Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 100.0%

Time: 18.0s

Alternatives: 11

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.5%

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

   1. add-sqr-sqrt 98.5%

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

   2. hypot-1-def 98.5%

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

   3. sqrt-prod 98.5%

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

   4. unpow2 98.5%

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

   5. sqrt-prod 54.5%

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

   6. add-sqr-sqrt 98.8%

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

   7. associate-/r/ 98.8%

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

   8. *-commutative 98.8%

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

   9. unpow2 98.8%

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

   10. unpow2 98.8%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 93.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.5%

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

3. Simplified 98.5%

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

   1. add-sqr-sqrt 98.5%

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

   2. hypot-1-def 98.5%

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

   3. sqrt-prod 98.5%

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

   4. unpow2 98.5%

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

   5. sqrt-prod 54.5%

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

   6. add-sqr-sqrt 98.8%

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

   7. associate-/r/ 98.8%

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

   8. *-commutative 98.8%

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

   9. unpow2 98.8%

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

   10. unpow2 98.8%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 94.0%

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

   1. *-un-lft-identity 94.0%

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

   2. un-div-inv 94.0%

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

8. Applied egg-rr 94.0%

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

   1. *-lft-identity 94.0%

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

   2. associate-*l* 94.0%

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

10. Simplified 94.0%

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

11. Final simplification 94.0%

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

Alternative 3: 73.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \left(\ell \cdot \frac{2}{Om}\right) \cdot ky\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 9e-178)
   1.0
   (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (* (* l (/ 2.0 Om)) ky))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 9e-178) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((l * (2.0 / Om)) * ky))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 9e-178) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, ((l * (2.0 / Om)) * ky))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 9e-178:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, ((l * (2.0 / Om)) * ky))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 9e-178)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(Float64(l * Float64(2.0 / Om)) * ky))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 9e-178)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((l * (2.0 / Om)) * ky))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 9e-178], 1.0, N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(l * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision] * ky), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.99999999999999957e-178

   1. Initial program 97.8%

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

   3. Simplified 97.8%

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

      1. add-sqr-sqrt 97.8%

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

      2. hypot-1-def 97.8%

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

      3. sqrt-prod 97.8%

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

      4. unpow2 97.8%

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

      5. sqrt-prod 59.8%

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

      6. add-sqr-sqrt 98.2%

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

      7. associate-/r/ 98.2%

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

      8. *-commutative 98.2%

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

      9. unpow2 98.2%

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

      10. unpow2 98.2%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in l around 0 71.4%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{1}} \]

   if 8.99999999999999957e-178 < l

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 99.3%

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

      2. hypot-1-def 99.3%

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

      3. sqrt-prod 99.3%

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

      4. unpow2 99.3%

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

      5. sqrt-prod 47.8%

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

      6. add-sqr-sqrt 99.6%

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

      7. associate-/r/ 99.6%

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

      8. *-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. unpow2 99.6%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 93.7%

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

   7. Taylor expanded in ky around 0 85.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \left(\ell \cdot \frac{2}{Om}\right) \cdot ky\right)}}\\ \end{array} \]

Alternative 4: 68.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 9.5 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 6 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 9.5e-170)
   (sqrt 0.5)
   (if (<= Om 6e+72)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* ky ky) (/ (* Om Om) (* l l)))))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 9.5e-170) {
		tmp = sqrt(0.5);
	} else if (Om <= 6e+72) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 9.5d-170) then
        tmp = sqrt(0.5d0)
    else if (om <= 6d+72) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((ky * ky) / ((om * om) / (l * l)))))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 9.5e-170) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 6e+72) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 9.5e-170:
		tmp = math.sqrt(0.5)
	elif Om <= 6e+72:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 9.5e-170)
		tmp = sqrt(0.5);
	elseif (Om <= 6e+72)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * ky) / Float64(Float64(Om * Om) / Float64(l * l)))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 9.5e-170)
		tmp = sqrt(0.5);
	elseif (Om <= 6e+72)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 9.5e-170], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 6e+72], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(ky * ky), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 9.5000000000000001e-170

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in Om around 0 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

      3. hypot-def 51.8%

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

   6. Simplified 51.8%

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

   7. Taylor expanded in l around inf 60.0%

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

   if 9.5000000000000001e-170 < Om < 6.00000000000000006e72

   1. Initial program 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in kx around 0 83.3%

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

      1. *-commutative 83.3%

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

      2. associate-/l* 80.8%

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

      3. associate-*l/ 80.8%

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

      4. unpow2 80.8%

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

      5. unpow2 80.8%

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

   6. Simplified 80.8%

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

   7. Taylor expanded in ky around 0 71.9%

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

      1. associate-/l* 71.4%

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

      2. unpow2 71.4%

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

      3. unpow2 71.4%

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

      4. unpow2 71.4%

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

   9. Simplified 71.4%

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

   if 6.00000000000000006e72 < Om

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

      1. add-sqr-sqrt 98.2%

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

      2. hypot-1-def 98.2%

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

      3. sqrt-prod 98.2%

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

      4. unpow2 98.2%

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

      5. sqrt-prod 67.9%

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

      6. add-sqr-sqrt 98.6%

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

      7. associate-/r/ 98.6%

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

      8. *-commutative 98.6%

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

      9. unpow2 98.6%

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

      10. unpow2 98.6%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in l around 0 91.9%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 9.5 \cdot 10^{-170}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 6 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 52.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+83} \lor \neg \left(\ell \leq 6.6 \cdot 10^{+142}\right) \land \ell \leq 1.22 \cdot 10^{+165}:\\ \;\;\;\;1 + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{kx}{\frac{Om}{kx}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (or (<= l 5.8e+83) (and (not (<= l 6.6e+142)) (<= l 1.22e+165)))
   (+ 1.0 (* (* (/ l (/ Om l)) (/ kx (/ Om kx))) -0.5))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((l <= 5.8e+83) || (!(l <= 6.6e+142) && (l <= 1.22e+165))) {
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if ((l <= 5.8d+83) .or. (.not. (l <= 6.6d+142)) .and. (l <= 1.22d+165)) then
        tmp = 1.0d0 + (((l / (om / l)) * (kx / (om / kx))) * (-0.5d0))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((l <= 5.8e+83) || (!(l <= 6.6e+142) && (l <= 1.22e+165))) {
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if (l <= 5.8e+83) or (not (l <= 6.6e+142) and (l <= 1.22e+165)):
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if ((l <= 5.8e+83) || (!(l <= 6.6e+142) && (l <= 1.22e+165)))
		tmp = Float64(1.0 + Float64(Float64(Float64(l / Float64(Om / l)) * Float64(kx / Float64(Om / kx))) * -0.5));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((l <= 5.8e+83) || (~((l <= 6.6e+142)) && (l <= 1.22e+165)))
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[Or[LessEqual[l, 5.8e+83], And[N[Not[LessEqual[l, 6.6e+142]], $MachinePrecision], LessEqual[l, 1.22e+165]]], N[(1.0 + N[(N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(kx / N[(Om / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.79999999999999999e83 or 6.6000000000000004e142 < l < 1.2199999999999999e165

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

      1. add-sqr-sqrt 98.2%

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

      2. hypot-1-def 98.2%

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

      3. sqrt-prod 98.2%

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

      4. unpow2 98.2%

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

      5. sqrt-prod 56.3%

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

      6. add-sqr-sqrt 98.6%

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

      7. associate-/r/ 98.6%

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

      8. *-commutative 98.6%

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

      9. unpow2 98.6%

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

      10. unpow2 98.6%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in ky around 0 92.8%

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

      1. associate-*r/ 92.8%

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

   8. Simplified 92.8%

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

   9. Taylor expanded in kx around 0 44.7%

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

       1. *-commutative 44.7%

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

       2. *-commutative 44.7%

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

       3. unpow2 44.7%

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

       4. unpow2 44.7%

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

       5. unpow2 44.7%

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

   11. Simplified 44.7%

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

   12. Taylor expanded in l around 0 44.7%

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

       1. unpow2 44.7%

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

       2. *-commutative 44.7%

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

       3. unpow2 44.7%

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

       4. times-frac 48.2%

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

       5. unpow2 48.2%

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

       6. associate-/l* 50.5%

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

       7. associate-/l* 53.7%

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

   14. Simplified 53.7%

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

   if 5.79999999999999999e83 < l < 6.6000000000000004e142 or 1.2199999999999999e165 < l

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in Om around 0 86.5%

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

      1. unpow2 86.5%

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

      2. unpow2 86.5%

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

      3. hypot-def 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in l around inf 88.2%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+83} \lor \neg \left(\ell \leq 6.6 \cdot 10^{+142}\right) \land \ell \leq 1.22 \cdot 10^{+165}:\\ \;\;\;\;1 + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{kx}{\frac{Om}{kx}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 69.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+88}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.04 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+165}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 6.5e+88)
   1.0
   (if (<= l 1.04e+142) (sqrt 0.5) (if (<= l 1.5e+165) 1.0 (sqrt 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 6.5e+88) {
		tmp = 1.0;
	} else if (l <= 1.04e+142) {
		tmp = sqrt(0.5);
	} else if (l <= 1.5e+165) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 6.5d+88) then
        tmp = 1.0d0
    else if (l <= 1.04d+142) then
        tmp = sqrt(0.5d0)
    else if (l <= 1.5d+165) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 6.5e+88) {
		tmp = 1.0;
	} else if (l <= 1.04e+142) {
		tmp = Math.sqrt(0.5);
	} else if (l <= 1.5e+165) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 6.5e+88:
		tmp = 1.0
	elif l <= 1.04e+142:
		tmp = math.sqrt(0.5)
	elif l <= 1.5e+165:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 6.5e+88)
		tmp = 1.0;
	elseif (l <= 1.04e+142)
		tmp = sqrt(0.5);
	elseif (l <= 1.5e+165)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 6.5e+88)
		tmp = 1.0;
	elseif (l <= 1.04e+142)
		tmp = sqrt(0.5);
	elseif (l <= 1.5e+165)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 6.5e+88], 1.0, If[LessEqual[l, 1.04e+142], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, 1.5e+165], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 6.5000000000000002e88 or 1.04e142 < l < 1.49999999999999995e165

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

      1. add-sqr-sqrt 98.2%

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

      2. hypot-1-def 98.2%

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

      3. sqrt-prod 98.2%

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

      4. unpow2 98.2%

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

      5. sqrt-prod 56.5%

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

      6. add-sqr-sqrt 98.6%

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

      7. associate-/r/ 98.6%

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

      8. *-commutative 98.6%

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

      9. unpow2 98.6%

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

      10. unpow2 98.6%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in l around 0 72.2%

      \[\leadsto \sqrt{0.5 + 0.5 \cdot \color{blue}{1}} \]

   if 6.5000000000000002e88 < l < 1.04e142 or 1.49999999999999995e165 < l

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in Om around 0 88.5%

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

      1. unpow2 88.5%

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

      2. unpow2 88.5%

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

      3. hypot-def 88.5%

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

   6. Simplified 88.5%

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

   7. Taylor expanded in l around inf 89.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.5 \cdot 10^{+88}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.04 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+165}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 42.1% accurate, 37.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{+186}:\\ \;\;\;\;1 + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{kx}{\frac{Om}{kx}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(kx \cdot kx\right)\right)\right) \cdot \frac{1}{Om \cdot Om}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.7e+186)
   (+ 1.0 (* (* (/ l (/ Om l)) (/ kx (/ Om kx))) -0.5))
   (+ 1.0 (* -0.5 (* (* l (* l (* kx kx))) (/ 1.0 (* Om Om)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.7e+186) {
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * (l * (kx * kx))) * (1.0 / (Om * Om))));
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 1.7d+186) then
        tmp = 1.0d0 + (((l / (om / l)) * (kx / (om / kx))) * (-0.5d0))
    else
        tmp = 1.0d0 + ((-0.5d0) * ((l * (l * (kx * kx))) * (1.0d0 / (om * om))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.7e+186) {
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * (l * (kx * kx))) * (1.0 / (Om * Om))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.7e+186:
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5)
	else:
		tmp = 1.0 + (-0.5 * ((l * (l * (kx * kx))) * (1.0 / (Om * Om))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.7e+186)
		tmp = Float64(1.0 + Float64(Float64(Float64(l / Float64(Om / l)) * Float64(kx / Float64(Om / kx))) * -0.5));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(l * Float64(l * Float64(kx * kx))) * Float64(1.0 / Float64(Om * Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.7e+186)
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	else
		tmp = 1.0 + (-0.5 * ((l * (l * (kx * kx))) * (1.0 / (Om * Om))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.7e+186], N[(1.0 + N[(N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(kx / N[(Om / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(l * N[(l * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.70000000000000003e186

   1. Initial program 98.3%

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

   3. Simplified 98.3%

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

      1. add-sqr-sqrt 98.3%

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

      2. hypot-1-def 98.3%

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

      3. sqrt-prod 98.3%

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

      4. unpow2 98.3%

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

      5. sqrt-prod 55.4%

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

      6. add-sqr-sqrt 98.6%

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

      7. associate-/r/ 98.6%

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

      8. *-commutative 98.6%

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

      9. unpow2 98.6%

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

      10. unpow2 98.6%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in ky around 0 92.9%

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

      1. associate-*r/ 92.9%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in kx around 0 42.1%

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

       1. *-commutative 42.1%

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

       2. *-commutative 42.1%

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

       3. unpow2 42.1%

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

       4. unpow2 42.1%

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

       5. unpow2 42.1%

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

   11. Simplified 42.1%

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

   12. Taylor expanded in l around 0 42.1%

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

       1. unpow2 42.1%

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

       2. *-commutative 42.1%

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

       3. unpow2 42.1%

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

       4. times-frac 45.4%

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

       5. unpow2 45.4%

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

       6. associate-/l* 47.6%

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

       7. associate-/l* 50.6%

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

   14. Simplified 50.6%

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

   if 1.70000000000000003e186 < l

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

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

      3. sqrt-prod 100.0%

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

      4. unpow2 100.0%

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

      5. sqrt-prod 46.2%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/r/ 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. unpow2 100.0%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in ky around 0 93.3%

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

      1. associate-*r/ 93.3%

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

   8. Simplified 93.3%

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

   9. Taylor expanded in kx around 0 0.4%

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

       1. *-commutative 0.4%

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

       2. *-commutative 0.4%

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

       3. unpow2 0.4%

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

       4. unpow2 0.4%

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

       5. unpow2 0.4%

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

   11. Simplified 0.4%

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

       1. div-inv 0.4%

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

       2. associate-*l* 2.7%

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

   13. Applied egg-rr 2.7%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{+186}:\\ \;\;\;\;1 + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{kx}{\frac{Om}{kx}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(kx \cdot kx\right)\right)\right) \cdot \frac{1}{Om \cdot Om}\right)\\ \end{array} \]

Alternative 8: 42.1% accurate, 42.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+185}:\\ \;\;\;\;1 + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{kx}{\frac{Om}{kx}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot \left(\ell \cdot \left(kx \cdot kx\right)\right)}{Om \cdot Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 7.5e+185)
   (+ 1.0 (* (* (/ l (/ Om l)) (/ kx (/ Om kx))) -0.5))
   (+ 1.0 (* -0.5 (/ (* l (* l (* kx kx))) (* Om Om))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 7.5e+185) {
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * (l * (kx * kx))) / (Om * Om)));
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 7.5d+185) then
        tmp = 1.0d0 + (((l / (om / l)) * (kx / (om / kx))) * (-0.5d0))
    else
        tmp = 1.0d0 + ((-0.5d0) * ((l * (l * (kx * kx))) / (om * om)))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 7.5e+185) {
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * (l * (kx * kx))) / (Om * Om)));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 7.5e+185:
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5)
	else:
		tmp = 1.0 + (-0.5 * ((l * (l * (kx * kx))) / (Om * Om)))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 7.5e+185)
		tmp = Float64(1.0 + Float64(Float64(Float64(l / Float64(Om / l)) * Float64(kx / Float64(Om / kx))) * -0.5));
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(l * Float64(l * Float64(kx * kx))) / Float64(Om * Om))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 7.5e+185)
		tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
	else
		tmp = 1.0 + (-0.5 * ((l * (l * (kx * kx))) / (Om * Om)));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 7.5e+185], N[(1.0 + N[(N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(kx / N[(Om / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(l * N[(l * N[(kx * kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.49999999999999955e185

   1. Initial program 98.3%

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

   3. Simplified 98.3%

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

      1. add-sqr-sqrt 98.3%

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

      2. hypot-1-def 98.3%

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

      3. sqrt-prod 98.3%

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

      4. unpow2 98.3%

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

      5. sqrt-prod 55.4%

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

      6. add-sqr-sqrt 98.6%

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

      7. associate-/r/ 98.6%

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

      8. *-commutative 98.6%

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

      9. unpow2 98.6%

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

      10. unpow2 98.6%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in ky around 0 92.9%

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

      1. associate-*r/ 92.9%

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

   8. Simplified 92.9%

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

   9. Taylor expanded in kx around 0 42.1%

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

       1. *-commutative 42.1%

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

       2. *-commutative 42.1%

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

       3. unpow2 42.1%

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

       4. unpow2 42.1%

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

       5. unpow2 42.1%

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

   11. Simplified 42.1%

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

   12. Taylor expanded in l around 0 42.1%

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

       1. unpow2 42.1%

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

       2. *-commutative 42.1%

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

       3. unpow2 42.1%

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

       4. times-frac 45.4%

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

       5. unpow2 45.4%

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

       6. associate-/l* 47.6%

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

       7. associate-/l* 50.6%

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

   14. Simplified 50.6%

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

   if 7.49999999999999955e185 < l

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

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

      3. sqrt-prod 100.0%

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

      4. unpow2 100.0%

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

      5. sqrt-prod 46.2%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/r/ 100.0%

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

      8. *-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. unpow2 100.0%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in ky around 0 93.3%

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

      1. associate-*r/ 93.3%

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

   8. Simplified 93.3%

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

   9. Taylor expanded in kx around 0 0.4%

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

       1. *-commutative 0.4%

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

       2. *-commutative 0.4%

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

       3. unpow2 0.4%

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

       4. unpow2 0.4%

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

       5. unpow2 0.4%

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

   11. Simplified 0.4%

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

   12. Taylor expanded in l around 0 0.4%

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

       1. unpow2 0.4%

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

       2. *-commutative 0.4%

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

       3. unpow2 0.4%

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

       4. associate-*r* 2.7%

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

   14. Simplified 2.7%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.5 \cdot 10^{+185}:\\ \;\;\;\;1 + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{kx}{\frac{Om}{kx}}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot \left(\ell \cdot \left(kx \cdot kx\right)\right)}{Om \cdot Om}\\ \end{array} \]

Alternative 9: 33.3% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}} \cdot -0.5 \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (+ 1.0 (* (/ (* ky ky) (/ (* Om Om) (* l l))) -0.5)))

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0 + (((ky * ky) / ((om * om) / (l * l))) * (-0.5d0))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0 + (((ky * ky) / ((Om * Om) / (l * l))) * -0.5);
}

Python

def code(l, Om, kx, ky):
	return 1.0 + (((ky * ky) / ((Om * Om) / (l * l))) * -0.5)

Julia

function code(l, Om, kx, ky)
	return Float64(1.0 + Float64(Float64(Float64(ky * ky) / Float64(Float64(Om * Om) / Float64(l * l))) * -0.5))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = 1.0 + (((ky * ky) / ((Om * Om) / (l * l))) * -0.5);
end

Wolfram

code[l_, Om_, kx_, ky_] := N[(1.0 + N[(N[(N[(ky * ky), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

3. Simplified 98.5%

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

   1. add-sqr-sqrt 98.5%

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

   2. hypot-1-def 98.5%

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

   3. sqrt-prod 98.5%

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

   4. unpow2 98.5%

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

   5. sqrt-prod 54.5%

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

   6. add-sqr-sqrt 98.8%

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

   7. associate-/r/ 98.8%

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

   8. *-commutative 98.8%

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

   9. unpow2 98.8%

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

   10. unpow2 98.8%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 94.0%

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

7. Taylor expanded in ky around 0 36.5%

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

   1. associate-/l* 36.6%

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

   2. unpow2 36.6%

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

   3. unpow2 36.6%

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

   4. unpow2 36.6%

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

9. Simplified 36.6%

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

10. Final simplification 36.6%

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

Alternative 10: 39.2% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot \left(\left(kx \cdot kx\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right) \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (+ 1.0 (* -0.5 (* (* kx kx) (* (/ l Om) (/ l Om))))))

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0 + ((-0.5d0) * ((kx * kx) * ((l / om) * (l / om))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0 + (-0.5 * ((kx * kx) * ((l / Om) * (l / Om))));
}

Python

def code(l, Om, kx, ky):
	return 1.0 + (-0.5 * ((kx * kx) * ((l / Om) * (l / Om))))

Julia

function code(l, Om, kx, ky)
	return Float64(1.0 + Float64(-0.5 * Float64(Float64(kx * kx) * Float64(Float64(l / Om) * Float64(l / Om)))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = 1.0 + (-0.5 * ((kx * kx) * ((l / Om) * (l / Om))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[(1.0 + N[(-0.5 * N[(N[(kx * kx), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

3. Simplified 98.5%

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

   1. add-sqr-sqrt 98.5%

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

   2. hypot-1-def 98.5%

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

   3. sqrt-prod 98.5%

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

   4. unpow2 98.5%

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

   5. sqrt-prod 54.5%

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

   6. add-sqr-sqrt 98.8%

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

   7. associate-/r/ 98.8%

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

   8. *-commutative 98.8%

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

   9. unpow2 98.8%

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

   10. unpow2 98.8%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in ky around 0 93.0%

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

   1. associate-*r/ 93.0%

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

8. Simplified 93.0%

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

9. Taylor expanded in kx around 0 37.9%

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

    1. *-commutative 37.9%

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

    2. *-commutative 37.9%

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

    3. unpow2 37.9%

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

    4. unpow2 37.9%

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

    5. unpow2 37.9%

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

11. Simplified 37.9%

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

    1. *-un-lft-identity 37.9%

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

    2. associate-/l* 36.7%

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

13. Applied egg-rr 36.7%

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

    1. *-lft-identity 36.7%

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

    2. unpow2 36.7%

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

    3. unpow2 36.7%

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

    4. associate-/r/ 37.7%

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

    5. unpow2 37.7%

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

    6. unpow2 37.7%

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

    7. times-frac 43.6%

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

15. Simplified 43.6%

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

16. Final simplification 43.6%

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

Alternative 11: 41.8% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{kx}{\frac{Om}{kx}}\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (+ 1.0 (* (* (/ l (/ Om l)) (/ kx (/ Om kx))) -0.5)))

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0 + (((l / (om / l)) * (kx / (om / kx))) * (-0.5d0))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
}

Python

def code(l, Om, kx, ky):
	return 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5)

Julia

function code(l, Om, kx, ky)
	return Float64(1.0 + Float64(Float64(Float64(l / Float64(Om / l)) * Float64(kx / Float64(Om / kx))) * -0.5))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = 1.0 + (((l / (Om / l)) * (kx / (Om / kx))) * -0.5);
end

Wolfram

code[l_, Om_, kx_, ky_] := N[(1.0 + N[(N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(kx / N[(Om / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

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

3. Simplified 98.5%

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

   1. add-sqr-sqrt 98.5%

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

   2. hypot-1-def 98.5%

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

   3. sqrt-prod 98.5%

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

   4. unpow2 98.5%

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

   5. sqrt-prod 54.5%

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

   6. add-sqr-sqrt 98.8%

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

   7. associate-/r/ 98.8%

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

   8. *-commutative 98.8%

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

   9. unpow2 98.8%

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

   10. unpow2 98.8%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in ky around 0 93.0%

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

   1. associate-*r/ 93.0%

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

8. Simplified 93.0%

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

9. Taylor expanded in kx around 0 37.9%

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

    1. *-commutative 37.9%

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

    2. *-commutative 37.9%

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

    3. unpow2 37.9%

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

    4. unpow2 37.9%

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

    5. unpow2 37.9%

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

11. Simplified 37.9%

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

12. Taylor expanded in l around 0 37.9%

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

    1. unpow2 37.9%

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

    2. *-commutative 37.9%

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

    3. unpow2 37.9%

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

    4. times-frac 40.8%

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

    5. unpow2 40.8%

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

    6. associate-/l* 42.8%

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

    7. associate-/l* 45.5%

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

14. Simplified 45.5%

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

15. Final simplification 45.5%

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

Reproduce

herbie shell --seed 2023293 
(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))))))))))