Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 24.6s

Alternatives: 9

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

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 (/ l Om)))))))))

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 * (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.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 * (1.0 / 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 * Float64(1.0 / 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 * (1.0 / 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[(1.0 / 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]], $MachinePrecision]

Derivation

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%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 90.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= kx 4.2e-199)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* ky l) Om))))))
   (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 (kx <= 4.2e-199) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} 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 (kx <= 4.2e-199) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} 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 kx <= 4.2e-199:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((ky * l) / Om))))))
	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 (kx <= 4.2e-199)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(ky * l) / Om))))));
	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 (kx <= 4.2e-199)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	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[kx, 4.2e-199], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $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 kx < 4.20000000000000004e-199

   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%

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

   4. Taylor expanded in kx around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. associate-*r* 76.5%

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

      3. unpow2 76.5%

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

      4. unpow2 76.5%

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

   6. Simplified 76.5%

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

      1. add-sqr-sqrt 76.5%

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

      2. hypot-1-def 76.5%

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

      3. sqrt-div 76.5%

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

      4. *-commutative 76.5%

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

      5. sqrt-prod 76.5%

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

      6. unpow2 76.5%

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

      7. sqrt-prod 38.0%

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

      8. add-sqr-sqrt 85.6%

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

      9. *-commutative 85.6%

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

      10. sqrt-prod 85.6%

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

      11. sqrt-prod 40.4%

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

      12. add-sqr-sqrt 91.6%

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

      13. metadata-eval 91.6%

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

      14. sqrt-prod 40.2%

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

      15. add-sqr-sqrt 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. un-div-inv 94.2%

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

      2. associate-/l* 94.2%

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

   10. Applied egg-rr 94.2%

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

   11. Taylor expanded in ky around 0 82.8%

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

   if 4.20000000000000004e-199 < kx

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in ky around 0 98.3%

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

      1. expm1-log1p-u 97.6%

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

      2. expm1-udef 97.6%

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

      3. un-div-inv 97.6%

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

   10. Applied egg-rr 97.6%

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

       1. expm1-def 97.6%

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

       2. expm1-log1p 98.3%

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

   12. Simplified 98.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 4.2 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \]

Alternative 3: 93.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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.sin(ky) / (Om / (2.0 * l)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

4. Taylor expanded in kx around 0 74.2%

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

   1. associate-*r/ 74.2%

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

   2. associate-*r* 74.2%

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

   3. unpow2 74.2%

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

   4. unpow2 74.2%

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

6. Simplified 74.2%

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

   1. add-sqr-sqrt 74.2%

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

   2. hypot-1-def 74.2%

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

   3. sqrt-div 74.2%

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

   4. *-commutative 74.2%

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

   5. sqrt-prod 74.3%

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

   6. unpow2 74.3%

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

   7. sqrt-prod 37.9%

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

   8. add-sqr-sqrt 81.8%

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

   9. *-commutative 81.8%

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

   10. sqrt-prod 81.8%

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

   11. sqrt-prod 39.8%

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

   12. add-sqr-sqrt 88.9%

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

   13. metadata-eval 88.9%

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

   14. sqrt-prod 43.7%

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

   15. add-sqr-sqrt 94.4%

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

8. Applied egg-rr 94.4%

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

   1. un-div-inv 94.4%

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

   2. associate-/l* 94.4%

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

10. Applied egg-rr 94.4%

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

11. Final simplification 94.4%

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

Alternative 4: 74.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 7.9 \cdot 10^{-56} \lor \neg \left(\ell \leq 2.9 \cdot 10^{-26}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 3.8e-159)
   1.0
   (if (or (<= l 7.9e-56) (not (<= l 2.9e-26)))
     (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* ky l) Om))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 3.8e-159) {
		tmp = 1.0;
	} else if ((l <= 7.9e-56) || !(l <= 2.9e-26)) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 3.8e-159) {
		tmp = 1.0;
	} else if ((l <= 7.9e-56) || !(l <= 2.9e-26)) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 3.8e-159:
		tmp = 1.0
	elif (l <= 7.9e-56) or not (l <= 2.9e-26):
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((ky * l) / Om))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 3.8e-159)
		tmp = 1.0;
	elseif ((l <= 7.9e-56) || !(l <= 2.9e-26))
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(ky * l) / Om))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 3.8e-159)
		tmp = 1.0;
	elseif ((l <= 7.9e-56) || ~((l <= 2.9e-26)))
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 3.8e-159], 1.0, If[Or[LessEqual[l, 7.9e-56], N[Not[LessEqual[l, 2.9e-26]], $MachinePrecision]], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if l < 3.8000000000000001e-159 or 7.90000000000000034e-56 < l < 2.8999999999999998e-26

   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%

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

   4. Taylor expanded in kx around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. associate-*r* 75.5%

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

      3. unpow2 75.5%

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

      4. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. add-sqr-sqrt 75.5%

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

      2. hypot-1-def 75.5%

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

      3. sqrt-div 75.5%

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

      4. *-commutative 75.5%

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

      5. sqrt-prod 75.6%

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

      6. unpow2 75.6%

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

      7. sqrt-prod 40.7%

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

      8. add-sqr-sqrt 80.7%

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

      9. *-commutative 80.7%

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

      10. sqrt-prod 80.7%

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

      11. sqrt-prod 16.9%

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

      12. add-sqr-sqrt 88.7%

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

      13. metadata-eval 88.7%

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

      14. sqrt-prod 47.8%

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

      15. add-sqr-sqrt 94.8%

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

   8. Applied egg-rr 94.8%

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

   9. Taylor expanded in ky around 0 67.6%

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

   if 3.8000000000000001e-159 < l < 7.90000000000000034e-56 or 2.8999999999999998e-26 < 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%

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

   4. Taylor expanded in kx around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. associate-*r* 71.4%

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

      3. unpow2 71.4%

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

      4. unpow2 71.4%

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

   6. Simplified 71.4%

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

      1. add-sqr-sqrt 71.4%

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

      2. hypot-1-def 71.4%

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

      3. sqrt-div 71.4%

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

      4. *-commutative 71.4%

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

      5. sqrt-prod 71.4%

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

      6. unpow2 71.4%

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

      7. sqrt-prod 31.7%

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

      8. add-sqr-sqrt 84.0%

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

      9. *-commutative 84.0%

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

      10. sqrt-prod 84.0%

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

      11. sqrt-prod 89.4%

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

      12. add-sqr-sqrt 89.4%

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

      13. metadata-eval 89.4%

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

      14. sqrt-prod 34.8%

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

      15. add-sqr-sqrt 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. un-div-inv 93.6%

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

      2. associate-/l* 93.6%

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

   10. Applied egg-rr 93.6%

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

   11. Taylor expanded in ky around 0 83.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.8 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 7.9 \cdot 10^{-56} \lor \neg \left(\ell \leq 2.9 \cdot 10^{-26}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 72.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.2e-26)
   1.0
   (if (<= l 1.8e+140)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* ky ky) (/ (* Om Om) (* l l)))))))))
     (sqrt 0.5))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.2e-26) {
		tmp = 1.0;
	} else if (l <= 1.8e+140) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 1.2d-26) then
        tmp = 1.0d0
    else if (l <= 1.8d+140) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((ky * ky) / ((om * om) / (l * l)))))))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.2e-26) {
		tmp = 1.0;
	} else if (l <= 1.8e+140) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.2e-26:
		tmp = 1.0
	elif l <= 1.8e+140:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.2e-26)
		tmp = 1.0;
	elseif (l <= 1.8e+140)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * ky) / Float64(Float64(Om * Om) / Float64(l * l)))))))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.2e-26)
		tmp = 1.0;
	elseif (l <= 1.8e+140)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * ky) / ((Om * Om) / (l * l)))))))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.2e-26], 1.0, If[LessEqual[l, 1.8e+140], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(ky * ky), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.2e-26

   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%

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

   4. Taylor expanded in kx around 0 76.8%

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

      1. associate-*r/ 76.8%

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

      2. associate-*r* 76.8%

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

      3. unpow2 76.8%

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

      4. unpow2 76.8%

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

   6. Simplified 76.8%

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

      1. add-sqr-sqrt 76.8%

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

      2. hypot-1-def 76.8%

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

      3. sqrt-div 76.8%

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

      4. *-commutative 76.8%

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

      5. sqrt-prod 76.9%

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

      6. unpow2 76.9%

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

      7. sqrt-prod 38.8%

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

      8. add-sqr-sqrt 81.7%

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

      9. *-commutative 81.7%

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

      10. sqrt-prod 81.7%

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

      11. sqrt-prod 22.6%

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

      12. add-sqr-sqrt 89.1%

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

      13. metadata-eval 89.1%

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

      14. sqrt-prod 46.9%

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

      15. add-sqr-sqrt 95.2%

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

   8. Applied egg-rr 95.2%

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

   9. Taylor expanded in ky around 0 68.7%

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

   if 1.2e-26 < l < 1.8e140

   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%

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

   4. Taylor expanded in kx around 0 89.8%

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

      1. associate-*r/ 89.8%

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

      2. associate-*r* 89.8%

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

      3. unpow2 89.8%

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

      4. unpow2 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in ky around 0 79.5%

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

      1. associate-/l* 79.5%

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

      2. unpow2 79.5%

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

      3. unpow2 79.5%

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

      4. unpow2 79.5%

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

   9. Simplified 79.5%

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

   if 1.8e140 < 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%

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

   4. Taylor expanded in Om around 0 82.3%

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

      1. associate-*r* 82.3%

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

      2. unpow2 82.3%

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

      3. unpow2 82.3%

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

      4. hypot-def 82.3%

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

      5. *-commutative 82.3%

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

   6. Simplified 82.3%

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

   7. Taylor expanded in l around inf 84.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot ky}{\frac{Om \cdot Om}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 53.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-24}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(kx \cdot kx\right)}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.9e-24)
   (+ 1.0 (/ (* -0.5 (* kx kx)) (* (/ Om l) (/ Om l))))
   (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.9e-24) {
		tmp = 1.0 + ((-0.5 * (kx * kx)) / ((Om / l) * (Om / l)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 1.9d-24) then
        tmp = 1.0d0 + (((-0.5d0) * (kx * kx)) / ((om / l) * (om / l)))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.9e-24) {
		tmp = 1.0 + ((-0.5 * (kx * kx)) / ((Om / l) * (Om / l)));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.9e-24:
		tmp = 1.0 + ((-0.5 * (kx * kx)) / ((Om / l) * (Om / l)))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.9e-24)
		tmp = Float64(1.0 + Float64(Float64(-0.5 * Float64(kx * kx)) / Float64(Float64(Om / l) * Float64(Om / l))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.9e-24)
		tmp = 1.0 + ((-0.5 * (kx * kx)) / ((Om / l) * (Om / l)));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.9e-24], N[(1.0 + N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] * N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.90000000000000013e-24

   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%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in ky around 0 94.6%

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

   9. Taylor expanded in kx around 0 37.1%

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

       1. associate-/l* 38.2%

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

       2. unpow2 38.2%

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

       3. unpow2 38.2%

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

       4. unpow2 38.2%

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

   11. Simplified 38.2%

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

   12. Taylor expanded in kx around 0 37.1%

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

       1. metadata-eval 37.1%

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

       2. unpow2 37.1%

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

       3. associate-/l* 38.2%

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

       4. unpow2 38.2%

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

       5. unpow2 38.2%

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

       6. times-frac 45.4%

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

       7. times-frac 45.4%

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

       8. *-lft-identity 45.4%

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

   14. Simplified 45.4%

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

   if 1.90000000000000013e-24 < 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%

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

   4. Taylor expanded in Om around 0 68.8%

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

      1. associate-*r* 68.8%

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

      2. unpow2 68.8%

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

      3. unpow2 68.8%

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

      4. hypot-def 68.8%

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

      5. *-commutative 68.8%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in l around inf 74.4%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{-24}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(kx \cdot kx\right)}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 70.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= l 1.6e+32) 1.0 (sqrt 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.6e+32) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (l <= 1.6d+32) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.6e+32) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.6e+32:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.6e+32)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.6e+32)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.6e+32], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5999999999999999e32

   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%

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

   4. Taylor expanded in kx around 0 77.5%

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

      1. associate-*r/ 77.5%

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

      2. associate-*r* 77.5%

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

      3. unpow2 77.5%

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

      4. unpow2 77.5%

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

   6. Simplified 77.5%

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

      1. add-sqr-sqrt 77.5%

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

      2. hypot-1-def 77.5%

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

      3. sqrt-div 77.5%

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

      4. *-commutative 77.5%

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

      5. sqrt-prod 77.6%

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

      6. unpow2 77.6%

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

      7. sqrt-prod 39.6%

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

      8. add-sqr-sqrt 82.7%

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

      9. *-commutative 82.7%

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

      10. sqrt-prod 82.7%

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

      11. sqrt-prod 26.9%

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

      12. add-sqr-sqrt 89.7%

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

      13. metadata-eval 89.7%

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

      14. sqrt-prod 45.4%

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

      15. add-sqr-sqrt 95.1%

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

   8. Applied egg-rr 95.1%

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

   9. Taylor expanded in ky around 0 68.5%

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

   if 1.5999999999999999e32 < 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%

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

   4. Taylor expanded in Om around 0 73.3%

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

      1. associate-*r* 73.3%

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

      2. unpow2 73.3%

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

      3. unpow2 73.3%

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

      4. hypot-def 73.3%

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

      5. *-commutative 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in l around inf 77.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.6 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8: 39.7% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{kx}{Om} \cdot \frac{kx}{Om}\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0 + ((-0.5d0) * ((l * l) * ((kx / om) * (kx / om))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in ky around 0 94.6%

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

9. Taylor expanded in kx around 0 31.6%

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

    1. associate-/l* 32.7%

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

    2. unpow2 32.7%

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

    3. unpow2 32.7%

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

    4. unpow2 32.7%

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

11. Simplified 32.7%

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

12. Taylor expanded in kx around 0 31.6%

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

    1. unpow2 31.6%

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

    2. associate-*l/ 30.5%

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

    3. unpow2 30.5%

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

    4. *-commutative 30.5%

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

    5. unpow2 30.5%

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

    6. times-frac 35.2%

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

14. Simplified 35.2%

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

15. Final simplification 35.2%

    \[\leadsto 1 + -0.5 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{kx}{Om} \cdot \frac{kx}{Om}\right)\right) \]

Alternative 9: 39.2% accurate, 48.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.5 \cdot \left(kx \cdot kx\right)}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = 1.0d0 + (((-0.5d0) * (kx * kx)) / ((om / l) * (om / l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in ky around 0 94.6%

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

9. Taylor expanded in kx around 0 31.6%

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

    1. associate-/l* 32.7%

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

    2. unpow2 32.7%

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

    3. unpow2 32.7%

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

    4. unpow2 32.7%

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

11. Simplified 32.7%

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

12. Taylor expanded in kx around 0 31.6%

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

    1. metadata-eval 31.6%

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

    2. unpow2 31.6%

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

    3. associate-/l* 32.7%

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

    4. unpow2 32.7%

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

    5. unpow2 32.7%

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

    6. times-frac 39.2%

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

    7. times-frac 39.2%

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

    8. *-lft-identity 39.2%

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

14. Simplified 39.2%

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

15. Final simplification 39.2%

    \[\leadsto 1 + \frac{-0.5 \cdot \left(kx \cdot kx\right)}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}} \]

Reproduce

herbie shell --seed 2023275 
(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))))))))))