Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 15.6s

Alternatives: 8

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: 98.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

   1. add-sqr-sqrt 98.4%

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

   2. hypot-1-def 98.4%

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

   3. sqrt-prod 98.4%

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

   4. sqrt-pow1 99.0%

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

   5. metadata-eval 99.0%

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

   6. div-inv 99.0%

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

   7. pow1 99.0%

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

   8. clear-num 99.0%

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

   9. unpow2 99.0%

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

   10. unpow2 99.0%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

   3. associate-*l/ 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*r/ 99.3%

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

7. Applied egg-rr 99.3%

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

   1. expm1-def 99.3%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-/l* 100.0%

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

   5. hypot-def 99.0%

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

   6. unpow2 99.0%

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

   7. unpow2 99.0%

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

   8. +-commutative 99.0%

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

   9. unpow2 99.0%

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

   10. unpow2 99.0%

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

   11. hypot-def 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 93.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in kx around 0 80.5%

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

   1. associate-/l* 80.7%

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

   2. associate-/r/ 81.4%

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

   3. unpow2 81.4%

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

   4. unpow2 81.4%

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

   5. times-frac 89.9%

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

   6. unpow2 89.9%

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

6. Simplified 89.9%

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

   1. pow-prod-down 93.8%

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

8. Applied egg-rr 93.8%

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

9. Final simplification 93.8%

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

Alternative 3: 93.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (cbrt (pow (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin ky) (* 2.0 (/ l Om)))))) 1.5)))

C

double code(double l, double Om, double kx, double ky) {
	return cbrt(pow((0.5 + (0.5 / hypot(1.0, (sin(ky) * (2.0 * (l / Om)))))), 1.5));
}

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.cbrt(Math.pow((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(ky) * (2.0 * (l / Om)))))), 1.5));
}

Julia

function code(l, Om, kx, ky)
	return cbrt((Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(ky) * Float64(2.0 * Float64(l / Om)))))) ^ 1.5))
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Power[N[Power[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[ky], $MachinePrecision] * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in kx around 0 80.5%

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

   1. associate-/l* 80.7%

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

   2. associate-/r/ 81.4%

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

   3. unpow2 81.4%

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

   4. unpow2 81.4%

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

   5. times-frac 89.9%

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

   6. unpow2 89.9%

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

6. Simplified 89.9%

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

   1. add-sqr-sqrt 89.9%

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

   2. hypot-1-def 89.9%

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

   3. associate-*r* 89.9%

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

   4. sqrt-prod 89.9%

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

   5. metadata-eval 89.9%

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

   6. unpow2 89.9%

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

   7. swap-sqr 89.9%

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

   8. sqrt-unprod 49.7%

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

   9. add-sqr-sqrt 90.9%

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

   10. associate-*r/ 90.9%

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

   11. unpow2 90.9%

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

   12. sqrt-prod 42.5%

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

   13. add-sqr-sqrt 93.8%

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

8. Applied egg-rr 93.8%

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

   1. add-cbrt-cube 93.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}}} \]

   2. pow1/3 93.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

10. Applied egg-rr 93.8%

    \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
11. Step-by-step derivation

    1. unpow1/3 93.8%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

12. Simplified 93.8%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

13. Final simplification 93.8%

    \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}} \]

Alternative 4: 89.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{\frac{Om}{ky}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot \sin kx\right)}{Om}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= kx 5e-70)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (/ (* 2.0 l) (/ Om ky))))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (/ (* 2.0 (* l (sin kx))) Om)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 5e-70) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * l) / (Om / ky))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * (l * sin(kx))) / Om)))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 5e-70) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 * l) / (Om / ky))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 * (l * Math.sin(kx))) / Om)))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if kx <= 5e-70:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 * l) / (Om / ky))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 * (l * math.sin(kx))) / Om)))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (kx <= 5e-70)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * l) / Float64(Om / ky))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * Float64(l * sin(kx))) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (kx <= 5e-70)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * l) / (Om / ky))))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * (l * sin(kx))) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[kx, 5e-70], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * l), $MachinePrecision] / N[(Om / ky), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[(l * N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 4.9999999999999998e-70

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in kx around 0 85.1%

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

      1. associate-/l* 84.2%

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

      2. associate-/r/ 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

      5. times-frac 91.3%

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

      6. unpow2 91.3%

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

   6. Simplified 91.3%

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

      1. add-sqr-sqrt 91.3%

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

      2. hypot-1-def 91.3%

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

      3. associate-*r* 91.3%

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

      4. sqrt-prod 91.3%

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

      5. metadata-eval 91.3%

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

      6. unpow2 91.3%

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

      7. swap-sqr 91.3%

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

      8. sqrt-unprod 51.3%

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

      9. add-sqr-sqrt 92.3%

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

      10. associate-*r/ 92.3%

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

      11. unpow2 92.3%

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

      12. sqrt-prod 44.3%

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

      13. add-sqr-sqrt 94.4%

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

   8. Applied egg-rr 94.4%

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

   9. Taylor expanded in ky around 0 86.6%

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

       1. *-commutative 86.6%

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

   11. Simplified 86.6%

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

       1. expm1-log1p-u 86.0%

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

       2. expm1-udef 86.0%

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

       3. associate-*l/ 86.0%

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

       4. metadata-eval 86.0%

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

       5. associate-/l* 86.0%

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

   13. Applied egg-rr 86.0%

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

       1. expm1-def 86.0%

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

       2. expm1-log1p 86.6%

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

       3. associate-*r/ 86.6%

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

   15. Simplified 86.6%

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

   if 4.9999999999999998e-70 < kx

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

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

      3. sqrt-prod 100.0%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

      6. div-inv 100.0%

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

      7. pow1 100.0%

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

      8. clear-num 100.0%

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

      9. unpow2 100.0%

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

      10. unpow2 100.0%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. associate-*l/ 99.3%

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

      4. metadata-eval 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r/ 99.3%

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

   7. Applied egg-rr 99.3%

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

      1. expm1-def 99.3%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

      5. hypot-def 100.0%

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

      6. unpow2 100.0%

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

      7. unpow2 100.0%

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

      8. +-commutative 100.0%

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

      9. unpow2 100.0%

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

      10. unpow2 100.0%

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

      11. hypot-def 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in ky around 0 100.0%

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

       1. associate-*r/ 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 5 \cdot 10^{-70}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{\frac{Om}{ky}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot \sin kx\right)}{Om}\right)}}\\ \end{array} \]

Alternative 5: 93.4% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt (+ 0.5 (* 0.5 (/ 1.0 (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 * (1.0 / 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 * (1.0 / 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 * (1.0 / 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 * Float64(1.0 / 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 * (1.0 / 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[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[ky], $MachinePrecision] * N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in kx around 0 80.5%

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

   1. associate-/l* 80.7%

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

   2. associate-/r/ 81.4%

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

   3. unpow2 81.4%

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

   4. unpow2 81.4%

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

   5. times-frac 89.9%

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

   6. unpow2 89.9%

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

6. Simplified 89.9%

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

   1. add-sqr-sqrt 89.9%

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

   2. hypot-1-def 89.9%

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

   3. associate-*r* 89.9%

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

   4. sqrt-prod 89.9%

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

   5. metadata-eval 89.9%

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

   6. unpow2 89.9%

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

   7. swap-sqr 89.9%

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

   8. sqrt-unprod 49.7%

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

   9. add-sqr-sqrt 90.9%

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

   10. associate-*r/ 90.9%

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

   11. unpow2 90.9%

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

   12. sqrt-prod 42.5%

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

   13. add-sqr-sqrt 93.8%

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

8. Applied egg-rr 93.8%

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

9. Final simplification 93.8%

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

Alternative 6: 72.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot \ell \leq 5 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{\frac{Om}{ky}}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (* 2.0 l) 5e-14)
   1.0
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (/ (* 2.0 l) (/ Om ky))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((2.0 * l) <= 5e-14) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * l) / (Om / ky))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((2.0 * l) <= 5e-14) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 * l) / (Om / ky))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if (2.0 * l) <= 5e-14:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 * l) / (Om / ky))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(2.0 * l) <= 5e-14)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * l) / Float64(Om / ky))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((2.0 * l) <= 5e-14)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * l) / (Om / ky))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(2.0 * l), $MachinePrecision], 5e-14], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * l), $MachinePrecision] / N[(Om / ky), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 2 l) < 5.0000000000000002e-14

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in kx around 0 83.9%

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

      1. associate-/l* 84.1%

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

      2. associate-/r/ 85.2%

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

      3. unpow2 85.2%

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

      4. unpow2 85.2%

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

      5. times-frac 91.3%

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

      6. unpow2 91.3%

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

   6. Simplified 91.3%

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

      1. add-sqr-sqrt 91.3%

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

      2. hypot-1-def 91.3%

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

      3. associate-*r* 91.3%

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

      4. sqrt-prod 91.3%

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

      5. metadata-eval 91.3%

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

      6. unpow2 91.3%

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

      7. swap-sqr 91.3%

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

      8. sqrt-unprod 52.2%

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

      9. add-sqr-sqrt 92.2%

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

      10. associate-*r/ 92.2%

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

      11. unpow2 92.2%

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

      12. sqrt-prod 39.8%

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

      13. add-sqr-sqrt 94.7%

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

   8. Applied egg-rr 94.7%

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

      1. add-cbrt-cube 94.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}}} \]

      2. pow1/3 94.7%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 94.7%

       \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. unpow1/3 94.7%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

   12. Simplified 94.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

   13. Taylor expanded in ky around 0 69.1%

       \[\leadsto \sqrt[3]{\color{blue}{1}} \]

   if 5.0000000000000002e-14 < (*.f64 2 l)

   1. Initial program 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in kx around 0 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate-/r/ 70.9%

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

      3. unpow2 70.9%

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

      4. unpow2 70.9%

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

      5. times-frac 86.3%

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

      6. unpow2 86.3%

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

   6. Simplified 86.3%

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

      1. add-sqr-sqrt 86.3%

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

      2. hypot-1-def 86.3%

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

      3. associate-*r* 86.3%

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

      4. sqrt-prod 86.3%

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

      5. metadata-eval 86.3%

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

      6. unpow2 86.3%

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

      7. swap-sqr 86.3%

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

      8. sqrt-unprod 42.8%

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

      9. add-sqr-sqrt 87.3%

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

      10. associate-*r/ 87.3%

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

      11. unpow2 87.3%

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

      12. sqrt-prod 49.8%

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

      13. add-sqr-sqrt 91.4%

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

   8. Applied egg-rr 91.4%

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

   9. Taylor expanded in ky around 0 84.9%

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

       1. *-commutative 84.9%

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

   11. Simplified 84.9%

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

       1. expm1-log1p-u 84.1%

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

       2. expm1-udef 84.1%

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

       3. associate-*l/ 84.1%

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

       4. metadata-eval 84.1%

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

       5. associate-/l* 84.1%

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

   13. Applied egg-rr 84.1%

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

       1. expm1-def 84.1%

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

       2. expm1-log1p 84.9%

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

       3. associate-*r/ 84.9%

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

   15. Simplified 84.9%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot \ell \leq 5 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{\frac{Om}{ky}}\right)}}\\ \end{array} \]

Alternative 7: 69.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.16 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 1.16e+106) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.16e+106) {
		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 <= 1.16d+106) 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 <= 1.16e+106) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.16e+106:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.16e+106)
		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 <= 1.16e+106)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.16e+106], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.16000000000000004e106

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in kx around 0 85.4%

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

      1. associate-/l* 85.6%

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

      2. associate-/r/ 86.5%

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

      3. unpow2 86.5%

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

      4. unpow2 86.5%

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

      5. times-frac 91.9%

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

      6. unpow2 91.9%

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

   6. Simplified 91.9%

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

      1. add-sqr-sqrt 91.9%

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

      2. hypot-1-def 91.9%

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

      3. associate-*r* 91.9%

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

      4. sqrt-prod 91.9%

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

      5. metadata-eval 91.9%

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

      6. unpow2 91.9%

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

      7. swap-sqr 91.9%

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

      8. sqrt-unprod 52.9%

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

      9. add-sqr-sqrt 92.7%

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

      10. associate-*r/ 92.7%

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

      11. unpow2 92.7%

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

      12. sqrt-prod 41.9%

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

      13. add-sqr-sqrt 94.9%

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

   8. Applied egg-rr 94.9%

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

      1. add-cbrt-cube 94.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}}} \]

      2. pow1/3 94.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 94.9%

       \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. unpow1/3 94.9%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

   12. Simplified 94.9%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

   13. Taylor expanded in ky around 0 70.0%

       \[\leadsto \sqrt[3]{\color{blue}{1}} \]

   if 1.16000000000000004e106 < l

   1. Initial program 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in Om around 0 77.3%

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

      1. unpow2 77.3%

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

      2. unpow2 77.3%

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

      3. hypot-def 81.8%

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

   6. Simplified 81.8%

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

   7. Taylor expanded in l around inf 85.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.16 \cdot 10^{+106}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8: 62.3% 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 98.4%

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

3. Simplified 98.4%

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

4. Taylor expanded in kx around 0 80.5%

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

   1. associate-/l* 80.7%

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

   2. associate-/r/ 81.4%

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

   3. unpow2 81.4%

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

   4. unpow2 81.4%

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

   5. times-frac 89.9%

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

   6. unpow2 89.9%

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

6. Simplified 89.9%

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

   1. add-sqr-sqrt 89.9%

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

   2. hypot-1-def 89.9%

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

   3. associate-*r* 89.9%

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

   4. sqrt-prod 89.9%

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

   5. metadata-eval 89.9%

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

   6. unpow2 89.9%

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

   7. swap-sqr 89.9%

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

   8. sqrt-unprod 49.7%

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

   9. add-sqr-sqrt 90.9%

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

   10. associate-*r/ 90.9%

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

   11. unpow2 90.9%

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

   12. sqrt-prod 42.5%

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

   13. add-sqr-sqrt 93.8%

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

8. Applied egg-rr 93.8%

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

   1. add-cbrt-cube 93.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}}} \]

   2. pow1/3 93.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5} \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right) \cdot \sqrt{0.5 + \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \ell}{Om} \cdot \sin ky\right)} \cdot 0.5}\right)}^{0.3333333333333333}} \]

10. Applied egg-rr 93.8%

    \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
11. Step-by-step derivation

    1. unpow1/3 93.8%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

12. Simplified 93.8%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin ky \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}\right)}^{1.5}}} \]

13. Taylor expanded in ky around 0 63.9%

    \[\leadsto \sqrt[3]{\color{blue}{1}} \]

14. Final simplification 63.9%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023325 
(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))))))))))