Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 100.0%

Time: 14.8s

Alternatives: 7

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

   1. *-un-lft-identity 98.6%

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

   2. add-sqr-sqrt 98.6%

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

   3. hypot-1-def 98.6%

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

   4. sqrt-prod 98.6%

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

   5. sqrt-pow1 98.9%

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

   6. metadata-eval 98.9%

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

   7. pow1 98.9%

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

   8. clear-num 98.9%

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

   9. un-div-inv 98.9%

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

   10. unpow2 98.9%

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

   11. unpow2 98.9%

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

   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. Step-by-step derivation

   1. un-div-inv 100.0%

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

   2. associate-*l/ 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. times-frac 100.0%

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

   5. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

Alternative 2: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{\frac{Om}{\ell}}{\sin ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      5.0)
   1.0
   (sqrt (+ 0.5 (* 0.25 (/ (/ Om l) (sin ky)))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 5.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.25 * ((Om / l) / sin(ky)))));
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))) <= 5.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + (0.25d0 * ((om / l) / sin(ky)))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))) <= 5.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.25 * ((Om / l) / Math.sin(ky)))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))) <= 5.0:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.25 * ((Om / l) / math.sin(ky)))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 5.0)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(Float64(Om / l) / sin(ky)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 5.0)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.25 * ((Om / l) / sin(ky)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[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], 5.0], 1.0, N[Sqrt[N[(0.5 + N[(0.25 * N[(N[(Om / l), $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 5

   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 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-/l* 84.0%

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

      3. unpow2 84.0%

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

      4. unpow2 84.0%

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

      5. times-frac 99.1%

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

      6. unpow2 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in ky around 0 99.1%

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

   if 5 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in kx around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. associate-/l* 71.3%

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

      3. unpow2 71.3%

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

      4. unpow2 71.3%

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

      5. times-frac 76.9%

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

      6. unpow2 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in l around inf 87.1%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   9. Step-by-step derivation

      1. associate-/r* 87.1%

         \[\leadsto \sqrt{0.5 + 0.25 \cdot \color{blue}{\frac{\frac{Om}{\ell}}{\sin ky}}} \]

   10. Simplified 87.1%

       \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{\frac{Om}{\ell}}{\sin ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 4 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{\frac{Om}{\ell}}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      4e-10)
   1.0
   (sqrt (+ 0.5 (* 0.25 (/ (/ Om l) ky))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 4e-10) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.25 * ((Om / l) / ky))));
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))) <= 4d-10) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + (0.25d0 * ((om / l) / ky))))
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))) <= 4e-10) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.25 * ((Om / l) / ky))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))) <= 4e-10:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.25 * ((Om / l) / ky))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 4e-10)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(Float64(Om / l) / ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 4e-10)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.25 * ((Om / l) / ky))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[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], 4e-10], 1.0, N[Sqrt[N[(0.5 + N[(0.25 * N[(N[(Om / l), $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 4.00000000000000015e-10

   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 84.6%

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

      1. *-commutative 84.6%

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

      2. associate-/l* 84.6%

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

      3. unpow2 84.6%

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

      4. unpow2 84.6%

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

      5. times-frac 99.6%

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

      6. unpow2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in ky around 0 99.6%

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

   if 4.00000000000000015e-10 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in kx around 0 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-/l* 70.7%

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

      3. unpow2 70.7%

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

      4. unpow2 70.7%

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

      5. times-frac 76.4%

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

      6. unpow2 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in l around inf 86.4%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   9. Step-by-step derivation

      1. associate-/r* 86.4%

         \[\leadsto \sqrt{0.5 + 0.25 \cdot \color{blue}{\frac{\frac{Om}{\ell}}{\sin ky}}} \]

   10. Simplified 86.4%

       \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{\frac{Om}{\ell}}{\sin ky}}} \]

   11. Taylor expanded in ky around 0 86.4%

       \[\leadsto \sqrt{0.5 + 0.25 \cdot \frac{\frac{Om}{\ell}}{\color{blue}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq 3.75:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))
      3.75)
   1.0
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 3.75) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0))) <= 3.75d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0))) <= 3.75) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0))) <= 3.75:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 3.75)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))) <= 3.75)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[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], 3.75], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))) < 3.75

   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 84.6%

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

      1. *-commutative 84.6%

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

      2. associate-/l* 84.6%

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

      3. unpow2 84.6%

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

      4. unpow2 84.6%

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

      5. times-frac 99.6%

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

      6. unpow2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in ky around 0 99.6%

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

   if 3.75 < (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))

   1. Initial program 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in kx around 0 68.8%

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

      1. *-commutative 68.8%

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

      2. associate-/l* 70.7%

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

      3. unpow2 70.7%

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

      4. unpow2 70.7%

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

      5. times-frac 76.4%

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

      6. unpow2 76.4%

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

   7. Simplified 76.4%

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

   8. Taylor expanded in l around inf 96.7%

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

Alternative 5: 93.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in kx around 0 77.6%

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

   1. *-commutative 77.6%

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

   2. associate-/l* 78.5%

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

   3. unpow2 78.5%

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

   4. unpow2 78.5%

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

   5. times-frac 89.5%

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

   6. unpow2 89.5%

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

7. Simplified 89.5%

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

   1. *-un-lft-identity 89.5%

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

   2. un-div-inv 89.5%

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

   3. add-sqr-sqrt 89.5%

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

   4. hypot-1-def 89.5%

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

   5. sqrt-prod 89.5%

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

   6. metadata-eval 89.5%

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

   7. pow-prod-down 95.1%

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

   8. sqrt-pow1 95.1%

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

   9. metadata-eval 95.1%

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

   10. pow1 95.1%

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

9. Applied egg-rr 95.1%

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

    1. *-lft-identity 95.1%

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

    2. associate-*r* 95.1%

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

11. Simplified 95.1%

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

12. Final simplification 95.1%

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

Alternative 6: 78.4% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= (/ (* 2.0 l) Om) 2e-5)
   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 (((2.0 * l) / Om) <= 2e-5) {
		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 (((2.0 * l) / Om) <= 2e-5) {
		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 ((2.0 * l) / Om) <= 2e-5:
		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 (Float64(Float64(2.0 * l) / Om) <= 2e-5)
		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 (((2.0 * l) / Om) <= 2e-5)
		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[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2e-5], 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 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) < 2.00000000000000016e-5

   1. Initial program 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in kx around 0 79.1%

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

      1. *-commutative 79.1%

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

      2. associate-/l* 80.4%

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

      3. unpow2 80.4%

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

      4. unpow2 80.4%

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

      5. times-frac 92.3%

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

      6. unpow2 92.3%

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

   7. Simplified 92.3%

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

   8. Taylor expanded in ky around 0 79.2%

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

   if 2.00000000000000016e-5 < (/.f64 (*.f64 #s(literal 2 binary64) l) Om)

   1. Initial program 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in kx around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-/l* 73.3%

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

      3. unpow2 73.3%

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

      4. unpow2 73.3%

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

      5. times-frac 81.8%

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

      6. unpow2 81.8%

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

   7. Simplified 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. un-div-inv 81.8%

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

      3. add-sqr-sqrt 81.8%

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

      4. hypot-1-def 81.8%

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

      5. sqrt-prod 81.8%

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

      6. metadata-eval 81.8%

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

      7. pow-prod-down 88.5%

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

      8. sqrt-pow1 88.5%

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

      9. metadata-eval 88.5%

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

      10. pow1 88.5%

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

   9. Applied egg-rr 88.5%

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

       1. *-lft-identity 88.5%

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

       2. associate-*r* 88.5%

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

   11. Simplified 88.5%

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

   12. Taylor expanded in ky around 0 88.5%

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

4. Final simplification 81.7%

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

Alternative 7: 62.6% 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 98.6%

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

3. Simplified 98.6%

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

5. Taylor expanded in kx around 0 77.6%

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

   1. *-commutative 77.6%

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

   2. associate-/l* 78.5%

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

   3. unpow2 78.5%

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

   4. unpow2 78.5%

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

   5. times-frac 89.5%

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

   6. unpow2 89.5%

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

7. Simplified 89.5%

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

8. Taylor expanded in ky around 0 64.8%

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

Reproduce

herbie shell --seed 2024191 
(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))))))))))