Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 20.9s

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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

Java

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. expm1-log1p-u 97.3%

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

   2. expm1-udef 97.2%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
7. 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%

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

   4. hypot-def 98.3%

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

   5. unpow2 98.3%

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

   6. unpow2 98.3%

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

   7. associate-*l* 98.3%

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

   8. unpow2 98.3%

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

   9. unpow2 98.3%

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

   10. hypot-def 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

   3. associate-*l/ 99.3%

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

   4. metadata-eval 99.3%

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

10. Applied egg-rr 99.3%

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

    1. expm1-def 99.3%

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

    2. expm1-log1p 100.0%

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

    3. associate-*r* 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 2: 90.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 2.65 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin kx\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= kx 2.65e-151)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (/ (* 2.0 (* l ky)) Om)))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (/ l Om) (* 2.0 (sin kx)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 2.65e-151) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * (l * ky)) / Om)))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((l / Om) * (2.0 * sin(kx)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 2.65e-151) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 * (l * ky)) / Om)))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((l / Om) * (2.0 * Math.sin(kx)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if kx <= 2.65e-151:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 * (l * ky)) / Om)))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((l / Om) * (2.0 * math.sin(kx)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (kx <= 2.65e-151)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * Float64(l * ky)) / Om)))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(l / Om) * Float64(2.0 * sin(kx)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (kx <= 2.65e-151)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * (l * ky)) / Om)))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((l / Om) * (2.0 * sin(kx)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[kx, 2.65e-151], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[(l * ky), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(l / Om), $MachinePrecision] * N[(2.0 * N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 2.64999999999999989e-151

   1. Initial program 95.6%

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

   3. Simplified 95.6%

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

      1. expm1-log1p-u 95.6%

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

      2. expm1-udef 95.6%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   7. 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%

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

      4. hypot-def 97.2%

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

      5. unpow2 97.2%

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

      6. unpow2 97.2%

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

      7. associate-*l* 97.2%

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

      8. unpow2 97.2%

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

      9. unpow2 97.2%

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

      10. hypot-def 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.3%

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

      3. associate-*l/ 99.3%

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

      4. metadata-eval 99.3%

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

   10. Applied egg-rr 99.3%

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

       1. expm1-def 99.3%

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

       2. expm1-log1p 100.0%

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

       3. associate-*r* 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in kx around 0 95.6%

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

       1. associate-*r/ 95.6%

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

   15. Simplified 95.6%

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

   16. Taylor expanded in ky around 0 89.5%

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

       1. *-commutative 89.5%

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

   18. Simplified 89.5%

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

   if 2.64999999999999989e-151 < kx

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in ky around 0 86.4%

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

      1. associate-/l* 84.4%

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

      2. associate-/r/ 86.4%

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

      3. associate-*l* 86.4%

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

      4. metadata-eval 86.4%

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

      5. unpow2 86.4%

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

      6. unpow2 86.4%

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

      7. times-frac 96.8%

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

      8. swap-sqr 96.8%

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

      9. unpow2 96.8%

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

      10. swap-sqr 96.8%

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

      11. *-commutative 96.8%

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

      12. *-commutative 96.8%

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

      13. hypot-1-def 96.8%

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

      14. *-commutative 96.8%

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

   7. Simplified 96.8%

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

      1. *-commutative 96.8%

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

      2. div-inv 96.8%

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

      3. *-commutative 96.8%

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

      4. associate-*l* 96.8%

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

   9. Applied egg-rr 96.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 2.65 \cdot 10^{-151}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin kx\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. expm1-log1p-u 97.3%

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

   2. expm1-udef 97.2%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
7. 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%

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

   4. hypot-def 98.3%

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

   5. unpow2 98.3%

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

   6. unpow2 98.3%

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

   7. associate-*l* 98.3%

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

   8. unpow2 98.3%

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

   9. unpow2 98.3%

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

   10. hypot-def 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

   3. associate-*l/ 99.3%

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

   4. metadata-eval 99.3%

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

10. Applied egg-rr 99.3%

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

    1. expm1-def 99.3%

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

    2. expm1-log1p 100.0%

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

    3. associate-*r* 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in kx around 0 94.1%

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

    1. associate-*r/ 94.1%

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

15. Simplified 94.1%

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

16. Final simplification 94.1%

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

Alternative 4: 74.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 5.5e-129)
   1.0
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (/ (* 2.0 (* l ky)) Om)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5.5e-129) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * (l * ky)) / Om)))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5.5e-129) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 * (l * ky)) / Om)))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 5.5e-129:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 * (l * ky)) / Om)))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 5.5e-129)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * Float64(l * ky)) / Om)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 5.5e-129)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * (l * ky)) / Om)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 5.5e-129], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[(l * ky), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.50000000000000023e-129

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. expm1-log1p-u 97.6%

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

      2. expm1-udef 97.6%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   7. 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%

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

      4. hypot-def 97.9%

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

      5. unpow2 97.9%

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

      6. unpow2 97.9%

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

      7. associate-*l* 97.9%

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

      8. unpow2 97.9%

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

      9. unpow2 97.9%

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

      10. hypot-def 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 99.4%

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

      2. expm1-udef 99.4%

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

      3. associate-*l/ 99.4%

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

      4. metadata-eval 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. expm1-def 99.4%

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

       2. expm1-log1p 100.0%

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

       3. associate-*r* 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in l around 0 68.0%

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

   if 5.50000000000000023e-129 < l

   1. Initial program 96.6%

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

   3. Simplified 96.6%

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

      1. expm1-log1p-u 96.6%

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

      2. expm1-udef 96.6%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   7. 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%

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

      4. hypot-def 98.9%

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

      5. unpow2 98.9%

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

      6. unpow2 98.9%

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

      7. associate-*l* 98.9%

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

      8. unpow2 98.9%

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

      9. unpow2 98.9%

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

      10. hypot-def 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-udef 99.1%

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

      3. associate-*l/ 99.1%

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

      4. metadata-eval 99.1%

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

   10. Applied egg-rr 99.1%

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

       1. expm1-def 99.1%

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

       2. expm1-log1p 100.0%

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

       3. associate-*r* 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in kx around 0 93.1%

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

       1. associate-*r/ 93.1%

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

   15. Simplified 93.1%

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

   16. Taylor expanded in ky around 0 86.7%

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

       1. *-commutative 86.7%

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

   18. Simplified 86.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-129}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.3e-45)
   1.0
   (if (<= l 8.2e-17) (sqrt 0.5) (if (<= l 6e+79) 1.0 (sqrt 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.3e-45) {
		tmp = 1.0;
	} else if (l <= 8.2e-17) {
		tmp = sqrt(0.5);
	} else if (l <= 6e+79) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 1.3d-45) then
        tmp = 1.0d0
    else if (l <= 8.2d-17) then
        tmp = sqrt(0.5d0)
    else if (l <= 6d+79) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.3e-45) {
		tmp = 1.0;
	} else if (l <= 8.2e-17) {
		tmp = Math.sqrt(0.5);
	} else if (l <= 6e+79) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.3e-45:
		tmp = 1.0
	elif l <= 8.2e-17:
		tmp = math.sqrt(0.5)
	elif l <= 6e+79:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.3e-45)
		tmp = 1.0;
	elseif (l <= 8.2e-17)
		tmp = sqrt(0.5);
	elseif (l <= 6e+79)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.3e-45)
		tmp = 1.0;
	elseif (l <= 8.2e-17)
		tmp = sqrt(0.5);
	elseif (l <= 6e+79)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.3e-45], 1.0, If[LessEqual[l, 8.2e-17], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, 6e+79], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 1.29999999999999993e-45 or 8.2000000000000001e-17 < l < 5.99999999999999948e79

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

      1. expm1-log1p-u 97.1%

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

      2. expm1-udef 97.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
   7. 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%

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

      4. hypot-def 98.3%

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

      5. unpow2 98.3%

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

      6. unpow2 98.3%

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

      7. associate-*l* 98.3%

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

      8. unpow2 98.3%

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

      9. unpow2 98.3%

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

      10. hypot-def 100.0%

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

   8. Simplified 100.0%

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

      1. expm1-log1p-u 99.4%

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

      2. expm1-udef 99.4%

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

      3. associate-*l/ 99.4%

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

      4. metadata-eval 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. expm1-def 99.4%

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

       2. expm1-log1p 100.0%

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

       3. associate-*r* 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in l around 0 69.7%

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

   if 1.29999999999999993e-45 < l < 8.2000000000000001e-17 or 5.99999999999999948e79 < l

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in Om around 0 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

      3. associate-*l* 79.3%

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

      4. unpow2 79.3%

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

      5. unpow2 79.3%

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

      6. hypot-def 81.2%

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

   7. Simplified 81.2%

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

   8. Taylor expanded in l around inf 83.9%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.3 \cdot 10^{-45}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

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 97.3%

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

3. Simplified 97.3%

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

   1. expm1-log1p-u 97.3%

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

   2. expm1-udef 97.2%

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

6. Applied egg-rr 100.0%

   \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)} - 1}} \cdot 0.5} \]
7. 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%

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

   4. hypot-def 98.3%

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

   5. unpow2 98.3%

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

   6. unpow2 98.3%

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

   7. associate-*l* 98.3%

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

   8. unpow2 98.3%

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

   9. unpow2 98.3%

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

   10. hypot-def 100.0%

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

8. Simplified 100.0%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.3%

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

   3. associate-*l/ 99.3%

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

   4. metadata-eval 99.3%

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

10. Applied egg-rr 99.3%

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

    1. expm1-def 99.3%

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

    2. expm1-log1p 100.0%

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

    3. associate-*r* 100.0%

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

12. Simplified 100.0%

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

13. Taylor expanded in l around 0 62.2%

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

14. Final simplification 62.2%

    \[\leadsto 1 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))))))))))