Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.1% → 100.0%

Time: 16.2s

Alternatives: 8

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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. add-sqr-sqrt 99.2%

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

   3. hypot-1-def 99.2%

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

   4. sqrt-prod 99.2%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

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

   7. pow1 99.3%

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

   8. clear-num 99.3%

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

   9. un-div-inv 99.3%

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

   10. unpow2 99.3%

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

   11. unpow2 99.3%

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

   12. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-/r/ 100.0%

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

8. Simplified 100.0%

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

Alternative 2: 93.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + 0.5 \cdot \frac{1}{\sqrt{1 + 4 \cdot {\left(\sin ky \cdot \frac{\ell}{Om}\right)}^{2}}}} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (* 0.5 (/ 1.0 (sqrt (+ 1.0 (* 4.0 (pow (* (sin ky) (/ l Om)) 2.0)))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (4.0 * pow((sin(ky) * (l / Om)), 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((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + (4.0d0 * ((sin(ky) * (l / om)) ** 2.0d0))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt((0.5 + (0.5 * (1.0 / Math.sqrt((1.0 + (4.0 * Math.pow((Math.sin(ky) * (l / Om)), 2.0))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + (4.0 * math.pow((math.sin(ky) * (l / Om)), 2.0))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / sqrt(Float64(1.0 + Float64(4.0 * (Float64(sin(ky) * Float64(l / Om)) ^ 2.0))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + (4.0 * ((sin(ky) * (l / Om)) ^ 2.0))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[N[(1.0 + N[(4.0 * N[Power[N[(N[Sin[ky], $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in kx around 0 79.7%

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

   1. *-commutative 79.7%

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

   2. associate-/l* 81.3%

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

   3. unpow2 81.3%

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

   4. unpow2 81.3%

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

   5. times-frac 92.6%

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

   6. unpow2 92.6%

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

7. Simplified 92.6%

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

   1. pow-prod-down 95.0%

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

9. Applied egg-rr 95.0%

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

Alternative 3: 87.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 3 \cdot 10^{+132}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell \cdot \left(ky \cdot 2\right)}{Om}\right)}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= ky 3e+132)
   (cbrt (pow (+ 0.5 (/ 0.5 (hypot 1.0 (/ (* l (* ky 2.0)) Om)))) 1.5))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin kx) (* 2.0 (/ l Om)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 3e+132) {
		tmp = cbrt(pow((0.5 + (0.5 / hypot(1.0, ((l * (ky * 2.0)) / Om)))), 1.5));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (2.0 * (l / Om)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 3e+132) {
		tmp = Math.cbrt(Math.pow((0.5 + (0.5 / Math.hypot(1.0, ((l * (ky * 2.0)) / Om)))), 1.5));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) * (2.0 * (l / Om)))))));
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (ky <= 3e+132)
		tmp = cbrt((Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(l * Float64(ky * 2.0)) / Om)))) ^ 1.5));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) * Float64(2.0 * Float64(l / Om)))))));
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[ky, 3e+132], N[Power[N[Power[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(l * N[(ky * 2.0), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.9999999999999998e132

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in kx around 0 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-/l* 80.0%

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

      3. unpow2 80.0%

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

      4. unpow2 80.0%

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

      5. times-frac 91.6%

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

      6. unpow2 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in ky around inf 78.2%

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

      1. metadata-eval 78.2%

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

      2. metadata-eval 78.2%

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

      3. associate-/l* 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

      6. unpow2 80.0%

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

      7. times-frac 84.0%

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

      8. swap-sqr 94.4%

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

      9. associate-/l* 94.4%

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

      10. associate-/l* 94.4%

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

      11. swap-sqr 94.4%

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

      12. unpow2 94.4%

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

      13. *-commutative 94.4%

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

      14. associate-*r/ 94.4%

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

      15. unpow2 94.4%

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

   10. Simplified 94.4%

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

   11. Taylor expanded in ky around 0 90.2%

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

       1. add-cbrt-cube 90.2%

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

       2. add-sqr-sqrt 90.2%

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

       3. pow1 90.2%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(ky \cdot \ell\right)}{Om}\right)}\right)}^{1}} \cdot \sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(ky \cdot \ell\right)}{Om}\right)}}} \]

       4. pow1/2 90.2%

          \[\leadsto \sqrt[3]{{\left(0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(ky \cdot \ell\right)}{Om}\right)}\right)}^{1} \cdot \color{blue}{{\left(0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(ky \cdot \ell\right)}{Om}\right)}\right)}^{0.5}}} \]

       5. pow-prod-up 90.2%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(ky \cdot \ell\right)}{Om}\right)}\right)}^{\left(1 + 0.5\right)}}} \]

       6. un-div-inv 90.2%

          \[\leadsto \sqrt[3]{{\left(0.5 + \color{blue}{\frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(ky \cdot \ell\right)}{Om}\right)}}\right)}^{\left(1 + 0.5\right)}} \]

       7. associate-*r* 90.2%

          \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\color{blue}{\left(2 \cdot ky\right) \cdot \ell}}{Om}\right)}\right)}^{\left(1 + 0.5\right)}} \]

       8. metadata-eval 90.2%

          \[\leadsto \sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\left(2 \cdot ky\right) \cdot \ell}{Om}\right)}\right)}^{\color{blue}{1.5}}} \]

   13. Applied egg-rr 90.2%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\left(2 \cdot ky\right) \cdot \ell}{Om}\right)}\right)}^{1.5}}} \]

   if 2.9999999999999998e132 < ky

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

      4. sqrt-prod 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1 100.0%

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

      8. clear-num 100.0%

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

      9. un-div-inv 100.0%

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

      10. unpow2 100.0%

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

      11. unpow2 100.0%

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

      12. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in ky around 0 94.3%

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

       1. *-un-lft-identity 94.3%

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

       2. un-div-inv 94.3%

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

       3. *-commutative 94.3%

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

   11. Applied egg-rr 94.3%

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

       1. *-lft-identity 94.3%

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

       2. associate-*r/ 94.3%

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

       3. *-commutative 94.3%

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

       4. associate-*r/ 94.3%

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

   13. Simplified 94.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3 \cdot 10^{+132}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell \cdot \left(ky \cdot 2\right)}{Om}\right)}\right)}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 7 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(ky \cdot 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= ky 7e+132)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (/ l Om) (* ky 2.0))))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin kx) (* 2.0 (/ l Om)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 7e+132) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((l / Om) * (ky * 2.0))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (2.0 * (l / Om)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 7e+132) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((l / Om) * (ky * 2.0))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) * (2.0 * (l / Om)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ky <= 7e+132:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((l / Om) * (ky * 2.0))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.sin(kx) * (2.0 * (l / Om)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (ky <= 7e+132)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(l / Om) * Float64(ky * 2.0))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) * Float64(2.0 * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (ky <= 7e+132)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((l / Om) * (ky * 2.0))))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (2.0 * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[ky, 7e+132], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(l / Om), $MachinePrecision] * N[(ky * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 7.00000000000000041e132

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in kx around 0 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-/l* 80.0%

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

      3. unpow2 80.0%

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

      4. unpow2 80.0%

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

      5. times-frac 91.6%

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

      6. unpow2 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in ky around inf 78.2%

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

      1. metadata-eval 78.2%

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

      2. metadata-eval 78.2%

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

      3. associate-/l* 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

      6. unpow2 80.0%

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

      7. times-frac 84.0%

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

      8. swap-sqr 94.4%

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

      9. associate-/l* 94.4%

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

      10. associate-/l* 94.4%

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

      11. swap-sqr 94.4%

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

      12. unpow2 94.4%

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

      13. *-commutative 94.4%

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

      14. associate-*r/ 94.4%

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

      15. unpow2 94.4%

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

   10. Simplified 94.4%

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

   11. Taylor expanded in ky around 0 90.2%

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

       1. *-un-lft-identity 90.2%

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

       2. un-div-inv 90.2%

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

       3. associate-*r* 90.2%

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

   13. Applied egg-rr 90.2%

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

       1. *-lft-identity 90.2%

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

       2. associate-/l* 90.2%

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

   15. Simplified 90.2%

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

   if 7.00000000000000041e132 < ky

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

      4. sqrt-prod 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1 100.0%

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

      8. clear-num 100.0%

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

      9. un-div-inv 100.0%

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

      10. unpow2 100.0%

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

      11. unpow2 100.0%

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

      12. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in ky around 0 94.3%

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

       1. *-un-lft-identity 94.3%

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

       2. un-div-inv 94.3%

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

       3. *-commutative 94.3%

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

   11. Applied egg-rr 94.3%

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

       1. *-lft-identity 94.3%

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

       2. associate-*r/ 94.3%

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

       3. *-commutative 94.3%

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

       4. associate-*r/ 94.3%

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

   13. Simplified 94.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 7 \cdot 10^{+132}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(ky \cdot 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in kx around 0 79.7%

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

   1. *-commutative 79.7%

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

   2. associate-/l* 81.3%

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

   3. unpow2 81.3%

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

   4. unpow2 81.3%

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

   5. times-frac 92.6%

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

   6. unpow2 92.6%

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

7. Simplified 92.6%

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

8. Taylor expanded in ky around inf 79.7%

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

   1. metadata-eval 79.7%

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

   2. metadata-eval 79.7%

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

   3. associate-/l* 81.3%

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

   4. unpow2 81.3%

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

   5. unpow2 81.3%

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

   6. unpow2 81.3%

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

   7. times-frac 84.8%

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

   8. swap-sqr 95.0%

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

   9. associate-/l* 95.0%

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

   10. associate-/l* 95.0%

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

   11. swap-sqr 95.0%

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

   12. unpow2 95.0%

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

   13. *-commutative 95.0%

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

   14. associate-*r/ 95.0%

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

   15. unpow2 95.0%

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

10. Simplified 95.0%

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

11. Final simplification 95.0%

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

Alternative 6: 74.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-76}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(ky \cdot 2\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 8.2e-76)
   1.0
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (/ l Om) (* ky 2.0))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 8.2e-76) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((l / Om) * (ky * 2.0))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 8.2e-76) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((l / Om) * (ky * 2.0))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 8.2e-76:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((l / Om) * (ky * 2.0))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 8.2e-76)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(l / Om) * Float64(ky * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 8.2e-76)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((l / Om) * (ky * 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 8.2e-76], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(l / Om), $MachinePrecision] * N[(ky * 2.0), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.1999999999999996e-76

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. add-sqr-sqrt 98.9%

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

      3. hypot-1-def 98.9%

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

      4. sqrt-prod 98.9%

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

      5. sqrt-pow1 99.0%

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

      6. metadata-eval 99.0%

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

      7. pow1 99.0%

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

      8. clear-num 99.0%

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

      9. un-div-inv 99.0%

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

      10. unpow2 99.0%

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

      11. unpow2 99.0%

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

      12. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in ky around 0 94.4%

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

       1. *-un-lft-identity 94.4%

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

       2. un-div-inv 94.4%

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

       3. *-commutative 94.4%

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

   11. Applied egg-rr 94.4%

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

       1. *-lft-identity 94.4%

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

       2. associate-*r/ 94.4%

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

       3. *-commutative 94.4%

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

       4. associate-*r/ 94.4%

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

   13. Simplified 94.4%

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

   14. Taylor expanded in kx around 0 68.3%

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

   if 8.1999999999999996e-76 < l

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in kx around 0 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-/l* 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

      5. times-frac 91.4%

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

      6. unpow2 91.4%

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

   7. Simplified 91.4%

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

   8. Taylor expanded in ky around inf 77.2%

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

      1. metadata-eval 77.2%

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

      2. metadata-eval 77.2%

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

      3. associate-/l* 77.2%

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

      4. unpow2 77.2%

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

      5. unpow2 77.2%

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

      6. unpow2 77.2%

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

      7. times-frac 78.5%

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

      8. swap-sqr 92.7%

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

      9. associate-/l* 92.7%

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

      10. associate-/l* 92.7%

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

      11. swap-sqr 92.7%

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

      12. unpow2 92.7%

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

      13. *-commutative 92.7%

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

      14. associate-*r/ 92.7%

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

      15. unpow2 92.7%

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

   10. Simplified 92.7%

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

   11. Taylor expanded in ky around 0 90.3%

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

       1. *-un-lft-identity 90.3%

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

       2. un-div-inv 90.3%

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

       3. associate-*r* 90.3%

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

   13. Applied egg-rr 90.3%

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

       1. *-lft-identity 90.3%

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

       2. associate-/l* 90.3%

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

   15. Simplified 90.3%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 8.2 \cdot 10^{-76}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(ky \cdot 2\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 4e-25) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 4e-25) {
		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 <= 4d-25) 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 <= 4e-25) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 4e-25:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 4e-25)
		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 <= 4e-25)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 4e-25], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 4.00000000000000015e-25

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in l around inf 56.5%

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

      1. *-commutative 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

      4. unpow2 56.5%

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

      5. unpow2 56.5%

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

      6. hypot-undefine 57.6%

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

      7. associate-*l/ 57.6%

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

      8. associate-/l* 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in l around inf 64.9%

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

   if 4.00000000000000015e-25 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

      4. sqrt-prod 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1 100.0%

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

      8. clear-num 100.0%

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

      9. un-div-inv 100.0%

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

      10. unpow2 100.0%

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

      11. unpow2 100.0%

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

      12. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in ky around 0 91.0%

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

       1. *-un-lft-identity 91.0%

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

       2. un-div-inv 91.0%

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

       3. *-commutative 91.0%

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

   11. Applied egg-rr 91.0%

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

       1. *-lft-identity 91.0%

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

       2. associate-*r/ 91.0%

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

       3. *-commutative 91.0%

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

       4. associate-*r/ 91.0%

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

   13. Simplified 91.0%

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

   14. Taylor expanded in kx around 0 81.3%

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

Alternative 8: 62.7% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. add-sqr-sqrt 99.2%

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

   3. hypot-1-def 99.2%

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

   4. sqrt-prod 99.2%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

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

   7. pow1 99.3%

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

   8. clear-num 99.3%

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

   9. un-div-inv 99.3%

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

   10. unpow2 99.3%

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

   11. unpow2 99.3%

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

   12. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-/r/ 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in ky around 0 92.8%

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

    1. *-un-lft-identity 92.8%

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

    2. un-div-inv 92.8%

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

    3. *-commutative 92.8%

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

11. Applied egg-rr 92.8%

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

    1. *-lft-identity 92.8%

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

    2. associate-*r/ 92.8%

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

    3. *-commutative 92.8%

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

    4. associate-*r/ 92.8%

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

13. Simplified 92.8%

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

14. Taylor expanded in kx around 0 61.5%

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

Reproduce

herbie shell --seed 2024089 
(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))))))))))