Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 15.1s

Alternatives: 6

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

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

FPCore

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 99.7%

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

   5. unpow2 99.7%

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

   6. unpow2 99.7%

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

   7. +-commutative 99.7%

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

   8. unpow2 99.7%

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

   9. unpow2 99.7%

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

   10. hypot-def 100.0%

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

   11. *-commutative 100.0%

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

   12. associate-*l/ 100.0%

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

   13. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 87.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 6 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= ky 6e+71)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* ky l) Om))))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin kx) (* l (/ 2.0 Om)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 6e+71) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (l * (2.0 / Om)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 6e+71) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) * (l * (2.0 / Om)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ky <= 6e+71:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((ky * l) / Om))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.sin(kx) * (l * (2.0 / Om)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (ky <= 6e+71)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(ky * l) / Om))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) * Float64(l * Float64(2.0 / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (ky <= 6e+71)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (l * (2.0 / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[ky, 6e+71], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] * N[(l * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 6.00000000000000025e71

   1. Initial program 98.5%

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

      1. distribute-rgt-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. metadata-eval 98.5%

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

      4. associate-/l* 98.5%

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

      5. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in kx around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 77.1%

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

      4. unpow2 77.1%

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

      5. unpow2 77.1%

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

   6. Simplified 77.1%

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

      1. add-sqr-sqrt 77.1%

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

      2. hypot-1-def 77.1%

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

      3. sqrt-div 77.1%

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

      4. *-commutative 77.1%

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

      5. sqrt-prod 78.4%

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

      6. sqrt-prod 39.3%

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

      7. add-sqr-sqrt 84.9%

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

      8. *-commutative 84.9%

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

      9. sqrt-prod 84.9%

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

      10. unpow2 84.9%

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

      11. sqrt-prod 43.2%

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

      12. add-sqr-sqrt 91.4%

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

      13. metadata-eval 91.4%

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

      14. sqrt-prod 49.4%

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

      15. add-sqr-sqrt 93.2%

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

   8. Applied egg-rr 93.2%

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

      1. *-un-lft-identity 93.2%

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

      2. associate-*l/ 93.2%

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

      3. metadata-eval 93.2%

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

      4. associate-/l* 93.2%

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

   10. Applied egg-rr 93.2%

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

       1. *-lft-identity 93.2%

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

       2. associate-/r/ 93.2%

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

       3. *-commutative 93.2%

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

   12. Simplified 93.2%

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

   13. Taylor expanded in ky around 0 87.9%

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

   if 6.00000000000000025e71 < ky

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 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. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 100.0%

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

      5. unpow2 100.0%

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

      6. unpow2 100.0%

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

      7. +-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-*l/ 100.0%

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

      13. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in ky around 0 90.0%

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

      1. *-un-lft-identity 90.0%

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

      2. associate-*l/ 90.0%

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

      3. metadata-eval 90.0%

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

   10. Applied egg-rr 90.0%

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

       1. *-lft-identity 90.0%

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

   12. Simplified 90.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 6 \cdot 10^{+71}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}}\\ \end{array} \]

Alternative 3: 94.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in kx around 0 79.0%

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

   1. associate-*r/ 79.0%

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

   2. *-commutative 79.0%

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

   3. associate-*r* 79.0%

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

   4. unpow2 79.0%

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

   5. unpow2 79.0%

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

6. Simplified 79.0%

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

   1. add-sqr-sqrt 79.0%

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

   2. hypot-1-def 79.0%

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

   3. sqrt-div 79.0%

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

   4. *-commutative 79.0%

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

   5. sqrt-prod 80.1%

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

   6. sqrt-prod 41.0%

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

   7. add-sqr-sqrt 87.1%

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

   8. *-commutative 87.1%

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

   9. sqrt-prod 87.1%

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

   10. unpow2 87.1%

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

   11. sqrt-prod 46.6%

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

   12. add-sqr-sqrt 92.4%

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

   13. metadata-eval 92.4%

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

   14. sqrt-prod 49.1%

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

   15. add-sqr-sqrt 94.5%

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

8. Applied egg-rr 94.5%

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

   1. *-un-lft-identity 94.5%

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

   2. associate-*l/ 94.5%

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

   3. metadata-eval 94.5%

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

   4. associate-/l* 94.5%

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

10. Applied egg-rr 94.5%

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

    1. *-lft-identity 94.5%

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

    2. associate-/r/ 94.5%

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

    3. *-commutative 94.5%

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

12. Simplified 94.5%

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

13. Final simplification 94.5%

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

Alternative 4: 84.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 4000000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 4000000000000.0)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* ky l) Om))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 4000000000000.0) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 4000000000000.0) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 4000000000000.0:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((ky * l) / Om))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 4000000000000.0)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(ky * l) / Om))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 4000000000000.0)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 4000000000000.0], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 4e12

   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 kx around 0 79.2%

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

      1. associate-*r/ 79.2%

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

      2. *-commutative 79.2%

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

      3. associate-*r* 79.2%

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

      4. unpow2 79.2%

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

      5. unpow2 79.2%

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

   6. Simplified 79.2%

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

      1. add-sqr-sqrt 79.2%

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

      2. hypot-1-def 79.2%

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

      3. sqrt-div 79.2%

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

      4. *-commutative 79.2%

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

      5. sqrt-prod 80.7%

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

      6. sqrt-prod 41.4%

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

      7. add-sqr-sqrt 84.4%

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

      8. *-commutative 84.4%

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

      9. sqrt-prod 84.4%

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

      10. unpow2 84.4%

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

      11. sqrt-prod 47.1%

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

      12. add-sqr-sqrt 91.5%

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

      13. metadata-eval 91.5%

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

      14. sqrt-prod 32.2%

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

      15. add-sqr-sqrt 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. *-un-lft-identity 94.0%

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

      2. associate-*l/ 94.0%

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

      3. metadata-eval 94.0%

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

      4. associate-/l* 94.0%

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

   10. Applied egg-rr 94.0%

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

       1. *-lft-identity 94.0%

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

       2. associate-/r/ 94.0%

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

       3. *-commutative 94.0%

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

   12. Simplified 94.0%

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

   13. Taylor expanded in ky around 0 87.9%

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

   if 4e12 < Om

   1. Initial program 98.5%

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

      1. distribute-rgt-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. metadata-eval 98.5%

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

      4. associate-/l* 98.5%

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

      5. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in kx around 0 78.6%

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

      1. associate-*r/ 78.6%

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

      2. *-commutative 78.6%

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

      3. associate-*r* 78.6%

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

      4. unpow2 78.6%

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

      5. unpow2 78.6%

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

   6. Simplified 78.6%

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

      1. add-sqr-sqrt 78.6%

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

      2. hypot-1-def 78.6%

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

      3. sqrt-div 78.6%

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

      4. *-commutative 78.6%

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

      5. sqrt-prod 78.6%

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

      6. sqrt-prod 40.0%

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

      7. add-sqr-sqrt 94.8%

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

      8. *-commutative 94.8%

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

      9. sqrt-prod 94.8%

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

      10. unpow2 94.8%

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

      11. sqrt-prod 45.1%

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

      12. add-sqr-sqrt 94.8%

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

      13. metadata-eval 94.8%

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

      14. sqrt-prod 95.9%

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

      15. add-sqr-sqrt 95.9%

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

   8. Applied egg-rr 95.9%

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

      1. *-un-lft-identity 95.9%

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

      2. associate-*l/ 95.9%

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

      3. metadata-eval 95.9%

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

      4. associate-/l* 95.9%

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

   10. Applied egg-rr 95.9%

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

       1. *-lft-identity 95.9%

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

       2. associate-/r/ 95.9%

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

       3. *-commutative 95.9%

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

   12. Simplified 95.9%

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

   13. Taylor expanded in l around 0 85.9%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 4000000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 66.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.2 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-69}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.2e-155)
   (sqrt 0.5)
   (if (<= Om 1.15e-69) 1.0 (if (<= Om 7e-40) (sqrt 0.5) 1.0))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.2e-155) {
		tmp = sqrt(0.5);
	} else if (Om <= 1.15e-69) {
		tmp = 1.0;
	} else if (Om <= 7e-40) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 1.2d-155) then
        tmp = sqrt(0.5d0)
    else if (om <= 1.15d-69) then
        tmp = 1.0d0
    else if (om <= 7d-40) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.2e-155) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 1.15e-69) {
		tmp = 1.0;
	} else if (Om <= 7e-40) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.2e-155:
		tmp = math.sqrt(0.5)
	elif Om <= 1.15e-69:
		tmp = 1.0
	elif Om <= 7e-40:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.2e-155)
		tmp = sqrt(0.5);
	elseif (Om <= 1.15e-69)
		tmp = 1.0;
	elseif (Om <= 7e-40)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1.2e-155)
		tmp = sqrt(0.5);
	elseif (Om <= 1.15e-69)
		tmp = 1.0;
	elseif (Om <= 7e-40)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.2e-155], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 1.15e-69], 1.0, If[LessEqual[Om, 7e-40], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if Om < 1.2e-155 or 1.15e-69 < Om < 7.0000000000000003e-40

   1. Initial program 98.7%

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

      1. distribute-rgt-in 98.7%

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

      2. metadata-eval 98.7%

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

      3. metadata-eval 98.7%

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

      4. associate-/l* 98.7%

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

      5. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in Om around 0 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*r* 56.9%

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

      3. unpow2 56.9%

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

      4. unpow2 56.9%

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

      5. hypot-def 57.6%

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

      6. associate-*l/ 57.6%

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

      7. associate-*r/ 57.6%

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

   6. Simplified 57.6%

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

   7. Taylor expanded in l around inf 64.8%

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

   if 1.2e-155 < Om < 1.15e-69 or 7.0000000000000003e-40 < Om

   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 kx around 0 82.4%

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

      1. associate-*r/ 82.4%

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

      2. *-commutative 82.4%

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

      3. associate-*r* 82.4%

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

      4. unpow2 82.4%

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

      5. unpow2 82.4%

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

   6. Simplified 82.4%

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

      1. add-sqr-sqrt 82.4%

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

      2. hypot-1-def 82.4%

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

      3. sqrt-div 82.4%

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

      4. *-commutative 82.4%

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

      5. sqrt-prod 82.4%

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

      6. sqrt-prod 39.6%

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

      7. add-sqr-sqrt 94.5%

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

      8. *-commutative 94.5%

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

      9. sqrt-prod 94.5%

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

      10. unpow2 94.5%

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

      11. sqrt-prod 46.9%

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

      12. add-sqr-sqrt 95.6%

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

      13. metadata-eval 95.6%

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

      14. sqrt-prod 96.5%

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

      15. add-sqr-sqrt 96.5%

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

   8. Applied egg-rr 96.5%

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

      1. *-un-lft-identity 96.5%

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

      2. associate-*l/ 96.5%

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

      3. metadata-eval 96.5%

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

      4. associate-/l* 96.5%

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

   10. Applied egg-rr 96.5%

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

       1. *-lft-identity 96.5%

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

       2. associate-/r/ 96.5%

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

       3. *-commutative 96.5%

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

   12. Simplified 96.5%

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

   13. Taylor expanded in l around 0 83.8%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.2 \cdot 10^{-155}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.15 \cdot 10^{-69}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 7 \cdot 10^{-40}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 62.2% 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 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in kx around 0 79.0%

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

   1. associate-*r/ 79.0%

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

   2. *-commutative 79.0%

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

   3. associate-*r* 79.0%

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

   4. unpow2 79.0%

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

   5. unpow2 79.0%

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

6. Simplified 79.0%

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

   1. add-sqr-sqrt 79.0%

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

   2. hypot-1-def 79.0%

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

   3. sqrt-div 79.0%

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

   4. *-commutative 79.0%

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

   5. sqrt-prod 80.1%

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

   6. sqrt-prod 41.0%

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

   7. add-sqr-sqrt 87.1%

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

   8. *-commutative 87.1%

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

   9. sqrt-prod 87.1%

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

   10. unpow2 87.1%

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

   11. sqrt-prod 46.6%

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

   12. add-sqr-sqrt 92.4%

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

   13. metadata-eval 92.4%

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

   14. sqrt-prod 49.1%

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

   15. add-sqr-sqrt 94.5%

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

8. Applied egg-rr 94.5%

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

   1. *-un-lft-identity 94.5%

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

   2. associate-*l/ 94.5%

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

   3. metadata-eval 94.5%

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

   4. associate-/l* 94.5%

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

10. Applied egg-rr 94.5%

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

    1. *-lft-identity 94.5%

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

    2. associate-/r/ 94.5%

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

    3. *-commutative 94.5%

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

12. Simplified 94.5%

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

13. Taylor expanded in l around 0 64.8%

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

14. Final simplification 64.8%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023229 
(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))))))))))