Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 12.8s

Alternatives: 5

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. add-sqr-sqrt 99.2%

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

   3. hypot-1-def 99.2%

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

   4. sqrt-prod 99.2%

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

   5. sqrt-pow1 99.6%

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

   6. metadata-eval 99.6%

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

   7. pow1 99.6%

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

   8. clear-num 99.6%

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

   9. un-div-inv 99.6%

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

   10. unpow2 99.6%

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

   11. unpow2 99.6%

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

   12. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-/r/ 100.0%

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

8. Simplified 100.0%

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

Alternative 2: 94.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot {\left(1 + {\left(\sin ky \cdot \frac{\ell}{Om}\right)}^{2} \cdot 4\right)}^{-0.5}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+ 0.5 (* 0.5 (pow (+ 1.0 (* (pow (* (sin ky) (/ l Om)) 2.0) 4.0)) -0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * pow((1.0 + (pow((sin(ky) * (l / Om)), 2.0) * 4.0)), -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 = sqrt((0.5d0 + (0.5d0 * ((1.0d0 + (((sin(ky) * (l / om)) ** 2.0d0) * 4.0d0)) ** (-0.5d0)))))
end function

Java

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

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * math.pow((1.0 + (math.pow((math.sin(ky) * (l / Om)), 2.0) * 4.0)), -0.5))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * (Float64(1.0 + Float64((Float64(sin(ky) * Float64(l / Om)) ^ 2.0) * 4.0)) ^ -0.5))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * ((1.0 + (((sin(ky) * (l / Om)) ^ 2.0) * 4.0)) ^ -0.5))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[(1.0 + N[(N[Power[N[(N[Sin[ky], $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in kx around 0 78.3%

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

   1. +-commutative 78.3%

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

   2. fma-define 78.3%

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

   3. *-commutative 78.3%

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

   4. associate-/l* 78.6%

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

   5. unpow2 78.6%

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

   6. unpow2 78.6%

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

   7. times-frac 91.5%

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

   8. unpow2 91.5%

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

7. Simplified 91.5%

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

   1. fma-undefine 91.5%

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

   2. pow-prod-down 95.5%

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

   3. clear-num 95.5%

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

   4. un-div-inv 95.5%

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

9. Applied egg-rr 95.5%

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

    1. sqrt-div 95.5%

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

    2. metadata-eval 95.5%

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

    3. +-commutative 95.5%

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

    4. add-sqr-sqrt 95.5%

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

    5. hypot-1-def 95.5%

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

    6. *-commutative 95.5%

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

    7. sqrt-prod 95.5%

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

    8. sqrt-pow1 95.5%

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

    9. metadata-eval 95.5%

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

    10. pow1 95.5%

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

    11. div-inv 95.5%

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

    12. clear-num 95.5%

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

    13. metadata-eval 95.5%

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

11. Applied egg-rr 95.5%

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

    1. inv-pow 95.5%

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

    2. hypot-undefine 95.5%

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

    3. sqrt-pow2 95.5%

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

    4. metadata-eval 95.5%

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

    5. swap-sqr 95.5%

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

    6. pow2 95.5%

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

    7. metadata-eval 95.5%

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

    8. metadata-eval 95.5%

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

13. Applied egg-rr 95.5%

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

Alternative 3: 94.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in kx around 0 78.3%

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

   1. +-commutative 78.3%

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

   2. fma-define 78.3%

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

   3. *-commutative 78.3%

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

   4. associate-/l* 78.6%

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

   5. unpow2 78.6%

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

   6. unpow2 78.6%

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

   7. times-frac 91.5%

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

   8. unpow2 91.5%

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

7. Simplified 91.5%

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

   1. fma-undefine 91.5%

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

   2. pow-prod-down 95.5%

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

   3. clear-num 95.5%

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

   4. un-div-inv 95.5%

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

9. Applied egg-rr 95.5%

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

    1. sqrt-div 95.5%

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

    2. metadata-eval 95.5%

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

    3. +-commutative 95.5%

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

    4. add-sqr-sqrt 95.5%

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

    5. hypot-1-def 95.5%

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

    6. *-commutative 95.5%

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

    7. sqrt-prod 95.5%

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

    8. sqrt-pow1 95.5%

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

    9. metadata-eval 95.5%

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

    10. pow1 95.5%

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

    11. div-inv 95.5%

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

    12. clear-num 95.5%

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

    13. metadata-eval 95.5%

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

11. Applied egg-rr 95.5%

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

    1. *-un-lft-identity 95.5%

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

    2. un-div-inv 95.5%

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

    3. associate-*l* 95.5%

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

13. Applied egg-rr 95.5%

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

    1. *-lft-identity 95.5%

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

    2. *-commutative 95.5%

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

    3. associate-*r/ 95.5%

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

15. Simplified 95.5%

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

Alternative 4: 70.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-70}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 1.2e-70) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.2e-70) {
		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 <= 1.2d-70) 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 <= 1.2e-70) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.2e-70:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.2e-70)
		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 <= 1.2e-70)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.2e-70], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.2000000000000001e-70

   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%

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

   5. Taylor expanded in kx around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. fma-define 76.0%

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

      3. *-commutative 76.0%

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

      4. associate-/l* 76.5%

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

      5. unpow2 76.5%

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

      6. unpow2 76.5%

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

      7. times-frac 92.3%

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

      8. unpow2 92.3%

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

   7. Simplified 92.3%

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

      1. fma-undefine 92.3%

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

      2. pow-prod-down 96.2%

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

      3. clear-num 96.2%

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

      4. un-div-inv 96.2%

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

   9. Applied egg-rr 96.2%

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

       1. sqrt-div 96.2%

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

       2. metadata-eval 96.2%

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

       3. +-commutative 96.2%

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

       4. add-sqr-sqrt 96.2%

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

       5. hypot-1-def 96.2%

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

       6. *-commutative 96.2%

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

       7. sqrt-prod 96.2%

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

       8. sqrt-pow1 96.2%

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

       9. metadata-eval 96.2%

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

       10. pow1 96.2%

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

       11. div-inv 96.2%

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

       12. clear-num 96.2%

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

       13. metadata-eval 96.2%

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

   11. Applied egg-rr 96.2%

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

   12. Taylor expanded in ky around 0 71.1%

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

   if 1.2000000000000001e-70 < l

   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%

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

      1. *-un-lft-identity 97.8%

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

      2. add-sqr-sqrt 97.8%

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

      3. hypot-1-def 97.8%

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

      4. sqrt-prod 97.8%

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

      5. sqrt-pow1 98.9%

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

      6. metadata-eval 98.9%

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

      7. pow1 98.9%

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

      8. clear-num 98.9%

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

      9. un-div-inv 98.9%

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

      10. unpow2 98.9%

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

      11. unpow2 98.9%

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

      12. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in Om around 0 74.4%

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

Alternative 5: 61.5% accurate, 722.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 1.0)

C

double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return 1.0;
}

Python

def code(l, Om, kx, ky):
	return 1.0

Julia

function code(l, Om, kx, ky)
	return 1.0
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = 1.0;
end

Wolfram

code[l_, Om_, kx_, ky_] := 1.0

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in kx around 0 78.3%

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

   1. +-commutative 78.3%

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

   2. fma-define 78.3%

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

   3. *-commutative 78.3%

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

   4. associate-/l* 78.6%

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

   5. unpow2 78.6%

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

   6. unpow2 78.6%

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

   7. times-frac 91.5%

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

   8. unpow2 91.5%

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

7. Simplified 91.5%

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

   1. fma-undefine 91.5%

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

   2. pow-prod-down 95.5%

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

   3. clear-num 95.5%

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

   4. un-div-inv 95.5%

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

9. Applied egg-rr 95.5%

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

    1. sqrt-div 95.5%

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

    2. metadata-eval 95.5%

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

    3. +-commutative 95.5%

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

    4. add-sqr-sqrt 95.5%

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

    5. hypot-1-def 95.5%

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

    6. *-commutative 95.5%

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

    7. sqrt-prod 95.5%

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

    8. sqrt-pow1 95.5%

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

    9. metadata-eval 95.5%

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

    10. pow1 95.5%

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

    11. div-inv 95.5%

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

    12. clear-num 95.5%

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

    13. metadata-eval 95.5%

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

11. Applied egg-rr 95.5%

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

12. Taylor expanded in ky around 0 61.7%

    \[\leadsto \color{blue}{1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(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))))))))))