Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 100.0%

Time: 30.3s

Alternatives: 8

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+ 0.5 (/ 0.5 (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 / 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 / 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 / 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 / hypot(1.0, Float64(2.0 * Float64(Float64(l / Om) * hypot(sin(kx), sin(ky))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 / 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[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l / Om), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

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%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. un-div-inv 100.0%

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

   4. associate-*r* 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 87.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.8 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot \sin kx}{Om}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= ky 2.8e+133)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* l ky) Om))))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* l (sin kx)) Om))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 2.8e+133) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * sin(kx)) / Om))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 2.8e+133) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((l * ky) / Om))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((l * Math.sin(kx)) / Om))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ky <= 2.8e+133:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((l * ky) / Om))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((l * math.sin(kx)) / Om))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (ky <= 2.8e+133)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l * ky) / Om))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l * sin(kx)) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (ky <= 2.8e+133)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * sin(kx)) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[ky, 2.8e+133], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l * N[Sin[kx], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.80000000000000016e133

   1. Initial program 98.6%

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

   3. Simplified 98.6%

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

      1. expm1-log1p-u 98.6%

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

      2. expm1-udef 98.6%

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

   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%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

      4. associate-*r* 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 90.0%

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

   13. Taylor expanded in ky around 0 83.9%

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

       1. *-commutative 83.9%

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

   15. Simplified 83.9%

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

   if 2.80000000000000016e133 < ky

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

      4. associate-*r* 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in ky around 0 87.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 2.8 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot \sin kx}{Om}\right)}}\\ \end{array} \]

Alternative 3: 93.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * sin(ky)) / 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[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

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%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. un-div-inv 100.0%

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

   4. associate-*r* 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in kx around 0 91.4%

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

13. Final simplification 91.4%

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

Alternative 4: 86.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.8 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.8e+143)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* l ky) Om))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.8e+143) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.8e+143) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((l * ky) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.8e+143:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((l * ky) / Om))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.8e+143)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l * ky) / Om))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1.8e+143)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.8e+143], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.8e143

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

      1. expm1-log1p-u 98.7%

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

      2. expm1-udef 98.7%

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

   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%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

      4. associate-*r* 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 91.2%

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

   13. Taylor expanded in ky around 0 82.8%

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

       1. *-commutative 82.8%

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

   15. Simplified 82.8%

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

   if 1.8e143 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

      4. associate-*r* 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 93.2%

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

   13. Taylor expanded in l around 0 90.3%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.8 \cdot 10^{+143}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 69.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 6.2 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{+58} \lor \neg \left(Om \leq 4.8 \cdot 10^{+107}\right) \land Om \leq 8.8 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + \frac{2 \cdot \left(\left(ky \cdot ky\right) \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 6.2e-137)
   (sqrt 0.5)
   (if (or (<= Om 1.6e+58) (and (not (<= Om 4.8e+107)) (<= Om 8.8e+142)))
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (/ (* 2.0 (* (* ky ky) (* l l))) (* Om Om)))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 6.2e-137) {
		tmp = sqrt(0.5);
	} else if ((Om <= 1.6e+58) || (!(Om <= 4.8e+107) && (Om <= 8.8e+142))) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + ((2.0 * ((ky * ky) * (l * l))) / (Om * Om)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 6.2d-137) then
        tmp = sqrt(0.5d0)
    else if ((om <= 1.6d+58) .or. (.not. (om <= 4.8d+107)) .and. (om <= 8.8d+142)) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + ((2.0d0 * ((ky * ky) * (l * l))) / (om * om)))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 6.2e-137) {
		tmp = Math.sqrt(0.5);
	} else if ((Om <= 1.6e+58) || (!(Om <= 4.8e+107) && (Om <= 8.8e+142))) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + ((2.0 * ((ky * ky) * (l * l))) / (Om * Om)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 6.2e-137:
		tmp = math.sqrt(0.5)
	elif (Om <= 1.6e+58) or (not (Om <= 4.8e+107) and (Om <= 8.8e+142)):
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + ((2.0 * ((ky * ky) * (l * l))) / (Om * Om)))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 6.2e-137)
		tmp = sqrt(0.5);
	elseif ((Om <= 1.6e+58) || (!(Om <= 4.8e+107) && (Om <= 8.8e+142)))
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(ky * ky) * Float64(l * l))) / Float64(Om * Om)))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 6.2e-137)
		tmp = sqrt(0.5);
	elseif ((Om <= 1.6e+58) || (~((Om <= 4.8e+107)) && (Om <= 8.8e+142)))
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + ((2.0 * ((ky * ky) * (l * l))) / (Om * Om)))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 6.2e-137], N[Sqrt[0.5], $MachinePrecision], If[Or[LessEqual[Om, 1.6e+58], And[N[Not[LessEqual[Om, 4.8e+107]], $MachinePrecision], LessEqual[Om, 8.8e+142]]], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(N[(2.0 * N[(N[(ky * ky), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 6.19999999999999955e-137

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

      1. expm1-log1p-u 98.2%

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

      2. expm1-udef 98.1%

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

   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%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 61.2%

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

   if 6.19999999999999955e-137 < Om < 1.60000000000000008e58 or 4.8000000000000001e107 < Om < 8.79999999999999947e142

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 93.6%

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

      1. associate-/l* 91.7%

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

      2. unpow2 91.7%

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

      3. unpow2 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in ky around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. unpow2 77.7%

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

      3. unpow2 77.7%

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

      4. unpow2 77.7%

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

   9. Simplified 77.7%

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

   if 1.60000000000000008e58 < Om < 4.8000000000000001e107 or 8.79999999999999947e142 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

      4. associate-*r* 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 95.1%

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

   13. Taylor expanded in l around 0 89.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 6.2 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.6 \cdot 10^{+58} \lor \neg \left(Om \leq 4.8 \cdot 10^{+107}\right) \land Om \leq 8.8 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + \frac{2 \cdot \left(\left(ky \cdot ky\right) \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 68.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 3.5 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{0.25 \cdot \frac{Om}{\ell \cdot ky} + 2 \cdot \frac{\ell \cdot ky}{Om}}}\\ \mathbf{elif}\;Om \leq 4.6 \cdot 10^{+58} \lor \neg \left(Om \leq 4.8 \cdot 10^{+108}\right) \land Om \leq 5.6 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + \frac{2 \cdot \left(\left(ky \cdot ky\right) \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 3.5e-137)
   (sqrt
    (+
     0.5
     (* 0.5 (/ 1.0 (+ (* 0.25 (/ Om (* l ky))) (* 2.0 (/ (* l ky) Om)))))))
   (if (or (<= Om 4.6e+58) (and (not (<= Om 4.8e+108)) (<= Om 5.6e+142)))
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (/ (* 2.0 (* (* ky ky) (* l l))) (* Om Om)))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 3.5e-137) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / ((0.25 * (Om / (l * ky))) + (2.0 * ((l * ky) / Om)))))));
	} else if ((Om <= 4.6e+58) || (!(Om <= 4.8e+108) && (Om <= 5.6e+142))) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + ((2.0 * ((ky * ky) * (l * l))) / (Om * Om)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 3.5d-137) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / ((0.25d0 * (om / (l * ky))) + (2.0d0 * ((l * ky) / om)))))))
    else if ((om <= 4.6d+58) .or. (.not. (om <= 4.8d+108)) .and. (om <= 5.6d+142)) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + ((2.0d0 * ((ky * ky) * (l * l))) / (om * om)))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 3.5e-137) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / ((0.25 * (Om / (l * ky))) + (2.0 * ((l * ky) / Om)))))));
	} else if ((Om <= 4.6e+58) || (!(Om <= 4.8e+108) && (Om <= 5.6e+142))) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + ((2.0 * ((ky * ky) * (l * l))) / (Om * Om)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 3.5e-137:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / ((0.25 * (Om / (l * ky))) + (2.0 * ((l * ky) / Om)))))))
	elif (Om <= 4.6e+58) or (not (Om <= 4.8e+108) and (Om <= 5.6e+142)):
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + ((2.0 * ((ky * ky) * (l * l))) / (Om * Om)))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 3.5e-137)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(Float64(0.25 * Float64(Om / Float64(l * ky))) + Float64(2.0 * Float64(Float64(l * ky) / Om)))))));
	elseif ((Om <= 4.6e+58) || (!(Om <= 4.8e+108) && (Om <= 5.6e+142)))
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(Float64(2.0 * Float64(Float64(ky * ky) * Float64(l * l))) / Float64(Om * Om)))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 3.5e-137)
		tmp = sqrt((0.5 + (0.5 * (1.0 / ((0.25 * (Om / (l * ky))) + (2.0 * ((l * ky) / Om)))))));
	elseif ((Om <= 4.6e+58) || (~((Om <= 4.8e+108)) && (Om <= 5.6e+142)))
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + ((2.0 * ((ky * ky) * (l * l))) / (Om * Om)))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 3.5e-137], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(N[(0.25 * N[(Om / N[(l * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(l * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[Om, 4.6e+58], And[N[Not[LessEqual[Om, 4.8e+108]], $MachinePrecision], LessEqual[Om, 5.6e+142]]], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(N[(2.0 * N[(N[(ky * ky), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 3.5000000000000001e-137

   1. Initial program 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in kx around 0 71.1%

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

      1. associate-/l* 70.4%

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

      2. unpow2 70.4%

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

      3. unpow2 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in ky around 0 61.1%

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

      1. unpow2 61.1%

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

      2. unpow2 61.1%

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

   9. Simplified 61.1%

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

   10. Taylor expanded in l around inf 59.8%

       \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{\color{blue}{0.25 \cdot \frac{Om}{ky \cdot \ell} + 2 \cdot \frac{ky \cdot \ell}{Om}}}} \]

   if 3.5000000000000001e-137 < Om < 4.60000000000000005e58 or 4.80000000000000037e108 < Om < 5.6e142

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in kx around 0 93.6%

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

      1. associate-/l* 91.7%

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

      2. unpow2 91.7%

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

      3. unpow2 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in ky around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. unpow2 77.7%

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

      3. unpow2 77.7%

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

      4. unpow2 77.7%

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

   9. Simplified 77.7%

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

   if 4.60000000000000005e58 < Om < 4.80000000000000037e108 or 5.6e142 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

      4. associate-*r* 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 95.1%

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

   13. Taylor expanded in l around 0 89.0%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 3.5 \cdot 10^{-137}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{0.25 \cdot \frac{Om}{\ell \cdot ky} + 2 \cdot \frac{\ell \cdot ky}{Om}}}\\ \mathbf{elif}\;Om \leq 4.6 \cdot 10^{+58} \lor \neg \left(Om \leq 4.8 \cdot 10^{+108}\right) \land Om \leq 5.6 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + \frac{2 \cdot \left(\left(ky \cdot ky\right) \cdot \left(\ell \cdot \ell\right)\right)}{Om \cdot Om}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 67.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 8.8 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.2 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 1000000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 8.8e-93)
   (sqrt 0.5)
   (if (<= Om 1.2e-77) 1.0 (if (<= Om 1000000000.0) (sqrt 0.5) 1.0))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8.8e-93) {
		tmp = sqrt(0.5);
	} else if (Om <= 1.2e-77) {
		tmp = 1.0;
	} else if (Om <= 1000000000.0) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 8.8d-93) then
        tmp = sqrt(0.5d0)
    else if (om <= 1.2d-77) then
        tmp = 1.0d0
    else if (om <= 1000000000.0d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8.8e-93) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 1.2e-77) {
		tmp = 1.0;
	} else if (Om <= 1000000000.0) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 8.8e-93:
		tmp = math.sqrt(0.5)
	elif Om <= 1.2e-77:
		tmp = 1.0
	elif Om <= 1000000000.0:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 8.8e-93)
		tmp = sqrt(0.5);
	elseif (Om <= 1.2e-77)
		tmp = 1.0;
	elseif (Om <= 1000000000.0)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 8.8e-93)
		tmp = sqrt(0.5);
	elseif (Om <= 1.2e-77)
		tmp = 1.0;
	elseif (Om <= 1000000000.0)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 8.8e-93], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 1.2e-77], 1.0, If[LessEqual[Om, 1000000000.0], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 2 regimes

2. if Om < 8.79999999999999983e-93 or 1.19999999999999995e-77 < Om < 1e9

   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%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-udef 98.5%

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

   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%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 62.3%

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

   if 8.79999999999999983e-93 < Om < 1.19999999999999995e-77 or 1e9 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

      4. associate-*r* 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 95.2%

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

   13. Taylor expanded in l around 0 84.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 8.8 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.2 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 1000000000:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 62.4% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

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%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. un-div-inv 100.0%

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

   4. associate-*r* 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in kx around 0 91.4%

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

13. Taylor expanded in l around 0 61.8%

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

14. Final simplification 61.8%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023282 
(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))))))))))