Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 19.0s

Alternatives: 9

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow2 98.8%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 87.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin ky}^{2}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-324}:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \frac{12 \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \left(\left(0.5 - \frac{\cos \left(ky + ky\right)}{2}\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (pow (sin ky) 2.0)))
   (if (<= t_0 5e-324)
     1.0
     (if (<= t_0 2e-33)
       (sqrt
        (+ 0.5 (* 0.5 (/ 1.0 (sqrt (+ 1.0 (/ (* 12.0 (* l l)) (* Om Om))))))))
       (sqrt
        (+
         0.5
         (*
          0.5
          (/
           1.0
           (+
            1.0
            (*
             2.0
             (*
              (- 0.5 (/ (cos (+ ky ky)) 2.0))
              (* (/ l Om) (/ l Om)))))))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = pow(sin(ky), 2.0);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = 1.0;
	} else if (t_0 <= 2e-33) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + ((12.0 * (l * l)) / (Om * Om))))))));
	} else {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((0.5 - (cos((ky + ky)) / 2.0)) * ((l / Om) * (l / Om)))))))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sin(ky) ** 2.0d0
    if (t_0 <= 5d-324) then
        tmp = 1.0d0
    else if (t_0 <= 2d-33) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + ((12.0d0 * (l * l)) / (om * om))))))))
    else
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((0.5d0 - (cos((ky + ky)) / 2.0d0)) * ((l / om) * (l / om)))))))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = Math.pow(Math.sin(ky), 2.0);
	double tmp;
	if (t_0 <= 5e-324) {
		tmp = 1.0;
	} else if (t_0 <= 2e-33) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.sqrt((1.0 + ((12.0 * (l * l)) / (Om * Om))))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((0.5 - (Math.cos((ky + ky)) / 2.0)) * ((l / Om) * (l / Om)))))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = math.pow(math.sin(ky), 2.0)
	tmp = 0
	if t_0 <= 5e-324:
		tmp = 1.0
	elif t_0 <= 2e-33:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + ((12.0 * (l * l)) / (Om * Om))))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((0.5 - (math.cos((ky + ky)) / 2.0)) * ((l / Om) * (l / Om)))))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = sin(ky) ^ 2.0
	tmp = 0.0
	if (t_0 <= 5e-324)
		tmp = 1.0;
	elseif (t_0 <= 2e-33)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(12.0 * Float64(l * l)) / Float64(Om * Om))))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(0.5 - Float64(cos(Float64(ky + ky)) / 2.0)) * Float64(Float64(l / Om) * Float64(l / Om)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = sin(ky) ^ 2.0;
	tmp = 0.0;
	if (t_0 <= 5e-324)
		tmp = 1.0;
	elseif (t_0 <= 2e-33)
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + ((12.0 * (l * l)) / (Om * Om))))))));
	else
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((0.5 - (cos((ky + ky)) / 2.0)) * ((l / Om) * (l / Om)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, 5e-324], 1.0, If[LessEqual[t$95$0, 2e-33], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(12.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 (sin.f64 ky) 2) < 4.94066e-324

   1. Initial program 95.0%

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

      1. distribute-rgt-in 95.0%

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

      2. metadata-eval 95.0%

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

      3. metadata-eval 95.0%

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

      4. associate-/l* 95.0%

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

      5. metadata-eval 95.0%

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

   3. Simplified 95.0%

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

      1. add-sqr-sqrt 95.0%

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

      2. pow2 95.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 84.0%

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

      1. expm1-log1p-u 84.0%

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

      2. expm1-udef 84.0%

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

      3. +-commutative 84.0%

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

      4. fma-def 84.0%

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

      5. pow-pow 84.0%

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

      6. metadata-eval 84.0%

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

      7. unpow-1 84.0%

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

   8. Applied egg-rr 84.0%

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

      1. expm1-def 84.0%

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

      2. expm1-log1p 84.0%

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

      3. fma-udef 84.0%

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

      4. associate-*l/ 84.0%

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

      5. metadata-eval 84.0%

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

      6. associate-*l* 84.0%

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

      7. *-commutative 84.0%

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

   10. Simplified 84.0%

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

   11. Taylor expanded in ky around 0 68.0%

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

   if 4.94066e-324 < (pow.f64 (sin.f64 ky) 2) < 2.0000000000000001e-33

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 90.3%

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

      1. associate-/l* 93.3%

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

      2. unpow2 93.3%

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

      3. unpow2 93.3%

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

   6. Simplified 93.3%

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

   7. Taylor expanded in ky around 0 93.3%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\color{blue}{0.3333333333333333 \cdot {Om}^{2} + \frac{{Om}^{2}}{{ky}^{2}}}}}} \cdot 0.5} \]
   8. Step-by-step derivation

      1. fma-def 93.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, {Om}^{2}, \frac{{Om}^{2}}{{ky}^{2}}\right)}}}} \cdot 0.5} \]

      2. unpow2 93.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(0.3333333333333333, \color{blue}{Om \cdot Om}, \frac{{Om}^{2}}{{ky}^{2}}\right)}}} \cdot 0.5} \]

      3. unpow2 93.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(0.3333333333333333, Om \cdot Om, \frac{\color{blue}{Om \cdot Om}}{{ky}^{2}}\right)}}} \cdot 0.5} \]

      4. unpow2 93.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(0.3333333333333333, Om \cdot Om, \frac{Om \cdot Om}{\color{blue}{ky \cdot ky}}\right)}}} \cdot 0.5} \]

   9. Simplified 93.3%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, Om \cdot Om, \frac{Om \cdot Om}{ky \cdot ky}\right)}}}} \cdot 0.5} \]

   10. Taylor expanded in ky around inf 96.9%

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

       1. associate-*r/ 96.9%

          \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\frac{12 \cdot {\ell}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]

       2. unpow2 96.9%

          \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \frac{12 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{Om}^{2}}}} \cdot 0.5} \]

       3. unpow2 96.9%

          \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \frac{12 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{Om \cdot Om}}}} \cdot 0.5} \]

   12. Simplified 96.9%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{1 + \frac{12 \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}} \cdot 0.5} \]

   if 2.0000000000000001e-33 < (pow.f64 (sin.f64 ky) 2)

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 85.7%

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

      1. associate-/l* 85.7%

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

      2. unpow2 85.7%

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

      3. unpow2 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in l around 0 85.4%

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

      1. associate-*l/ 85.4%

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

      2. unpow2 85.4%

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

      3. unpow2 85.4%

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

      4. *-commutative 85.4%

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

      5. times-frac 97.9%

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

   9. Simplified 97.9%

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

       1. unpow2 97.9%

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

       2. sin-mult 97.0%

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

   11. Applied egg-rr 97.0%

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

       1. div-sub 97.0%

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

       2. +-inverses 97.0%

          \[\leadsto \sqrt{0.5 + \frac{1}{1 + 2 \cdot \left(\left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(ky + ky\right)}{2}\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)} \cdot 0.5} \]

       3. cos-0 97.0%

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

       4. metadata-eval 97.0%

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

   13. Simplified 97.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\sin ky}^{2} \leq 5 \cdot 10^{-324}:\\ \;\;\;\;1\\ \mathbf{elif}\;{\sin ky}^{2} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \frac{12 \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \left(\left(0.5 - \frac{\cos \left(ky + ky\right)}{2}\right) \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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, ((2.0 * (l / Om)) * Math.hypot(Math.sin(ky), Math.sin(kx))))))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 99.2%

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

   5. unpow2 99.2%

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

   6. unpow2 99.2%

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

   7. +-commutative 99.2%

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

   8. unpow2 99.2%

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

   9. unpow2 99.2%

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

   10. hypot-def 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 4: 93.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow2 98.8%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 94.2%

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

7. Final simplification 94.2%

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

Alternative 5: 93.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow2 98.8%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 94.2%

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

   1. expm1-log1p-u 94.2%

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

   2. expm1-udef 94.2%

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

   3. +-commutative 94.2%

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

   4. fma-def 94.2%

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

   5. pow-pow 94.2%

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

   6. metadata-eval 94.2%

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

   7. unpow-1 94.2%

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

8. Applied egg-rr 94.2%

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

   1. expm1-def 94.2%

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

   2. expm1-log1p 94.2%

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

   3. fma-udef 94.2%

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

   4. associate-*l/ 94.2%

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

   5. metadata-eval 94.2%

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

   6. associate-*l* 94.2%

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

   7. *-commutative 94.2%

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

10. Simplified 94.2%

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

11. Final simplification 94.2%

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

Alternative 6: 76.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{-124}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \frac{12 \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 5.6e-124)
   1.0
   (if (<= l 6.2e+152)
     (sqrt
      (+ 0.5 (* 0.5 (/ 1.0 (sqrt (+ 1.0 (/ (* 12.0 (* l l)) (* Om Om))))))))
     (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5.6e-124) {
		tmp = 1.0;
	} else if (l <= 6.2e+152) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + ((12.0 * (l * l)) / (Om * Om))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 5.6d-124) then
        tmp = 1.0d0
    else if (l <= 6.2d+152) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + ((12.0d0 * (l * l)) / (om * om))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5.6e-124) {
		tmp = 1.0;
	} else if (l <= 6.2e+152) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.sqrt((1.0 + ((12.0 * (l * l)) / (Om * Om))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 5.6e-124:
		tmp = 1.0
	elif l <= 6.2e+152:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + ((12.0 * (l * l)) / (Om * Om))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 5.6e-124)
		tmp = 1.0;
	elseif (l <= 6.2e+152)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(12.0 * Float64(l * l)) / Float64(Om * Om))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 5.6e-124)
		tmp = 1.0;
	elseif (l <= 6.2e+152)
		tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + ((12.0 * (l * l)) / (Om * Om))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 5.6e-124], 1.0, If[LessEqual[l, 6.2e+152], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(12.0 * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 5.59999999999999996e-124

   1. Initial program 98.8%

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

      1. distribute-rgt-in 98.8%

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

      2. metadata-eval 98.8%

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

      3. metadata-eval 98.8%

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

      4. associate-/l* 98.8%

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

      5. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. add-sqr-sqrt 98.8%

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

      2. pow2 98.8%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 95.7%

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

      1. expm1-log1p-u 95.7%

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

      2. expm1-udef 95.7%

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

      3. +-commutative 95.7%

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

      4. fma-def 95.7%

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

      5. pow-pow 95.7%

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

      6. metadata-eval 95.7%

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

      7. unpow-1 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. expm1-def 95.7%

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

      2. expm1-log1p 95.7%

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

      3. fma-udef 95.7%

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

      4. associate-*l/ 95.7%

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

      5. metadata-eval 95.7%

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

      6. associate-*l* 95.7%

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

      7. *-commutative 95.7%

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

   10. Simplified 95.7%

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

   11. Taylor expanded in ky around 0 72.7%

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

   if 5.59999999999999996e-124 < l < 6.2e152

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 88.3%

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

      1. associate-/l* 88.3%

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

      2. unpow2 88.3%

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

      3. unpow2 88.3%

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

   6. Simplified 88.3%

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

   7. Taylor expanded in ky around 0 80.3%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\color{blue}{0.3333333333333333 \cdot {Om}^{2} + \frac{{Om}^{2}}{{ky}^{2}}}}}} \cdot 0.5} \]
   8. Step-by-step derivation

      1. fma-def 80.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, {Om}^{2}, \frac{{Om}^{2}}{{ky}^{2}}\right)}}}} \cdot 0.5} \]

      2. unpow2 80.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(0.3333333333333333, \color{blue}{Om \cdot Om}, \frac{{Om}^{2}}{{ky}^{2}}\right)}}} \cdot 0.5} \]

      3. unpow2 80.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(0.3333333333333333, Om \cdot Om, \frac{\color{blue}{Om \cdot Om}}{{ky}^{2}}\right)}}} \cdot 0.5} \]

      4. unpow2 80.3%

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(0.3333333333333333, Om \cdot Om, \frac{Om \cdot Om}{\color{blue}{ky \cdot ky}}\right)}}} \cdot 0.5} \]

   9. Simplified 80.3%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + 4 \cdot \frac{\ell \cdot \ell}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, Om \cdot Om, \frac{Om \cdot Om}{ky \cdot ky}\right)}}}} \cdot 0.5} \]

   10. Taylor expanded in ky around inf 95.8%

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

       1. associate-*r/ 95.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \color{blue}{\frac{12 \cdot {\ell}^{2}}{{Om}^{2}}}}} \cdot 0.5} \]

       2. unpow2 95.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \frac{12 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{Om}^{2}}}} \cdot 0.5} \]

       3. unpow2 95.8%

          \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{1 + \frac{12 \cdot \left(\ell \cdot \ell\right)}{\color{blue}{Om \cdot Om}}}} \cdot 0.5} \]

   12. Simplified 95.8%

       \[\leadsto \sqrt{0.5 + \frac{1}{\color{blue}{\sqrt{1 + \frac{12 \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}} \cdot 0.5} \]

   if 6.2e152 < l

   1. Initial program 96.4%

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

      1. distribute-rgt-in 96.4%

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

      2. metadata-eval 96.4%

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

      3. metadata-eval 96.4%

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

      4. associate-/l* 96.4%

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

      5. metadata-eval 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in Om around 0 87.6%

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

      1. *-commutative 87.6%

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

      2. associate-*r* 87.6%

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

      3. associate-*l/ 87.6%

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

      4. unpow2 87.6%

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

      5. unpow2 87.6%

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

      6. hypot-def 91.2%

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

      7. associate-*l/ 91.2%

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

      8. *-commutative 91.2%

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

   6. Simplified 91.2%

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

   7. Taylor expanded in l around inf 91.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.6 \cdot 10^{-124}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + \frac{12 \cdot \left(\ell \cdot \ell\right)}{Om \cdot Om}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 62.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 3.1 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om}{ky} \cdot \frac{Om}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 3.1e+56)
   (sqrt 0.5)
   (+ 1.0 (* -0.5 (/ (* l l) (* (/ Om ky) (/ Om ky)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 3.1e+56) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * l) / ((Om / ky) * (Om / ky))));
	}
	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 <= 3.1d+56) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0 + ((-0.5d0) * ((l * l) / ((om / ky) * (om / ky))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 3.1e+56) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0 + (-0.5 * ((l * l) / ((Om / ky) * (Om / ky))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 3.1e+56:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0 + (-0.5 * ((l * l) / ((Om / ky) * (Om / ky))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 3.1e+56)
		tmp = sqrt(0.5);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(l * l) / Float64(Float64(Om / ky) * Float64(Om / ky)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 3.1e+56)
		tmp = sqrt(0.5);
	else
		tmp = 1.0 + (-0.5 * ((l * l) / ((Om / ky) * (Om / ky))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 3.1e+56], N[Sqrt[0.5], $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om / ky), $MachinePrecision] * N[(Om / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 3.10000000000000005e56

   1. Initial program 98.5%

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

      1. distribute-rgt-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. metadata-eval 98.5%

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

      4. associate-/l* 98.5%

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

      5. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in Om around 0 51.3%

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

      1. *-commutative 51.3%

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

      2. associate-*r* 51.3%

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

      3. associate-*l/ 51.3%

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

      4. unpow2 51.3%

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

      5. unpow2 51.3%

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

      6. hypot-def 52.3%

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

      7. associate-*l/ 52.3%

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

      8. *-commutative 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in l around inf 60.3%

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

   if 3.10000000000000005e56 < Om

   1. Initial program 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. pow2 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 97.0%

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

      1. expm1-log1p-u 97.0%

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

      2. expm1-udef 97.0%

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

      3. +-commutative 97.0%

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

      4. fma-def 97.0%

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

      5. pow-pow 97.0%

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

      6. metadata-eval 97.0%

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

      7. unpow-1 97.0%

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

   8. Applied egg-rr 97.0%

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

      1. expm1-def 97.0%

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

      2. expm1-log1p 97.0%

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

      3. fma-udef 97.0%

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

      4. associate-*l/ 97.0%

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

      5. metadata-eval 97.0%

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

      6. associate-*l* 97.0%

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

      7. *-commutative 97.0%

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

   10. Simplified 97.0%

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

   11. Taylor expanded in ky around 0 63.2%

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

       1. *-commutative 63.2%

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

       2. associate-/l* 67.1%

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

       3. unpow2 67.1%

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

       4. unpow2 67.1%

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

       5. unpow2 67.1%

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

       6. times-frac 73.1%

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

   13. Simplified 73.1%

       \[\leadsto \color{blue}{1 + \frac{\ell \cdot \ell}{\frac{Om}{ky} \cdot \frac{Om}{ky}} \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 3.1 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om}{ky} \cdot \frac{Om}{ky}}\\ \end{array} \]

Alternative 8: 69.8% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+94}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 1.4e+94) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.4e+94) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 1.4d+94) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.4e+94) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.4e+94:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.4e+94)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.4e+94)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.4e+94], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.39999999999999999e94

   1. Initial program 99.1%

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

      1. distribute-rgt-in 99.1%

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

      2. metadata-eval 99.1%

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

      3. metadata-eval 99.1%

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

      4. associate-/l* 99.1%

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

      5. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. add-sqr-sqrt 99.1%

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

      2. pow2 99.1%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 94.2%

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

      1. expm1-log1p-u 94.2%

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

      2. expm1-udef 94.2%

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

      3. +-commutative 94.2%

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

      4. fma-def 94.2%

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

      5. pow-pow 94.2%

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

      6. metadata-eval 94.2%

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

      7. unpow-1 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. expm1-def 94.2%

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

      2. expm1-log1p 94.2%

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

      3. fma-udef 94.2%

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

      4. associate-*l/ 94.2%

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

      5. metadata-eval 94.2%

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

      6. associate-*l* 94.2%

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

      7. *-commutative 94.2%

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

   10. Simplified 94.2%

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

   11. Taylor expanded in ky around 0 70.3%

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

   if 1.39999999999999999e94 < l

   1. Initial program 97.6%

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

      1. distribute-rgt-in 97.6%

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

      2. metadata-eval 97.6%

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

      3. metadata-eval 97.6%

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

      4. associate-/l* 97.6%

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

      5. metadata-eval 97.6%

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

   3. Simplified 97.6%

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

   4. Taylor expanded in Om around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. associate-*r* 75.1%

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

      3. associate-*l/ 75.1%

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

      4. unpow2 75.1%

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

      5. unpow2 75.1%

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

      6. hypot-def 77.5%

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

      7. associate-*l/ 77.5%

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

      8. *-commutative 77.5%

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

   6. Simplified 77.5%

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

   7. Taylor expanded in l around inf 80.7%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+94}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 9: 39.5% accurate, 48.1× speedup

Math

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

FPCore

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

C

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

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 + ((-0.5d0) * ((l * l) / ((om / ky) * (om / ky))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. add-sqr-sqrt 98.8%

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

   2. pow2 98.8%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 94.2%

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

   1. expm1-log1p-u 94.2%

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

   2. expm1-udef 94.2%

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

   3. +-commutative 94.2%

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

   4. fma-def 94.2%

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

   5. pow-pow 94.2%

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

   6. metadata-eval 94.2%

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

   7. unpow-1 94.2%

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

8. Applied egg-rr 94.2%

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

   1. expm1-def 94.2%

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

   2. expm1-log1p 94.2%

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

   3. fma-udef 94.2%

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

   4. associate-*l/ 94.2%

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

   5. metadata-eval 94.2%

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

   6. associate-*l* 94.2%

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

   7. *-commutative 94.2%

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

10. Simplified 94.2%

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

11. Taylor expanded in ky around 0 37.9%

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

    1. *-commutative 37.9%

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

    2. associate-/l* 38.3%

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

    3. unpow2 38.3%

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

    4. unpow2 38.3%

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

    5. unpow2 38.3%

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

    6. times-frac 43.3%

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

13. Simplified 43.3%

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

14. Final simplification 43.3%

    \[\leadsto 1 + -0.5 \cdot \frac{\ell \cdot \ell}{\frac{Om}{ky} \cdot \frac{Om}{ky}} \]

Reproduce

herbie shell --seed 2023216 
(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))))))))))