Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.5% → 100.0%

Time: 16.1s

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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Simplified 97.7%

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

   1. expm1-log1p-u 97.7%

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

   2. expm1-udef 97.6%

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

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%

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

   3. *-commutative 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. un-div-inv 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

    3. hypot-def 99.4%

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

    4. unpow2 99.4%

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

    5. unpow2 99.4%

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

    6. +-commutative 99.4%

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

    7. unpow2 99.4%

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

    8. unpow2 99.4%

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

    9. hypot-def 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 87.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 8.2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= ky 8.2e+94)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* ky l) Om))))))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin kx) (* 2.0 (/ l Om)))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 8.2e+94) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (2.0 * (l / Om)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (ky <= 8.2e+94) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((ky * l) / Om))))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) * (2.0 * (l / Om)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if ky <= 8.2e+94:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((ky * l) / Om))))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.sin(kx) * (2.0 * (l / Om)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (ky <= 8.2e+94)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(ky * l) / Om))))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) * Float64(2.0 * Float64(l / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (ky <= 8.2e+94)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (2.0 * (l / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[ky, 8.2e+94], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] * N[(2.0 * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 8.20000000000000061e94

   1. Initial program 97.2%

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

   3. Simplified 97.2%

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

      1. expm1-log1p-u 97.2%

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

      2. expm1-udef 97.2%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. hypot-def 99.3%

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

       4. unpow2 99.3%

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

       5. unpow2 99.3%

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

       6. +-commutative 99.3%

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

       7. unpow2 99.3%

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

       8. unpow2 99.3%

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

       9. hypot-def 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 93.2%

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

   13. Taylor expanded in ky around 0 86.3%

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

   if 8.20000000000000061e94 < 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%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. hypot-def 100.0%

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

       4. unpow2 100.0%

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

       5. unpow2 100.0%

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

       6. +-commutative 100.0%

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

       7. unpow2 100.0%

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

       8. unpow2 100.0%

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

       9. hypot-def 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in ky around 0 92.1%

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

       1. *-commutative 92.1%

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

       2. *-commutative 92.1%

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

       3. associate-*r/ 92.1%

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

       4. associate-*l* 92.1%

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

       5. *-commutative 92.1%

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

   14. Simplified 92.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 8.2 \cdot 10^{+94}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(2 \cdot \frac{\ell}{Om}\right)\right)}}\\ \end{array} \]

Alternative 3: 93.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Simplified 97.7%

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

   1. expm1-log1p-u 97.7%

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

   2. expm1-udef 97.6%

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

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%

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

   3. *-commutative 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. un-div-inv 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

    3. hypot-def 99.4%

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

    4. unpow2 99.4%

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

    5. unpow2 99.4%

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

    6. +-commutative 99.4%

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

    7. unpow2 99.4%

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

    8. unpow2 99.4%

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

    9. hypot-def 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in kx around 0 94.0%

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

13. Final simplification 94.0%

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

Alternative 4: 73.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 10^{+24}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{ky}}}}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 6.6e-62)
   1.0
   (if (<= l 1e+24)
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* ky l) (* (/ Om l) (/ Om ky)))))))))
     (if (<= l 2.5e+34)
       1.0
       (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* ky l) Om))))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 6.6e-62) {
		tmp = 1.0;
	} else if (l <= 1e+24) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * l) / ((Om / l) * (Om / ky)))))))));
	} else if (l <= 2.5e+34) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 6.6e-62) {
		tmp = 1.0;
	} else if (l <= 1e+24) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * l) / ((Om / l) * (Om / ky)))))))));
	} else if (l <= 2.5e+34) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((ky * l) / Om))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 6.6e-62:
		tmp = 1.0
	elif l <= 1e+24:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * l) / ((Om / l) * (Om / ky)))))))))
	elif l <= 2.5e+34:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((ky * l) / Om))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 6.6e-62)
		tmp = 1.0;
	elseif (l <= 1e+24)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * l) / Float64(Float64(Om / l) * Float64(Om / ky)))))))));
	elseif (l <= 2.5e+34)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(ky * l) / Om))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 6.6e-62)
		tmp = 1.0;
	elseif (l <= 1e+24)
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * l) / ((Om / l) * (Om / ky)))))))));
	elseif (l <= 2.5e+34)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((ky * l) / Om))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 6.6e-62], 1.0, If[LessEqual[l, 1e+24], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(ky * l), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] * N[(Om / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.5e+34], 1.0, N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(ky * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < 6.60000000000000009e-62 or 9.9999999999999998e23 < l < 2.4999999999999999e34

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. expm1-log1p-u 99.0%

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

      2. expm1-udef 99.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. hypot-def 99.7%

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

       4. unpow2 99.7%

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

       5. unpow2 99.7%

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

       6. +-commutative 99.7%

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

       7. unpow2 99.7%

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

       8. unpow2 99.7%

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

       9. hypot-def 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 93.6%

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

   13. Taylor expanded in ky around 0 72.3%

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

   if 6.60000000000000009e-62 < l < 9.9999999999999998e23

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 98.1%

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

      1. associate-*r/ 98.1%

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

      2. associate-*r* 98.1%

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

      3. unpow2 98.1%

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

      4. unpow2 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in ky around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

      4. unpow2 100.0%

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

      5. associate-/l* 100.0%

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

      6. unpow2 100.0%

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

   9. Simplified 100.0%

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

       1. associate-/l* 100.0%

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

       2. frac-times 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 2.4999999999999999e34 < l

   1. Initial program 92.3%

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

   3. Simplified 92.3%

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

      1. expm1-log1p-u 92.3%

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

      2. expm1-udef 92.3%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. hypot-def 98.1%

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

       4. unpow2 98.1%

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

       5. unpow2 98.1%

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

       6. +-commutative 98.1%

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

       7. unpow2 98.1%

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

       8. unpow2 98.1%

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

       9. hypot-def 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 94.5%

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

   13. Taylor expanded in ky around 0 89.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6.6 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 10^{+24}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{ky}}}}\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{ky \cdot \ell}{Om}\right)}}\\ \end{array} \]

Alternative 5: 73.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+24} \lor \neg \left(\ell \leq 4 \cdot 10^{+34}\right):\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{ky}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 1.1e-61)
   1.0
   (if (or (<= l 1.3e+24) (not (<= l 4e+34)))
     (sqrt
      (+
       0.5
       (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (/ (* ky l) (* (/ Om l) (/ Om ky)))))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 1.1e-61) {
		tmp = 1.0;
	} else if ((l <= 1.3e+24) || !(l <= 4e+34)) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * l) / ((Om / l) * (Om / ky)))))))));
	} 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 (l <= 1.1d-61) then
        tmp = 1.0d0
    else if ((l <= 1.3d+24) .or. (.not. (l <= 4d+34))) then
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((ky * l) / ((om / l) * (om / ky)))))))))
    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 (l <= 1.1e-61) {
		tmp = 1.0;
	} else if ((l <= 1.3e+24) || !(l <= 4e+34)) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * l) / ((Om / l) * (Om / ky)))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 1.1e-61:
		tmp = 1.0
	elif (l <= 1.3e+24) or not (l <= 4e+34):
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * l) / ((Om / l) * (Om / ky)))))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 1.1e-61)
		tmp = 1.0;
	elseif ((l <= 1.3e+24) || !(l <= 4e+34))
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(ky * l) / Float64(Float64(Om / l) * Float64(Om / ky)))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 1.1e-61)
		tmp = 1.0;
	elseif ((l <= 1.3e+24) || ~((l <= 4e+34)))
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((ky * l) / ((Om / l) * (Om / ky)))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 1.1e-61], 1.0, If[Or[LessEqual[l, 1.3e+24], N[Not[LessEqual[l, 4e+34]], $MachinePrecision]], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(ky * l), $MachinePrecision] / N[(N[(Om / l), $MachinePrecision] * N[(Om / ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if l < 1.10000000000000004e-61 or 1.2999999999999999e24 < l < 3.99999999999999978e34

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. expm1-log1p-u 99.0%

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

      2. expm1-udef 99.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. hypot-def 99.7%

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

       4. unpow2 99.7%

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

       5. unpow2 99.7%

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

       6. +-commutative 99.7%

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

       7. unpow2 99.7%

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

       8. unpow2 99.7%

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

       9. hypot-def 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 93.6%

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

   13. Taylor expanded in ky around 0 72.3%

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

   if 1.10000000000000004e-61 < l < 1.2999999999999999e24 or 3.99999999999999978e34 < l

   1. Initial program 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in kx around 0 65.6%

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

      1. associate-*r/ 65.6%

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

      2. associate-*r* 65.6%

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

      3. unpow2 65.6%

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

      4. unpow2 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in ky around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. unpow2 61.6%

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

      3. times-frac 70.7%

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

      4. unpow2 70.7%

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

      5. associate-/l* 75.6%

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

      6. unpow2 75.6%

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

   9. Simplified 75.6%

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

       1. associate-/l* 80.6%

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

       2. frac-times 90.7%

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

   11. Applied egg-rr 90.7%

       \[\leadsto \sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \color{blue}{\frac{\ell \cdot ky}{\frac{Om}{\ell} \cdot \frac{Om}{ky}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.1 \cdot 10^{-61}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+24} \lor \neg \left(\ell \leq 4 \cdot 10^{+34}\right):\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \frac{ky \cdot \ell}{\frac{Om}{\ell} \cdot \frac{Om}{ky}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 69.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{ky \cdot ky}{Om}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 7.6e+34)
   1.0
   (sqrt
    (+
     0.5
     (* 0.5 (/ 1.0 (+ 1.0 (* 2.0 (* (/ l (/ Om l)) (/ (* ky ky) Om))))))))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 7.6e+34) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l / (Om / l)) * ((ky * ky) / Om))))))));
	}
	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+34) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + (0.5d0 * (1.0d0 / (1.0d0 + (2.0d0 * ((l / (om / l)) * ((ky * ky) / om))))))))
    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+34) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l / (Om / l)) * ((ky * ky) / Om))))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 7.6e+34:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l / (Om / l)) * ((ky * ky) / Om))))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 7.6e+34)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(l / Float64(Om / l)) * Float64(Float64(ky * ky) / Om))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (l <= 7.6e+34)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.5 * (1.0 / (1.0 + (2.0 * ((l / (Om / l)) * ((ky * ky) / Om))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 7.6e+34], 1.0, N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(l / N[(Om / l), $MachinePrecision]), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.6000000000000003e34

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. expm1-log1p-u 99.0%

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

      2. expm1-udef 99.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. hypot-def 99.7%

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

       4. unpow2 99.7%

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

       5. unpow2 99.7%

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

       6. +-commutative 99.7%

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

       7. unpow2 99.7%

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

       8. unpow2 99.7%

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

       9. hypot-def 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 93.8%

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

   13. Taylor expanded in ky around 0 71.1%

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

   if 7.6000000000000003e34 < l

   1. Initial program 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in kx around 0 60.0%

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

      1. associate-*r/ 60.0%

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

      2. associate-*r* 60.0%

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

      3. unpow2 60.0%

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

      4. unpow2 60.0%

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

   6. Simplified 60.0%

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

   7. Taylor expanded in ky around 0 55.0%

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

      1. *-commutative 55.0%

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

      2. unpow2 55.0%

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

      3. times-frac 65.7%

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

      4. unpow2 65.7%

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

      5. associate-/l* 71.4%

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

      6. unpow2 71.4%

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

   9. Simplified 71.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{+34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + 2 \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \frac{ky \cdot ky}{Om}\right)}}\\ \end{array} \]

Alternative 7: 70.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 10^{+70}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= l 5e+35)
   1.0
   (if (<= l 2e+47) (sqrt 0.5) (if (<= l 1e+70) 1.0 (sqrt 0.5)))))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (l <= 5e+35) {
		tmp = 1.0;
	} else if (l <= 2e+47) {
		tmp = sqrt(0.5);
	} else if (l <= 1e+70) {
		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 <= 5d+35) then
        tmp = 1.0d0
    else if (l <= 2d+47) then
        tmp = sqrt(0.5d0)
    else if (l <= 1d+70) 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 <= 5e+35) {
		tmp = 1.0;
	} else if (l <= 2e+47) {
		tmp = Math.sqrt(0.5);
	} else if (l <= 1e+70) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if l <= 5e+35:
		tmp = 1.0
	elif l <= 2e+47:
		tmp = math.sqrt(0.5)
	elif l <= 1e+70:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (l <= 5e+35)
		tmp = 1.0;
	elseif (l <= 2e+47)
		tmp = sqrt(0.5);
	elseif (l <= 1e+70)
		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 <= 5e+35)
		tmp = 1.0;
	elseif (l <= 2e+47)
		tmp = sqrt(0.5);
	elseif (l <= 1e+70)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[l, 5e+35], 1.0, If[LessEqual[l, 2e+47], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[l, 1e+70], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000021e35 or 2.0000000000000001e47 < l < 1.00000000000000007e70

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

      1. expm1-log1p-u 99.0%

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

      2. expm1-udef 99.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. expm1-def 100.0%

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

       2. expm1-log1p 100.0%

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

       3. hypot-def 99.7%

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

       4. unpow2 99.7%

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

       5. unpow2 99.7%

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

       6. +-commutative 99.7%

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

       7. unpow2 99.7%

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

       8. unpow2 99.7%

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

       9. hypot-def 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in kx around 0 93.9%

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

   13. Taylor expanded in ky around 0 71.7%

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

   if 5.00000000000000021e35 < l < 2.0000000000000001e47 or 1.00000000000000007e70 < l

   1. Initial program 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in Om around 0 72.5%

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

      1. associate-*r* 72.5%

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

      2. unpow2 72.5%

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

      3. unpow2 72.5%

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

      4. hypot-def 74.8%

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

      5. *-commutative 74.8%

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

   6. Simplified 74.8%

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

   7. Taylor expanded in l around inf 77.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;\ell \leq 10^{+70}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

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 97.7%

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

3. Simplified 97.7%

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

   1. expm1-log1p-u 97.7%

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

   2. expm1-udef 97.6%

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

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%

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

   3. *-commutative 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. un-div-inv 100.0%

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

9. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

    3. hypot-def 99.4%

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

    4. unpow2 99.4%

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

    5. unpow2 99.4%

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

    6. +-commutative 99.4%

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

    7. unpow2 99.4%

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

    8. unpow2 99.4%

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

    9. hypot-def 100.0%

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

11. Simplified 100.0%

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

12. Taylor expanded in kx around 0 94.0%

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

13. Taylor expanded in ky around 0 65.5%

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

14. Final simplification 65.5%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023272 
(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))))))))))