Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.2% → 100.0%

Time: 18.0s

Alternatives: 9

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

   1. pow1/2 98.0%

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

   2. pow-flip 98.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-undefine 98.0%

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

6. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. expm1-define 100.0%

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

   3. expm1-log1p-u 100.0%

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

   4. un-div-inv 100.0%

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

   5. div-inv 100.0%

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

   6. clear-num 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*r/ 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-commutative 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 3: 93.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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((0.5d0 + (0.5d0 * ((1.0d0 + (((2.0d0 / (om / l)) * sin(ky)) ** 2.0d0)) ** (-0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

   1. pow1/2 98.0%

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

   2. pow-flip 98.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in kx around 0 92.3%

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

8. Final simplification 92.3%

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

Alternative 4: 87.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ky \cdot \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;ky \leq 3.6 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + t\_0 \cdot t\_0\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (* ky (/ (* 2.0 l) Om))))
   (if (<= ky 3.6e+76)
     (sqrt (+ 0.5 (* 0.5 (pow (+ 1.0 (* t_0 t_0)) -0.5))))
     (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* (sin kx) (* l (/ 2.0 Om))))))))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = ky * ((2.0 * l) / Om);
	double tmp;
	if (ky <= 3.6e+76) {
		tmp = sqrt((0.5 + (0.5 * pow((1.0 + (t_0 * t_0)), -0.5))));
	} else {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (l * (2.0 / Om)))))));
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = ky * ((2.0 * l) / Om);
	double tmp;
	if (ky <= 3.6e+76) {
		tmp = Math.sqrt((0.5 + (0.5 * Math.pow((1.0 + (t_0 * t_0)), -0.5))));
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (Math.sin(kx) * (l * (2.0 / Om)))))));
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = ky * ((2.0 * l) / Om)
	tmp = 0
	if ky <= 3.6e+76:
		tmp = math.sqrt((0.5 + (0.5 * math.pow((1.0 + (t_0 * t_0)), -0.5))))
	else:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (math.sin(kx) * (l * (2.0 / Om)))))))
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = Float64(ky * Float64(Float64(2.0 * l) / Om))
	tmp = 0.0
	if (ky <= 3.6e+76)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * (Float64(1.0 + Float64(t_0 * t_0)) ^ -0.5))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(sin(kx) * Float64(l * Float64(2.0 / Om)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = ky * ((2.0 * l) / Om);
	tmp = 0.0;
	if (ky <= 3.6e+76)
		tmp = sqrt((0.5 + (0.5 * ((1.0 + (t_0 * t_0)) ^ -0.5))));
	else
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (sin(kx) * (l * (2.0 / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(ky * N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ky, 3.6e+76], N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[(1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[Sin[kx], $MachinePrecision] * N[(l * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 3.6000000000000003e76

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. pow1/2 97.6%

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

      2. pow-flip 97.6%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in kx around 0 90.6%

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

   8. Taylor expanded in ky around 0 83.0%

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

      1. unpow2 83.0%

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

      2. *-commutative 83.0%

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

      3. *-commutative 83.0%

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

      4. associate-/r/ 83.0%

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

      5. *-commutative 83.0%

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

      6. associate-*r/ 83.0%

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

      7. associate-/r/ 83.0%

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

      8. *-commutative 83.0%

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

      9. associate-*r/ 83.0%

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

   10. Applied egg-rr 83.0%

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

   if 3.6000000000000003e76 < 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%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. expm1-define 100.0%

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

      3. expm1-log1p-u 100.0%

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

      4. un-div-inv 100.0%

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

      5. div-inv 100.0%

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

      6. clear-num 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in ky around 0 96.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ky \leq 3.6 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + \left(ky \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left(ky \cdot \frac{2 \cdot \ell}{Om}\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \sin kx \cdot \left(\ell \cdot \frac{2}{Om}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-undefine 98.0%

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

6. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. expm1-define 100.0%

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

   3. expm1-log1p-u 100.0%

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

   4. un-div-inv 100.0%

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

   5. div-inv 100.0%

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

   6. clear-num 100.0%

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

8. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*r/ 100.0%

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

   3. associate-*l/ 100.0%

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

   4. *-commutative 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in kx around 0 92.3%

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

    1. associate-*r/ 92.3%

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

13. Simplified 92.3%

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

Alternative 6: 85.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ky \cdot \frac{2 \cdot \ell}{Om}\\ \mathbf{if}\;Om \leq 3 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + t\_0 \cdot t\_0\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (* ky (/ (* 2.0 l) Om))))
   (if (<= Om 3e+116)
     (sqrt (+ 0.5 (* 0.5 (pow (+ 1.0 (* t_0 t_0)) -0.5))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = ky * ((2.0 * l) / Om);
	double tmp;
	if (Om <= 3e+116) {
		tmp = sqrt((0.5 + (0.5 * pow((1.0 + (t_0 * t_0)), -0.5))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ky * ((2.0d0 * l) / om)
    if (om <= 3d+116) then
        tmp = sqrt((0.5d0 + (0.5d0 * ((1.0d0 + (t_0 * t_0)) ** (-0.5d0)))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = ky * ((2.0 * l) / Om);
	double tmp;
	if (Om <= 3e+116) {
		tmp = Math.sqrt((0.5 + (0.5 * Math.pow((1.0 + (t_0 * t_0)), -0.5))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = ky * ((2.0 * l) / Om)
	tmp = 0
	if Om <= 3e+116:
		tmp = math.sqrt((0.5 + (0.5 * math.pow((1.0 + (t_0 * t_0)), -0.5))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = Float64(ky * Float64(Float64(2.0 * l) / Om))
	tmp = 0.0
	if (Om <= 3e+116)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * (Float64(1.0 + Float64(t_0 * t_0)) ^ -0.5))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = ky * ((2.0 * l) / Om);
	tmp = 0.0;
	if (Om <= 3e+116)
		tmp = sqrt((0.5 + (0.5 * ((1.0 + (t_0 * t_0)) ^ -0.5))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(ky * N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[Om, 3e+116], N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[(1.0 + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if Om < 2.9999999999999999e116

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. pow1/2 97.6%

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

      2. pow-flip 97.6%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in kx around 0 91.3%

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

   8. Taylor expanded in ky around 0 83.0%

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

      1. unpow2 83.0%

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

      2. *-commutative 83.0%

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

      3. *-commutative 83.0%

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

      4. associate-/r/ 83.0%

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

      5. *-commutative 83.0%

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

      6. associate-*r/ 83.0%

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

      7. associate-/r/ 83.0%

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

      8. *-commutative 83.0%

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

      9. associate-*r/ 83.0%

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

   10. Applied egg-rr 83.0%

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

   if 2.9999999999999999e116 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in Om around inf 88.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 3 \cdot 10^{+116}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot {\left(1 + \left(ky \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left(ky \cdot \frac{2 \cdot \ell}{Om}\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 10^{+116}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{ky \cdot \left(2 \cdot \ell\right)}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1e+116)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (/ (* ky (* 2.0 l)) Om)))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1e+116) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((ky * (2.0 * l)) / Om)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1e+116) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, ((ky * (2.0 * l)) / Om)))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1e+116:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, ((ky * (2.0 * l)) / Om)))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1e+116)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(Float64(ky * Float64(2.0 * l)) / Om)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1e+116)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, ((ky * (2.0 * l)) / Om)))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1e+116], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(N[(ky * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.00000000000000002e116

   1. Initial program 97.6%

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

   3. Simplified 97.6%

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

      1. *-un-lft-identity 97.6%

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

      2. add-sqr-sqrt 97.6%

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

      3. hypot-1-def 97.6%

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

      4. sqrt-prod 97.6%

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

      5. sqrt-pow1 98.3%

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

      6. metadata-eval 98.3%

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

      7. pow1 98.3%

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

      8. clear-num 98.3%

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

      9. un-div-inv 98.3%

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

      10. unpow2 98.3%

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

      11. unpow2 98.3%

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

      12. hypot-define 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in kx around 0 91.3%

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

   10. Taylor expanded in ky around 0 83.0%

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

       1. associate-*r/ 83.0%

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

       2. *-commutative 83.0%

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

   12. Simplified 83.0%

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

       1. un-div-inv 83.0%

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

       2. associate-*r* 83.0%

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

   14. Applied egg-rr 83.0%

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

   if 1.00000000000000002e116 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in Om around inf 88.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 10^{+116}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{ky \cdot \left(2 \cdot \ell\right)}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 8.8 \cdot 10^{+33}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 8.8e+33) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8.8e+33) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if (om <= 8.8d+33) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 8.8e+33) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 8.8e+33:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 8.8e+33)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 8.8e+33)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 8.8e+33], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 8.79999999999999975e33

   1. Initial program 97.9%

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

   3. Simplified 97.9%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-undefine 97.9%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in Om around 0 63.7%

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

   if 8.79999999999999975e33 < Om

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

      1. expm1-log1p-u 98.5%

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

      2. expm1-undefine 98.5%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in Om around inf 84.2%

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

Alternative 9: 62.4% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

3. Simplified 98.0%

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

   1. expm1-log1p-u 98.0%

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

   2. expm1-undefine 98.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in Om around inf 62.3%

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

Reproduce

herbie shell --seed 2024145 
(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))))))))))