Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 100.0%

Time: 11.7s

Alternatives: 6

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt 98.0%

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

   2. hypot-1-def 98.0%

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

   3. sqrt-prod 98.0%

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

   4. unpow2 98.0%

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

   5. sqrt-prod 57.9%

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

   6. add-sqr-sqrt 98.5%

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

   7. associate-/r/ 98.5%

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

   8. *-commutative 98.5%

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

   9. unpow2 98.5%

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

   10. unpow2 98.5%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 90.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(ky \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \ell \cdot \left(\frac{2}{Om} \cdot \sin kx\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= kx 6e-96)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (* ky (/ l Om)))))))
   (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (* l (* (/ 2.0 Om) (sin kx))))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 6e-96) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (ky * (l / Om)))))));
	} else {
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, (l * ((2.0 / Om) * sin(kx))))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (kx <= 6e-96) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * (ky * (l / Om)))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, (l * ((2.0 / Om) * Math.sin(kx))))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if kx <= 6e-96:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * (ky * (l / Om)))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, (l * ((2.0 / Om) * math.sin(kx))))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (kx <= 6e-96)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(ky * Float64(l / Om)))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(l * Float64(Float64(2.0 / Om) * sin(kx))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (kx <= 6e-96)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (ky * (l / Om)))))));
	else
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, (l * ((2.0 / Om) * sin(kx))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[kx, 6e-96], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(ky * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(l * N[(N[(2.0 / Om), $MachinePrecision] * N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6e-96

   1. Initial program 97.2%

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

   3. Simplified 97.2%

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

      1. add-sqr-sqrt 97.2%

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

      2. hypot-1-def 97.2%

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

      3. sqrt-prod 97.2%

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

      4. unpow2 97.2%

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

      5. sqrt-prod 58.6%

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

      6. add-sqr-sqrt 97.8%

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

      7. associate-/r/ 97.8%

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

      8. *-commutative 97.8%

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

      9. unpow2 97.8%

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

      10. unpow2 97.8%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 93.5%

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

   7. Taylor expanded in ky around 0 87.8%

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

      1. *-commutative 87.8%

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

   9. Simplified 87.8%

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

       1. expm1-log1p-u 87.3%

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

       2. expm1-udef 87.3%

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

       3. associate-*l/ 87.3%

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

       4. metadata-eval 87.3%

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

       5. *-commutative 87.3%

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

       6. associate-/l* 87.3%

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

   11. Applied egg-rr 87.3%

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

       1. expm1-def 87.3%

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

       2. expm1-log1p 87.8%

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

       3. *-commutative 87.8%

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

       4. associate-/r/ 87.8%

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

       5. *-commutative 87.8%

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

   13. Simplified 87.8%

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

   if 6e-96 < kx

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in ky around 0 81.0%

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

      1. associate-/l* 81.0%

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

      2. associate-/r/ 81.0%

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

      3. associate-*l* 81.0%

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

      4. *-commutative 81.0%

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

      5. unpow2 81.0%

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

      6. unpow2 81.0%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

      9. swap-sqr 99.5%

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

      10. associate-*l/ 99.5%

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

      11. associate-*r/ 99.5%

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

      12. associate-*l/ 99.5%

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

      13. associate-*r/ 99.5%

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

      14. unpow2 99.5%

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

      15. swap-sqr 99.5%

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

      16. *-commutative 99.5%

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

      17. *-commutative 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 6 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(ky \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \ell \cdot \left(\frac{2}{Om} \cdot \sin kx\right)\right)}}\\ \end{array} \]

Alternative 3: 93.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt 98.0%

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

   2. hypot-1-def 98.0%

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

   3. sqrt-prod 98.0%

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

   4. unpow2 98.0%

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

   5. sqrt-prod 57.9%

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

   6. add-sqr-sqrt 98.5%

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

   7. associate-/r/ 98.5%

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

   8. *-commutative 98.5%

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

   9. unpow2 98.5%

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

   10. unpow2 98.5%

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

   11. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 91.8%

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

7. Final simplification 91.8%

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

Alternative 4: 85.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 3.5 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(ky \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 3.5e+220)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (* ky (/ l Om)))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 3.5e+220) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (ky * (l / Om)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 3.5e+220) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * (ky * (l / Om)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 3.5e+220:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * (ky * (l / Om)))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 3.5e+220)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(ky * Float64(l / Om)))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 3.5e+220)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (ky * (l / Om)))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 3.5e+220], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(ky * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 3.49999999999999986e220

   1. Initial program 97.9%

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

   3. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. hypot-1-def 97.9%

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

      3. sqrt-prod 97.9%

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

      4. unpow2 97.9%

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

      5. sqrt-prod 57.6%

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

      6. add-sqr-sqrt 98.4%

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

      7. associate-/r/ 98.4%

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

      8. *-commutative 98.4%

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

      9. unpow2 98.4%

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

      10. unpow2 98.4%

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

      11. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 91.6%

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

   7. Taylor expanded in ky around 0 86.8%

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

      1. *-commutative 86.8%

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

   9. Simplified 86.8%

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

       1. expm1-log1p-u 86.2%

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

       2. expm1-udef 86.2%

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

       3. associate-*l/ 86.2%

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

       4. metadata-eval 86.2%

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

       5. *-commutative 86.2%

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

       6. associate-/l* 86.2%

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

   11. Applied egg-rr 86.2%

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

       1. expm1-def 86.2%

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

       2. expm1-log1p 86.8%

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

       3. *-commutative 86.8%

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

       4. associate-/r/ 86.8%

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

       5. *-commutative 86.8%

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

   13. Simplified 86.8%

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

   if 3.49999999999999986e220 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in ky around 0 81.3%

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

      1. associate-/l* 81.3%

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

      2. associate-/r/ 81.3%

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

      3. associate-*l* 81.3%

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

      4. *-commutative 81.3%

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

      5. unpow2 81.3%

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

      6. unpow2 81.3%

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

      7. times-frac 100.0%

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

      8. metadata-eval 100.0%

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

      9. swap-sqr 100.0%

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

      10. associate-*l/ 100.0%

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

      11. associate-*r/ 100.0%

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

      12. associate-*l/ 100.0%

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

      13. associate-*r/ 100.0%

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

      14. unpow2 100.0%

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

      15. swap-sqr 100.0%

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

      16. *-commutative 100.0%

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

      17. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in l around 0 95.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 3.5 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(ky \cdot \frac{\ell}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 67.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 10^{-35}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 1e-35) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1e-35) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 1d-35) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1e-35) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1e-35:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1e-35)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1e-35)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1e-35], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.00000000000000001e-35

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in Om around 0 54.0%

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

      1. unpow2 54.0%

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

      2. unpow2 54.0%

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

      3. hypot-def 56.1%

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

   6. Simplified 56.1%

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

   7. Taylor expanded in l around inf 63.0%

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

   if 1.00000000000000001e-35 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in ky around 0 86.0%

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

      1. associate-/l* 84.4%

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

      2. associate-/r/ 86.0%

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

      3. associate-*l* 86.0%

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

      4. *-commutative 86.0%

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

      5. unpow2 86.0%

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

      6. unpow2 86.0%

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

      7. times-frac 97.1%

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

      8. metadata-eval 97.1%

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

      9. swap-sqr 97.1%

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

      10. associate-*l/ 97.1%

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

      11. associate-*r/ 97.1%

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

      12. associate-*l/ 97.1%

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

      13. associate-*r/ 97.1%

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

      14. unpow2 97.1%

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

      15. swap-sqr 98.7%

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

      16. *-commutative 98.7%

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

      17. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in l around 0 80.9%

      \[\leadsto \sqrt{0.5 + \color{blue}{1} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 10^{-35}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 56.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (sqrt 0.5))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

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(0.5d0)
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(0.5);
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

function code(l, Om, kx, ky)
	return sqrt(0.5)
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(0.5);
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in Om around 0 46.7%

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

   1. unpow2 46.7%

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

   2. unpow2 46.7%

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

   3. hypot-def 48.3%

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

6. Simplified 48.3%

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

7. Taylor expanded in l around inf 56.8%

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

8. Final simplification 56.8%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023338 
(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))))))))))