypot-1-def 93.9%

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

    6. associate-/r/ 93.9%

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

14. Simplified 93.9%

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

    1. associate-*l/ 93.9%

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

    2. metadata-eval 93.9%

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

    3. div-inv 93.9%

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

    4. associate-*l* 93.9%

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

    5. *-commutative 93.9%

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

    6. div-inv 93.9%

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

    7. associate-/r/ 93.9%

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

    8. *-commutative 93.9%

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

16. Applied egg-rr 93.9%

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

    1. associate-*r* 93.9%

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

18. Simplified 93.9%

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

19. Final simplification 93.9%

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

Alternative 2: 67.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \begin{array}{l} \mathbf{if}\;Om \leq 2.85 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 2.85e+23) (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 <= 2.85e+23) {
		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 <= 2.85d+23) 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 <= 2.85e+23) {
		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 <= 2.85e+23:
		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 <= 2.85e+23)
		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 <= 2.85e+23)
		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, 2.85e+23], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 2.85e23

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   4. Taylor expanded in l around inf 55.8%

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

      1. associate-*r* 55.8%

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

      2. *-commutative 55.8%

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

      3. associate-*l* 55.8%

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

      4. unpow2 55.8%

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

      5. unpow2 55.8%

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

      6. hypot-def 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in l around inf 64.5%

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

   if 2.85e23 < 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%

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

   4. Taylor expanded in kx around 0 87.6%

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

      1. associate-*r/ 87.6%

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

      2. unpow2 87.6%

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

      3. times-frac 93.3%

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

      4. unpow2 93.3%

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

      5. associate-*l* 94.2%

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

   6. Simplified 94.2%

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

      1. *-commutative 94.2%

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

      2. clear-num 94.2%

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

      3. un-div-inv 94.2%

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

      4. associate-*r* 93.3%

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

      5. pow2 93.3%

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

      6. pow-prod-down 94.2%

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

      7. div-inv 94.2%

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

      8. metadata-eval 94.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \frac{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om}}{Om \cdot \color{blue}{0.25}}}} \cdot 0.5} \]

   8. Applied egg-rr 94.2%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\frac{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om}}{Om \cdot 0.25}}}} \cdot 0.5} \]
   9. Step-by-step derivation

      1. associate-/r* 90.2%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)}}}} \cdot 0.5} \]

   10. Simplified 90.2%

       \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)}}}} \cdot 0.5} \]
   11. Step-by-step derivation

       1. remove-double-neg 90.2%

          \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{-\left(-\sqrt{1 + \frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)}}\right)}} \cdot 0.5} \]

       2. neg-sub0 90.2%

          \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{0 - \left(-\sqrt{1 + \frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)}}\right)}} \cdot 0.5} \]

       3. +-commutative 90.2%

          \[\leadsto \sqrt{0.5 + \frac{1}{0 - \left(-\sqrt{\color{blue}{\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot \left(Om \cdot 0.25\right)} + 1}}\right)} \cdot 0.5} \]

   12. Applied egg-rr 95.6%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{0 - \left(-\sqrt{{\left(\frac{-2}{\frac{\frac{Om}{\ell}}{\sin ky}}\right)}^{2} + 1}\right)}} \cdot 0.5} \]
   13. Step-by-step derivation

       1. neg-sub0 95.6%

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

       2. remove-double-neg 95.6%

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

       3. +-commutative 95.6%

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

       4. unpow2 95.6%

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

       5. hypot-1-def 95.6%

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

       6. associate-/r/ 95.6%

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

   14. Simplified 95.6%

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

   15. Taylor expanded in Om around inf 81.1%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 2.85 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 55.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} kx = |kx|\\ ky = |ky|\\ [kx, ky] = \mathsf{sort}([kx, ky])\\ \\ \sqrt{0.5} \end{array} \]

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))

C

kx = abs(kx);
ky = abs(ky);
assert(kx < ky);
double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

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)
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);
}

Python

kx = abs(kx)
ky = abs(ky)
[kx, ky] = sort([kx, ky])
def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

kx = abs(kx)
ky = abs(ky)
kx, ky = sort([kx, ky])
function code(l, Om, kx, ky)
	return sqrt(0.5)
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);
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[0.5], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in l around inf 48.9%

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

   1. associate-*r* 48.9%

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

   2. *-commutative 48.9%

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

   3. associate-*l* 48.9%

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

   4. unpow2 48.9%

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

   5. unpow2 48.9%

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

   6. hypot-def 50.5%

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

6. Simplified 50.5%

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

7. Taylor expanded in l around inf 58.6%

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

8. Final simplification 58.6%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023297 
(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))))))))))