Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 16.6s

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: 98.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

6. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l/ 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 66.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 7.4 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{0.5 + \frac{Om \cdot 0.25}{\ell \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\sin kx \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 7.4e-165)
   (sqrt (+ 0.5 (/ (* Om 0.25) (* l ky))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (* (sin kx) (/ l Om)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 7.4e-165) {
		tmp = sqrt((0.5 + ((Om * 0.25) / (l * ky))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (sin(kx) * (l / Om)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 7.4e-165) {
		tmp = Math.sqrt((0.5 + ((Om * 0.25) / (l * ky))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * (Math.sin(kx) * (l / Om)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 7.4e-165:
		tmp = math.sqrt((0.5 + ((Om * 0.25) / (l * ky))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * (math.sin(kx) * (l / Om)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 7.4e-165)
		tmp = sqrt(Float64(0.5 + Float64(Float64(Om * 0.25) / Float64(l * ky))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(sin(kx) * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 7.4e-165)
		tmp = sqrt((0.5 + ((Om * 0.25) / (l * ky))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * (sin(kx) * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 7.4e-165], N[Sqrt[N[(0.5 + N[(N[(Om * 0.25), $MachinePrecision] / N[(l * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[Sin[kx], $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 7.40000000000000003e-165

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in Om around 0 56.4%

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

      1. unpow2 56.4%

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

      2. unpow2 56.4%

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

      3. hypot-def 58.0%

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

   7. Simplified 58.0%

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

      1. add-cbrt-cube 58.0%

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

      2. add-sqr-sqrt 58.0%

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

      3. pow1 58.0%

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

      4. pow1/2 58.0%

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

      5. pow-prod-up 58.1%

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

   9. Applied egg-rr 58.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{0.25}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\frac{Om}{\ell}}}\right)}^{1.5}}} \]

   10. Taylor expanded in kx around 0 50.9%

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

       1. associate-*r/ 50.9%

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

       2. *-commutative 50.9%

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

   12. Simplified 50.9%

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

   13. Taylor expanded in ky around 0 52.0%

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

       1. associate-*r/ 52.0%

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

       2. *-commutative 52.0%

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

       3. *-commutative 52.0%

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

   15. Simplified 52.0%

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

   if 7.40000000000000003e-165 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in ky around 0 85.2%

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

      1. associate-/l* 84.3%

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

      2. associate-/r/ 85.0%

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

      3. associate-*l* 85.0%

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

      4. *-commutative 85.0%

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

      5. unpow2 85.0%

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

      6. unpow2 85.0%

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

      7. times-frac 91.4%

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

      8. metadata-eval 91.4%

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

      9. swap-sqr 91.4%

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

      10. associate-*l/ 91.4%

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

      11. associate-*r/ 91.4%

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

      12. associate-*l/ 91.4%

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

      13. associate-*r/ 91.4%

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

      14. unpow2 91.4%

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

      15. swap-sqr 94.2%

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

   7. Simplified 94.2%

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

      1. expm1-log1p-u 94.2%

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

      2. expm1-udef 94.2%

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

      3. *-commutative 94.2%

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

      4. div-inv 94.2%

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

      5. associate-*l/ 94.2%

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

      6. associate-/l* 94.2%

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

   9. Applied egg-rr 94.2%

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

       1. expm1-def 94.2%

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

       2. expm1-log1p 94.2%

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

       3. associate-/r/ 94.2%

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

   11. Simplified 94.2%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 7.4 \cdot 10^{-165}:\\ \;\;\;\;\sqrt{0.5 + \frac{Om \cdot 0.25}{\ell \cdot ky}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\sin kx \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

6. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l/ 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in kx around 0 92.9%

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

    1. expm1-log1p-u 92.3%

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

    2. expm1-udef 92.3%

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

    3. associate-*l/ 92.3%

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

    4. metadata-eval 92.3%

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

    5. *-un-lft-identity 92.3%

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

    6. times-frac 92.3%

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

    7. metadata-eval 92.3%

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

11. Applied egg-rr 92.3%

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

    1. expm1-def 92.3%

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

    2. expm1-log1p 92.9%

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

    3. associate-*l* 92.9%

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

13. Simplified 92.9%

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

14. Final simplification 92.9%

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

Alternative 4: 69.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 7.6e-73)
   1.0
   (if (<= l 5e-29) (sqrt 0.5) (if (<= l 500.0) 1.0 (sqrt 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 7.6e-73) {
		tmp = 1.0;
	} else if (l <= 5e-29) {
		tmp = sqrt(0.5);
	} else if (l <= 500.0) {
		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 <= 7.6d-73) then
        tmp = 1.0d0
    else if (l <= 5d-29) then
        tmp = sqrt(0.5d0)
    else if (l <= 500.0d0) 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 <= 7.6e-73) {
		tmp = 1.0;
	} else if (l <= 5e-29) {
		tmp = Math.sqrt(0.5);
	} else if (l <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 7.6e-73:
		tmp = 1.0
	elif l <= 5e-29:
		tmp = math.sqrt(0.5)
	elif l <= 500.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 7.6e-73)
		tmp = 1.0;
	elseif (l <= 5e-29)
		tmp = sqrt(0.5);
	elseif (l <= 500.0)
		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 <= 7.6e-73)
		tmp = 1.0;
	elseif (l <= 5e-29)
		tmp = sqrt(0.5);
	elseif (l <= 500.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 7.6e-73], 1.0, If[LessEqual[l, 5e-29], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, 500.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 7.6000000000000005e-73 or 4.99999999999999986e-29 < l < 500

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

      1. expm1-log1p-u 98.9%

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

      2. expm1-udef 98.9%

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

   6. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in kx around 0 95.8%

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

       1. expm1-log1p-u 95.3%

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

       2. expm1-udef 95.3%

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

       3. associate-*l/ 95.3%

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

       4. metadata-eval 95.3%

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

       5. *-un-lft-identity 95.3%

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

       6. times-frac 95.3%

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

       7. metadata-eval 95.3%

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

   11. Applied egg-rr 95.3%

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

       1. expm1-def 95.3%

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

       2. expm1-log1p 95.8%

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

       3. associate-*l* 95.8%

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

   13. Simplified 95.8%

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

   14. Taylor expanded in l around 0 69.0%

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

   if 7.6000000000000005e-73 < l < 4.99999999999999986e-29 or 500 < l

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in Om around 0 81.3%

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

      1. unpow2 81.3%

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

      2. unpow2 81.3%

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

      3. hypot-def 82.8%

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

   7. Simplified 82.8%

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

   8. Taylor expanded in l around inf 83.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-73}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.1% 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%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

6. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. associate-*l/ 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in kx around 0 92.9%

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

    1. expm1-log1p-u 92.3%

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

    2. expm1-udef 92.3%

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

    3. associate-*l/ 92.3%

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

    4. metadata-eval 92.3%

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

    5. *-un-lft-identity 92.3%

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

    6. times-frac 92.3%

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

    7. metadata-eval 92.3%

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

11. Applied egg-rr 92.3%

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

    1. expm1-def 92.3%

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

    2. expm1-log1p 92.9%

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

    3. associate-*l* 92.9%

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

13. Simplified 92.9%

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

14. Taylor expanded in l around 0 59.5%

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

15. Final simplification 59.5%

    \[\leadsto 1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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))))))))))