Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 16.1s

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: 98.4% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in 99.2%

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

   2. metadata-eval 99.2%

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

   3. metadata-eval 99.2%

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

   4. associate-/l* 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (*
    0.5
    (/ 1.0 (hypot 1.0 (* (hypot (sin ky) (sin kx)) (* 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, (hypot(sin(ky), sin(kx)) * (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.hypot(Math.sin(ky), Math.sin(kx)) * (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.hypot(math.sin(ky), math.sin(kx)) * (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(hypot(sin(ky), sin(kx)) * Float64(2.0 * Float64(l / Om))))))))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in 99.2%

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

   2. metadata-eval 99.2%

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

   3. metadata-eval 99.2%

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

   4. associate-/l* 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 99.3%

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

   5. unpow2 99.3%

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

   6. unpow2 99.3%

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

   7. +-commutative 99.3%

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

   8. unpow2 99.3%

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

   9. unpow2 99.3%

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

   10. hypot-def 100.0%

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

   11. *-commutative 100.0%

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

   12. associate-*l/ 100.0%

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

   13. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 93.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in 99.2%

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

   2. metadata-eval 99.2%

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

   3. metadata-eval 99.2%

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

   4. associate-/l* 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 94.9%

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

   1. *-commutative 94.9%

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

   2. associate-*r/ 94.9%

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

   3. associate-*r* 94.9%

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

8. Simplified 94.9%

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

   1. pow-flip 94.9%

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

   2. metadata-eval 94.9%

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

10. Applied egg-rr 94.9%

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

11. Final simplification 94.9%

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

Alternative 4: 94.2% accurate, 2.3× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in 99.2%

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

   2. metadata-eval 99.2%

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

   3. metadata-eval 99.2%

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

   4. associate-/l* 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 99.2%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 99.3%

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

   5. unpow2 99.3%

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

   6. unpow2 99.3%

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

   7. +-commutative 99.3%

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

   8. unpow2 99.3%

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

   9. unpow2 99.3%

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

   10. hypot-def 100.0%

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

   11. *-commutative 100.0%

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

   12. associate-*l/ 100.0%

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

   13. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in ky around 0 96.6%

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

   1. expm1-log1p-u 96.0%

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

   2. expm1-udef 96.0%

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

   3. associate-*l/ 96.0%

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

   4. metadata-eval 96.0%

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

10. Applied egg-rr 96.0%

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

    1. expm1-def 96.0%

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

    2. expm1-log1p 96.6%

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

12. Simplified 96.6%

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

13. Final simplification 96.6%

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

Alternative 5: 93.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in 99.2%

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

   2. metadata-eval 99.2%

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

   3. metadata-eval 99.2%

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

   4. associate-/l* 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 94.9%

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

   1. *-commutative 94.9%

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

   2. associate-*r/ 94.9%

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

   3. associate-*r* 94.9%

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

8. Simplified 94.9%

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

   1. pow-flip 94.9%

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

   2. metadata-eval 94.9%

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

10. Applied egg-rr 94.9%

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

    1. expm1-log1p-u 94.3%

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

    2. expm1-udef 94.3%

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

    3. sqrt-pow2 94.3%

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

    4. metadata-eval 94.3%

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

    5. inv-pow 94.3%

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

    6. associate-*l/ 94.3%

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

    7. metadata-eval 94.3%

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

12. Applied egg-rr 94.3%

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

    1. expm1-def 94.3%

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

    2. expm1-log1p 94.9%

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

14. Simplified 94.9%

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

15. Final simplification 94.9%

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

Alternative 6: 70.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.3 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 5.3e-20)
   1.0
   (if (<= l 200.0) (sqrt 0.5) (if (<= l 8.2e+50) 1.0 (sqrt 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5.3e-20) {
		tmp = 1.0;
	} else if (l <= 200.0) {
		tmp = sqrt(0.5);
	} else if (l <= 8.2e+50) {
		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 <= 5.3d-20) then
        tmp = 1.0d0
    else if (l <= 200.0d0) then
        tmp = sqrt(0.5d0)
    else if (l <= 8.2d+50) 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 <= 5.3e-20) {
		tmp = 1.0;
	} else if (l <= 200.0) {
		tmp = Math.sqrt(0.5);
	} else if (l <= 8.2e+50) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 5.3e-20:
		tmp = 1.0
	elif l <= 200.0:
		tmp = math.sqrt(0.5)
	elif l <= 8.2e+50:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 5.3e-20)
		tmp = 1.0;
	elseif (l <= 200.0)
		tmp = sqrt(0.5);
	elseif (l <= 8.2e+50)
		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 <= 5.3e-20)
		tmp = 1.0;
	elseif (l <= 200.0)
		tmp = sqrt(0.5);
	elseif (l <= 8.2e+50)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 5.3e-20], 1.0, If[LessEqual[l, 200.0], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, 8.2e+50], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.3000000000000002e-20 or 200 < l < 8.2000000000000002e50

   1. Initial program 99.5%

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

      1. distribute-rgt-in 99.5%

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

      2. metadata-eval 99.5%

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

      3. metadata-eval 99.5%

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

      4. associate-/l* 99.5%

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

      5. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 99.5%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 99.6%

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

      5. unpow2 99.6%

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

      6. unpow2 99.6%

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

      7. +-commutative 99.6%

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

      8. unpow2 99.6%

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

      9. unpow2 99.6%

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

      10. hypot-def 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-*l/ 100.0%

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

      13. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in ky around 0 97.6%

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

      1. expm1-log1p-u 97.0%

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

      2. expm1-udef 97.0%

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

      3. associate-*l/ 97.0%

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

      4. metadata-eval 97.0%

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

   10. Applied egg-rr 97.0%

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

       1. expm1-def 97.0%

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

       2. expm1-log1p 97.6%

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

   12. Simplified 97.6%

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

   13. Taylor expanded in kx around 0 70.3%

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

   if 5.3000000000000002e-20 < l < 200 or 8.2000000000000002e50 < l

   1. Initial program 98.2%

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

      1. distribute-rgt-in 98.2%

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

      2. metadata-eval 98.2%

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

      3. metadata-eval 98.2%

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

      4. associate-/l* 98.2%

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

      5. metadata-eval 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in Om around 0 72.7%

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

      1. associate-*r* 72.7%

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

      2. *-commutative 72.7%

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

      3. associate-*r* 72.7%

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

      4. unpow2 72.7%

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

      5. unpow2 72.7%

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

      6. hypot-def 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in l around inf 78.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.3 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 51.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-109}:\\ \;\;\;\;1 + \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{ky \cdot ky}{Om}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 8.5e-109)
   (+ 1.0 (* (* (/ (* l l) Om) (/ (* ky ky) Om)) -0.5))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 8.5e-109) {
		tmp = 1.0 + ((((l * l) / Om) * ((ky * ky) / Om)) * -0.5);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 8.5d-109) then
        tmp = 1.0d0 + ((((l * l) / om) * ((ky * ky) / om)) * (-0.5d0))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 8.5e-109) {
		tmp = 1.0 + ((((l * l) / Om) * ((ky * ky) / Om)) * -0.5);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 8.5e-109:
		tmp = 1.0 + ((((l * l) / Om) * ((ky * ky) / Om)) * -0.5)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 8.5e-109)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(l * l) / Om) * Float64(Float64(ky * ky) / Om)) * -0.5));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 8.5e-109)
		tmp = 1.0 + ((((l * l) / Om) * ((ky * ky) / Om)) * -0.5);
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 8.5e-109], N[(1.0 + N[(N[(N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.50000000000000005e-109

   1. Initial program 99.4%

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

      1. distribute-rgt-in 99.4%

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

      2. metadata-eval 99.4%

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

      3. metadata-eval 99.4%

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

      4. associate-/l* 99.4%

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

      5. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 96.8%

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

      1. *-commutative 96.8%

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

      2. associate-*r/ 96.8%

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

      3. associate-*r* 96.8%

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

   8. Simplified 96.8%

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

      1. pow-flip 96.8%

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

      2. metadata-eval 96.8%

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

   10. Applied egg-rr 96.8%

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

   11. Taylor expanded in ky around 0 41.7%

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

       1. *-commutative 41.7%

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

       2. unpow2 41.7%

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

       3. times-frac 44.4%

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

       4. unpow2 44.4%

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

       5. unpow2 44.4%

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

   13. Simplified 44.4%

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

   if 8.50000000000000005e-109 < l

   1. Initial program 98.9%

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

      1. distribute-rgt-in 98.9%

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

      2. metadata-eval 98.9%

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

      3. metadata-eval 98.9%

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

      4. associate-/l* 98.9%

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

      5. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in Om around 0 64.8%

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

      1. associate-*r* 64.8%

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

      2. *-commutative 64.8%

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

      3. associate-*r* 64.8%

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

      4. unpow2 64.8%

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

      5. unpow2 64.8%

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

      6. hypot-def 65.9%

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

   6. Simplified 65.9%

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

   7. Taylor expanded in l around inf 71.3%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-109}:\\ \;\;\;\;1 + \left(\frac{\ell \cdot \ell}{Om} \cdot \frac{ky \cdot ky}{Om}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8: 37.3% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. distribute-rgt-in 99.2%

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

   2. metadata-eval 99.2%

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

   3. metadata-eval 99.2%

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

   4. associate-/l* 99.2%

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

   5. metadata-eval 99.2%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. pow2 99.2%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 94.9%

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

   1. *-commutative 94.9%

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

   2. associate-*r/ 94.9%

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

   3. associate-*r* 94.9%

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

8. Simplified 94.9%

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

   1. pow-flip 94.9%

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

   2. metadata-eval 94.9%

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

10. Applied egg-rr 94.9%

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

11. Taylor expanded in ky around 0 33.4%

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

    1. *-commutative 33.4%

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

    2. unpow2 33.4%

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

    3. times-frac 35.6%

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

    4. unpow2 35.6%

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

    5. unpow2 35.6%

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

13. Simplified 35.6%

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

14. Final simplification 35.6%

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

Reproduce

herbie shell --seed 2023230 
(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))))))))))