Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 18.8s

Alternatives: 7

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. expm1-log1p-u 97.3%

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

   2. expm1-udef 97.3%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 93.3% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 l) Om)))
   (if (<= t_0 10.0)
     (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin kx) t_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 t_0 = (2.0 * l) / Om;
	double tmp;
	if (t_0 <= 10.0) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * t_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 t_0 = (2.0 * l) / Om;
	double tmp;
	if (t_0 <= 10.0) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) * t_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):
	t_0 = (2.0 * l) / Om
	tmp = 0
	if t_0 <= 10.0:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.sin(kx) * t_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)
	t_0 = Float64(Float64(2.0 * l) / Om)
	tmp = 0.0
	if (t_0 <= 10.0)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) * t_0)))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * l) * Float64(ky / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = (2.0 * l) / Om;
	tmp = 0.0;
	if (t_0 <= 10.0)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * t_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_] := Block[{t$95$0 = N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]}, If[LessEqual[t$95$0, 10.0], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] * t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * l), $MachinePrecision] * N[(ky / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 2 l) Om) < 10

   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%

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

      1. expm1-log1p-u 98.9%

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

      2. expm1-udef 98.9%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in ky around 0 96.1%

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

      1. expm1-log1p-u 96.1%

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

      2. expm1-udef 96.1%

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

      3. un-div-inv 96.1%

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

   10. Applied egg-rr 96.1%

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

       1. expm1-def 96.1%

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

       2. expm1-log1p 96.1%

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

       3. associate-*r/ 96.1%

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

   12. Simplified 96.1%

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

   if 10 < (/.f64 (*.f64 2 l) Om)

   1. Initial program 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in kx around 0 63.4%

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

      1. associate-*r/ 63.4%

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

      2. associate-*r* 63.4%

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

      3. unpow2 63.4%

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

      4. unpow2 63.4%

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

   6. Simplified 63.4%

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

      1. add-sqr-sqrt 63.4%

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

      2. hypot-1-def 63.4%

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

      3. sqrt-div 63.4%

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

      4. *-commutative 63.4%

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

      5. sqrt-prod 64.9%

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

      6. unpow2 64.9%

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

      7. sqrt-prod 34.9%

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

      8. add-sqr-sqrt 78.0%

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

      9. *-commutative 78.0%

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

      10. sqrt-prod 78.0%

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

      11. sqrt-prod 37.7%

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

      12. add-sqr-sqrt 86.5%

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

      13. metadata-eval 86.5%

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

      14. sqrt-prod 40.1%

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

      15. add-sqr-sqrt 94.1%

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

   8. Applied egg-rr 94.1%

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

      1. expm1-log1p-u 92.9%

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

      2. expm1-udef 92.8%

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

      3. un-div-inv 92.8%

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

      4. associate-/l* 92.8%

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

   10. Applied egg-rr 92.8%

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

       1. expm1-def 92.9%

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

       2. expm1-log1p 94.1%

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

       3. associate-/r/ 94.1%

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

       4. *-commutative 94.1%

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

   12. Simplified 94.1%

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

   13. Taylor expanded in ky around 0 92.1%

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

4. Final simplification 95.0%

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

Alternative 3: 93.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in kx around 0 77.3%

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

   1. associate-*r/ 77.3%

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

   2. associate-*r* 77.3%

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

   3. unpow2 77.3%

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

   4. unpow2 77.3%

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

6. Simplified 77.3%

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

   1. add-sqr-sqrt 77.3%

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

   2. hypot-1-def 77.3%

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

   3. sqrt-div 77.3%

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

   4. *-commutative 77.3%

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

   5. sqrt-prod 77.7%

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

   6. unpow2 77.7%

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

   7. sqrt-prod 42.2%

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

   8. add-sqr-sqrt 83.5%

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

   9. *-commutative 83.5%

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

   10. sqrt-prod 83.5%

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

   11. sqrt-prod 37.6%

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

   12. add-sqr-sqrt 89.6%

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

   13. metadata-eval 89.6%

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

   14. sqrt-prod 46.3%

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

   15. add-sqr-sqrt 96.0%

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

8. Applied egg-rr 96.0%

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

   1. expm1-log1p-u 95.3%

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

   2. expm1-udef 95.3%

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

   3. un-div-inv 95.3%

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

   4. associate-/l* 95.3%

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

10. Applied egg-rr 95.3%

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

    1. expm1-def 95.3%

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

    2. expm1-log1p 96.0%

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

    3. associate-/r/ 96.0%

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

    4. *-commutative 96.0%

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

12. Simplified 96.0%

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

13. Final simplification 96.0%

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

Alternative 4: 85.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 5.5 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \ell\right) \cdot \frac{ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 5.5e+96)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (* 2.0 l) (/ ky Om))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5.5e+96) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * l) * (ky / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 5.5e+96) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((2.0 * l) * (ky / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 5.5e+96:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((2.0 * l) * (ky / Om))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 5.5e+96)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(2.0 * l) * Float64(ky / Om))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 5.5e+96)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((2.0 * l) * (ky / Om))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 5.5e+96], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * l), $MachinePrecision] * N[(ky / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 5.5000000000000002e96

   1. Initial program 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in kx around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. associate-*r* 76.5%

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

      3. unpow2 76.5%

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

      4. unpow2 76.5%

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

   6. Simplified 76.5%

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

      1. add-sqr-sqrt 76.5%

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

      2. hypot-1-def 76.5%

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

      3. sqrt-div 76.5%

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

      4. *-commutative 76.5%

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

      5. sqrt-prod 77.0%

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

      6. unpow2 77.0%

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

      7. sqrt-prod 42.5%

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

      8. add-sqr-sqrt 83.5%

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

      9. *-commutative 83.5%

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

      10. sqrt-prod 83.5%

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

      11. sqrt-prod 35.8%

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

      12. add-sqr-sqrt 88.9%

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

      13. metadata-eval 88.9%

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

      14. sqrt-prod 39.2%

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

      15. add-sqr-sqrt 95.4%

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

   8. Applied egg-rr 95.4%

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

      1. expm1-log1p-u 94.7%

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

      2. expm1-udef 94.7%

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

      3. un-div-inv 94.7%

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

      4. associate-/l* 94.7%

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

   10. Applied egg-rr 94.7%

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

       1. expm1-def 94.7%

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

       2. expm1-log1p 95.4%

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

       3. associate-/r/ 95.4%

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

       4. *-commutative 95.4%

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

   12. Simplified 95.4%

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

   13. Taylor expanded in ky around 0 87.7%

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

   if 5.5000000000000002e96 < 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%

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

   4. Taylor expanded in kx around 0 83.3%

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

      1. associate-*r/ 83.3%

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

      2. associate-*r* 83.3%

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

      3. unpow2 83.3%

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

      4. unpow2 83.3%

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

   6. Simplified 83.3%

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

      1. add-sqr-sqrt 83.3%

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

      2. hypot-1-def 83.3%

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

      3. sqrt-div 83.3%

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

      4. *-commutative 83.3%

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

      5. sqrt-prod 83.3%

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

      6. unpow2 83.3%

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

      7. sqrt-prod 40.0%

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

      8. add-sqr-sqrt 83.3%

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

      9. *-commutative 83.3%

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

      10. sqrt-prod 83.3%

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

      11. sqrt-prod 51.3%

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

      12. add-sqr-sqrt 94.7%

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

      13. metadata-eval 94.7%

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

      14. sqrt-prod 100.0%

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

      15. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 99.9%

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

      2. expm1-udef 99.9%

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

      3. un-div-inv 99.9%

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

      4. associate-/l* 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. expm1-def 99.9%

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

       2. expm1-log1p 100.0%

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

       3. associate-/r/ 100.0%

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

       4. *-commutative 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in ky around 0 94.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 5.5 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \ell\right) \cdot \frac{ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 71.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.6e-23)
   1.0
   (if (<= l 1.06e+127)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* ky ky) (/ (* Om Om) (* l l)))))))))
     (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.6e-23) {
		tmp = 1.0;
	} else if (l <= 1.06e+127) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	} 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 <= 1.6d-23) then
        tmp = 1.0d0
    else if (l <= 1.06d+127) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((ky * ky) / ((om * om) / (l * l)))))))))
    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 <= 1.6e-23) {
		tmp = 1.0;
	} else if (l <= 1.06e+127) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.6e-23:
		tmp = 1.0
	elif l <= 1.06e+127:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.6e-23)
		tmp = 1.0;
	elseif (l <= 1.06e+127)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * ky) / Float64(Float64(Om * Om) / Float64(l * l)))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.6e-23)
		tmp = 1.0;
	elseif (l <= 1.06e+127)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.6e-23], 1.0, If[LessEqual[l, 1.06e+127], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(ky * ky), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.59999999999999988e-23

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in kx around 0 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-*r* 77.3%

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

      3. unpow2 77.3%

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

      4. unpow2 77.3%

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

   6. Simplified 77.3%

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

      1. add-sqr-sqrt 77.3%

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

      2. hypot-1-def 77.3%

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

      3. sqrt-div 77.3%

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

      4. *-commutative 77.3%

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

      5. sqrt-prod 77.8%

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

      6. unpow2 77.8%

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

      7. sqrt-prod 40.8%

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

      8. add-sqr-sqrt 82.5%

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

      9. *-commutative 82.5%

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

      10. sqrt-prod 82.5%

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

      11. sqrt-prod 20.1%

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

      12. add-sqr-sqrt 89.1%

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

      13. metadata-eval 89.1%

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

      14. sqrt-prod 47.4%

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

      15. add-sqr-sqrt 97.3%

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

   8. Applied egg-rr 97.3%

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

      1. expm1-log1p-u 96.7%

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

      2. expm1-udef 96.7%

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

      3. un-div-inv 96.7%

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

      4. associate-/l* 96.7%

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

   10. Applied egg-rr 96.7%

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

       1. expm1-def 96.7%

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

       2. expm1-log1p 97.3%

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

       3. associate-/r/ 97.3%

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

       4. *-commutative 97.3%

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

   12. Simplified 97.3%

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

   13. Taylor expanded in ky around 0 65.6%

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

   if 1.59999999999999988e-23 < l < 1.06000000000000006e127

   1. Initial program 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in kx around 0 77.0%

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

      1. associate-*r/ 77.0%

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

      2. associate-*r* 77.0%

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

      3. unpow2 77.0%

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

      4. unpow2 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in ky around 0 68.2%

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

      1. associate-/l* 67.4%

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

      2. unpow2 67.4%

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

      3. unpow2 67.4%

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

      4. unpow2 67.4%

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

   9. Simplified 67.4%

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

   if 1.06000000000000006e127 < 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%

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

   4. Taylor expanded in Om around 0 85.6%

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

      1. associate-*r* 85.6%

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

      2. unpow2 85.6%

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

      3. unpow2 85.6%

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

      4. hypot-def 85.6%

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

      5. *-commutative 85.6%

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

   6. Simplified 85.6%

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

   7. Taylor expanded in l around inf 87.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 70.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 24500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 24500000.0) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 24500000.0) {
		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 <= 24500000.0d0) 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 <= 24500000.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 24500000.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 24500000.0)
		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 <= 24500000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 24500000.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.45e7

   1. Initial program 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in kx around 0 77.5%

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

      1. associate-*r/ 77.5%

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

      2. associate-*r* 77.5%

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

      3. unpow2 77.5%

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

      4. unpow2 77.5%

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

   6. Simplified 77.5%

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

      1. add-sqr-sqrt 77.5%

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

      2. hypot-1-def 77.5%

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

      3. sqrt-div 77.5%

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

      4. *-commutative 77.5%

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

      5. sqrt-prod 78.0%

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

      6. unpow2 78.0%

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

      7. sqrt-prod 40.8%

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

      8. add-sqr-sqrt 82.5%

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

      9. *-commutative 82.5%

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

      10. sqrt-prod 82.5%

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

      11. sqrt-prod 21.7%

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

      12. add-sqr-sqrt 89.0%

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

      13. metadata-eval 89.0%

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

      14. sqrt-prod 46.7%

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

      15. add-sqr-sqrt 97.0%

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

   8. Applied egg-rr 97.0%

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

      1. expm1-log1p-u 96.4%

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

      2. expm1-udef 96.4%

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

      3. un-div-inv 96.4%

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

      4. associate-/l* 96.4%

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

   10. Applied egg-rr 96.4%

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

       1. expm1-def 96.4%

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

       2. expm1-log1p 97.0%

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

       3. associate-/r/ 97.0%

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

       4. *-commutative 97.0%

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

   12. Simplified 97.0%

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

   13. Taylor expanded in ky around 0 66.0%

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

   if 2.45e7 < 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%

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

   4. Taylor expanded in Om around 0 73.8%

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

      1. associate-*r* 73.8%

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

      2. unpow2 73.8%

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

      3. unpow2 73.8%

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

      4. hypot-def 75.9%

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

      5. *-commutative 75.9%

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

   6. Simplified 75.9%

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

   7. Taylor expanded in l around inf 77.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 24500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 62.0% 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%

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

4. Taylor expanded in kx around 0 77.3%

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

   1. associate-*r/ 77.3%

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

   2. associate-*r* 77.3%

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

   3. unpow2 77.3%

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

   4. unpow2 77.3%

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

6. Simplified 77.3%

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

   1. add-sqr-sqrt 77.3%

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

   2. hypot-1-def 77.3%

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

   3. sqrt-div 77.3%

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

   4. *-commutative 77.3%

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

   5. sqrt-prod 77.7%

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

   6. unpow2 77.7%

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

   7. sqrt-prod 42.2%

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

   8. add-sqr-sqrt 83.5%

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

   9. *-commutative 83.5%

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

   10. sqrt-prod 83.5%

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

   11. sqrt-prod 37.6%

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

   12. add-sqr-sqrt 89.6%

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

   13. metadata-eval 89.6%

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

   14. sqrt-prod 46.3%

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

   15. add-sqr-sqrt 96.0%

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

8. Applied egg-rr 96.0%

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

   1. expm1-log1p-u 95.3%

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

   2. expm1-udef 95.3%

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

   3. un-div-inv 95.3%

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

   4. associate-/l* 95.3%

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

10. Applied egg-rr 95.3%

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

    1. expm1-def 95.3%

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

    2. expm1-log1p 96.0%

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

    3. associate-/r/ 96.0%

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

    4. *-commutative 96.0%

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

12. Simplified 96.0%

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

13. Taylor expanded in ky around 0 59.7%

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

14. Final simplification 59.7%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023287 
(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))))))))))