Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 23.0s

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

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. log1p-define 99.2%

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

   3. add-log-exp 98.6%

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

   4. log1p-define 98.6%

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

   5. expm1-log1p-u 98.6%

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

   6. add-sqr-sqrt 98.6%

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

   7. pow2 98.6%

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

6. Applied egg-rr 99.4%

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

   1. un-div-inv 99.4%

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

   2. metadata-eval 99.4%

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

   3. rem-log-exp 100.0%

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

   4. unpow2 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-/r/ 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/r/ 100.0%

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

   9. hypot-undefine 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

10. Simplified 100.0%

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

Alternative 2: 87.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\ell \cdot 2}{Om}\\ \mathbf{if}\;ky \leq 10^{+42}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, t\_0 \cdot ky\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, t\_0 \cdot \sin kx\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (/ (* l 2.0) Om)))
   (if (<= ky 1e+42)
     (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* t_0 ky)))))
     (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* t_0 (sin kx)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = (l * 2.0) / Om;
	double tmp;
	if (ky <= 1e+42) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (t_0 * ky)))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (t_0 * sin(kx))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = (l * 2.0) / Om;
	double tmp;
	if (ky <= 1e+42) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (t_0 * ky)))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (t_0 * Math.sin(kx))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = (l * 2.0) / Om
	tmp = 0
	if ky <= 1e+42:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (t_0 * ky)))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (t_0 * math.sin(kx))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = Float64(Float64(l * 2.0) / Om)
	tmp = 0.0
	if (ky <= 1e+42)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(t_0 * ky)))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(t_0 * sin(kx))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = (l * 2.0) / Om;
	tmp = 0.0;
	if (ky <= 1e+42)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (t_0 * ky)))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (t_0 * sin(kx))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[ky, 1e+42], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(t$95$0 * ky), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(t$95$0 * N[Sin[kx], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.00000000000000004e42

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. expm1-log1p-u 99.0%

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

      2. log1p-define 99.0%

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

      3. add-log-exp 98.5%

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

      4. log1p-define 98.5%

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

      5. expm1-log1p-u 98.5%

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

      6. add-sqr-sqrt 98.5%

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

      7. pow2 98.5%

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

   6. Applied egg-rr 99.5%

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

      1. un-div-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. rem-log-exp 100.0%

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

      4. unpow2 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/r/ 100.0%

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

      9. hypot-undefine 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in kx around 0 93.5%

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

   12. Taylor expanded in ky around 0 83.2%

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

   if 1.00000000000000004e42 < ky

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. log1p-define 100.0%

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

      3. add-log-exp 99.0%

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

      4. log1p-define 99.0%

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

      5. expm1-log1p-u 99.0%

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

      6. add-sqr-sqrt 99.0%

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

      7. pow2 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. un-div-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. rem-log-exp 100.0%

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

      4. unpow2 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/r/ 100.0%

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

      9. hypot-undefine 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in ky around 0 95.9%

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

Alternative 3: 93.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. log1p-define 99.2%

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

   3. add-log-exp 98.6%

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

   4. log1p-define 98.6%

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

   5. expm1-log1p-u 98.6%

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

   6. add-sqr-sqrt 98.6%

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

   7. pow2 98.6%

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

6. Applied egg-rr 99.4%

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

   1. un-div-inv 99.4%

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

   2. metadata-eval 99.4%

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

   3. rem-log-exp 100.0%

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

   4. unpow2 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-/r/ 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/r/ 100.0%

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

   9. hypot-undefine 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in kx around 0 95.0%

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

Alternative 4: 74.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-88}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell \cdot 2}{Om} \cdot ky\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 4e-88)
   1.0
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (/ (* l 2.0) Om) ky)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 4e-88) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (((l * 2.0) / Om) * ky)))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 4e-88) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (((l * 2.0) / Om) * ky)))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 4e-88:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (((l * 2.0) / Om) * ky)))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 4e-88)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(Float64(l * 2.0) / Om) * ky)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 4e-88)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (((l * 2.0) / Om) * ky)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 4e-88], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision] * ky), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.99999999999999974e-88

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. expm1-log1p-u 98.9%

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

      2. log1p-define 98.9%

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

      3. add-log-exp 98.1%

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

      4. log1p-define 98.1%

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

      5. expm1-log1p-u 98.1%

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

      6. add-sqr-sqrt 98.1%

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

      7. pow2 98.1%

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

   6. Applied egg-rr 99.2%

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

      1. un-div-inv 99.2%

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

      2. metadata-eval 99.2%

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

      3. rem-log-exp 100.0%

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

      4. unpow2 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/r/ 100.0%

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

      9. hypot-undefine 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in l around 0 71.9%

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

   if 3.99999999999999974e-88 < l

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. log1p-define 100.0%

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

      3. add-log-exp 100.0%

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

      4. log1p-define 100.0%

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

      5. expm1-log1p-u 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. un-div-inv 100.0%

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

      2. metadata-eval 100.0%

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

      3. rem-log-exp 100.0%

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

      4. unpow2 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/r/ 100.0%

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

      9. hypot-undefine 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in kx around 0 96.4%

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

   12. Taylor expanded in ky around 0 89.6%

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

Alternative 5: 67.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.1 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 1.1e-32) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.1e-32) {
		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 <= 1.1d-32) 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 <= 1.1e-32) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.1e-32:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.1e-32)
		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 <= 1.1e-32)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.1e-32], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.1e-32

   1. Initial program 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in l around inf 51.0%

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

      1. unpow2 51.0%

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

      2. unpow2 51.0%

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

      3. hypot-undefine 52.2%

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

   7. Simplified 52.2%

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

   8. Taylor expanded in l around inf 59.6%

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

   if 1.1e-32 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. log1p-define 100.0%

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

      3. add-log-exp 99.9%

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

      4. log1p-define 99.9%

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

      5. expm1-log1p-u 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. un-div-inv 99.9%

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

      2. metadata-eval 99.9%

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

      3. rem-log-exp 100.0%

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

      4. unpow2 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/r/ 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/r/ 100.0%

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

      9. hypot-undefine 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in l around 0 86.2%

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

Alternative 6: 62.4% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. log1p-define 99.2%

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

   3. add-log-exp 98.6%

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

   4. log1p-define 98.6%

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

   5. expm1-log1p-u 98.6%

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

   6. add-sqr-sqrt 98.6%

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

   7. pow2 98.6%

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

6. Applied egg-rr 99.4%

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

   1. un-div-inv 99.4%

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

   2. metadata-eval 99.4%

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

   3. rem-log-exp 100.0%

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

   4. unpow2 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-/r/ 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/r/ 100.0%

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

   9. hypot-undefine 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*r/ 100.0%

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

   3. *-commutative 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in l around 0 68.1%

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

Reproduce

herbie shell --seed 2024114 
(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))))))))))