Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 100.0%

Time: 15.6s

Alternatives: 5

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: 98.3% 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 98.8%

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

3. Simplified 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. add-sqr-sqrt 98.8%

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

   3. hypot-1-def 98.8%

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

   4. sqrt-prod 98.8%

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

   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. 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: 74.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.6 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.6e-136)
   (sqrt 0.5)
   (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 (Om <= 1.6e-136) {
		tmp = sqrt(0.5);
	} 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 (Om <= 1.6e-136) {
		tmp = Math.sqrt(0.5);
	} 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 Om <= 1.6e-136:
		tmp = math.sqrt(0.5)
	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 (Om <= 1.6e-136)
		tmp = sqrt(0.5);
	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

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1.6e-136)
		tmp = sqrt(0.5);
	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[Om, 1.6e-136], N[Sqrt[0.5], $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 Om < 1.59999999999999996e-136

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in kx around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-/l* 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 86.4%

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

      6. unpow2 86.4%

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

   7. Simplified 86.4%

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

   8. Taylor expanded in l around inf 63.2%

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

   if 1.59999999999999996e-136 < 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. 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. Taylor expanded in ky around 0 90.9%

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

Alternative 3: 93.6% 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in kx around 0 75.8%

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

   1. *-commutative 75.8%

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

   2. associate-/l* 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 88.7%

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

   6. unpow2 88.7%

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

7. Simplified 88.7%

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

   1. *-un-lft-identity 88.7%

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

   2. un-div-inv 88.7%

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

   3. add-sqr-sqrt 88.7%

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

   4. hypot-1-def 88.7%

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

   5. sqrt-prod 88.7%

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

   6. metadata-eval 88.7%

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

   7. pow-prod-down 92.9%

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

   8. sqrt-pow1 92.9%

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

   9. metadata-eval 92.9%

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

   10. pow1 92.9%

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

9. Applied egg-rr 92.9%

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

    1. *-lft-identity 92.9%

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

    2. associate-*r* 92.9%

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

11. Simplified 92.9%

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

12. Final simplification 92.9%

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

Alternative 4: 67.7% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 2e-20) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 2e-20) {
		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 <= 2d-20) 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 <= 2e-20) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 2e-20:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 2e-20)
		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 <= 2e-20)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 2e-20], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.99999999999999989e-20

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in kx around 0 73.5%

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

      1. *-commutative 73.5%

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

      2. associate-/l* 75.5%

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

      3. unpow2 75.5%

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

      4. unpow2 75.5%

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

      5. times-frac 86.2%

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

      6. unpow2 86.2%

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

   7. Simplified 86.2%

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

   8. Taylor expanded in l around inf 66.2%

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

   if 1.99999999999999989e-20 < 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. Taylor expanded in kx around 0 82.6%

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

      1. *-commutative 82.6%

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

      2. associate-/l* 82.6%

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

      3. unpow2 82.6%

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

      4. unpow2 82.6%

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

      5. times-frac 96.3%

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

      6. unpow2 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in ky around 0 79.9%

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

Alternative 5: 62.5% 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 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in kx around 0 75.8%

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

   1. *-commutative 75.8%

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

   2. associate-/l* 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 88.7%

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

   6. unpow2 88.7%

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

7. Simplified 88.7%

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

8. Taylor expanded in ky around 0 59.2%

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

Reproduce

herbie shell --seed 2024141 
(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))))))))))