Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 17.4s

Alternatives: 9

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-*r/ 98.4%

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

   5. metadata-eval 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. fma-def 98.4%

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

   9. *-commutative 98.4%

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

   10. associate-*l/ 98.4%

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

   11. *-commutative 98.4%

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

3. Simplified 98.4%

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

   1. fma-udef 98.4%

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

   2. clear-num 98.4%

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

   3. div-inv 98.4%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-*r/ 98.4%

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

   5. metadata-eval 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. fma-def 98.4%

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

   9. *-commutative 98.4%

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

   10. associate-*l/ 98.4%

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

   11. *-commutative 98.4%

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

3. Simplified 98.4%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 98.9%

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

   5. unpow2 98.9%

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

   6. unpow2 98.9%

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

   7. +-commutative 98.9%

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

   8. unpow2 98.9%

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

   9. unpow2 98.9%

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

   10. hypot-def 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 93.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-*r/ 98.4%

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

   5. metadata-eval 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. fma-def 98.4%

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

   9. *-commutative 98.4%

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

   10. associate-*l/ 98.4%

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

   11. *-commutative 98.4%

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

3. Simplified 98.4%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 98.9%

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

   5. unpow2 98.9%

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

   6. unpow2 98.9%

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

   7. +-commutative 98.9%

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

   8. unpow2 98.9%

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

   9. unpow2 98.9%

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

   10. hypot-def 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 94.9%

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

9. Final simplification 94.9%

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

Alternative 4: 93.5% accurate, 2.3× speedup

Math

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-*r/ 98.4%

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

   5. metadata-eval 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. fma-def 98.4%

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

   9. *-commutative 98.4%

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

   10. associate-*l/ 98.4%

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

   11. *-commutative 98.4%

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

3. Simplified 98.4%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 98.9%

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

   5. unpow2 98.9%

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

   6. unpow2 98.9%

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

   7. +-commutative 98.9%

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

   8. unpow2 98.9%

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

   9. unpow2 98.9%

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

   10. hypot-def 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 94.9%

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

9. Taylor expanded in ky around inf 94.9%

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

    1. associate-*r/ 94.9%

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

    2. associate-*r* 94.9%

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

11. Simplified 94.9%

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

12. Final simplification 94.9%

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

Alternative 5: 84.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 0.00028 \lor \neg \left(Om \leq 2.7 \cdot 10^{+69}\right) \land Om \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (or (<= Om 0.00028) (and (not (<= Om 2.7e+69)) (<= Om 2e+169)))
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* l ky) Om))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((Om <= 0.00028) || (!(Om <= 2.7e+69) && (Om <= 2e+169))) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((Om <= 0.00028) || (!(Om <= 2.7e+69) && (Om <= 2e+169))) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((l * ky) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if (Om <= 0.00028) or (not (Om <= 2.7e+69) and (Om <= 2e+169)):
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((l * ky) / Om))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if ((Om <= 0.00028) || (!(Om <= 2.7e+69) && (Om <= 2e+169)))
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l * ky) / Om))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((Om <= 0.00028) || (~((Om <= 2.7e+69)) && (Om <= 2e+169)))
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[Or[LessEqual[Om, 0.00028], And[N[Not[LessEqual[Om, 2.7e+69]], $MachinePrecision], LessEqual[Om, 2e+169]]], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 2.7999999999999998e-4 or 2.6999999999999998e69 < Om < 1.99999999999999987e169

   1. Initial program 98.1%

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

      1. distribute-lft-in 98.1%

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

      2. metadata-eval 98.1%

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

      3. metadata-eval 98.1%

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

      4. associate-*r/ 98.1%

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

      5. metadata-eval 98.1%

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

      6. metadata-eval 98.1%

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

      7. +-commutative 98.1%

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

      8. fma-def 98.1%

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

      9. *-commutative 98.1%

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

      10. associate-*l/ 98.1%

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

      11. *-commutative 98.1%

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

   3. Simplified 98.1%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 98.7%

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

      5. unpow2 98.7%

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

      6. unpow2 98.7%

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

      7. +-commutative 98.7%

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

      8. unpow2 98.7%

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

      9. unpow2 98.7%

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

      10. hypot-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 94.0%

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

   9. Taylor expanded in ky around 0 84.1%

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

   if 2.7999999999999998e-4 < Om < 2.6999999999999998e69 or 1.99999999999999987e169 < Om

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l/ 100.0%

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

      11. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 100.0%

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

      5. unpow2 100.0%

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

      6. unpow2 100.0%

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

      7. +-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 98.9%

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

   9. Taylor expanded in ky around 0 89.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 0.00028 \lor \neg \left(Om \leq 2.7 \cdot 10^{+69}\right) \land Om \leq 2 \cdot 10^{+169}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 84.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{if}\;Om \leq 10500000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;1 + -0.5 \cdot \left({\sin ky}^{2} \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\\ \mathbf{elif}\;Om \leq 1.12 \cdot 10^{+170}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* 2.0 (/ (* l ky) Om))))))))
   (if (<= Om 10500000.0)
     t_0
     (if (<= Om 8.2e+68)
       (+ 1.0 (* -0.5 (* (pow (sin ky) 2.0) (* (/ l Om) (/ l Om)))))
       (if (<= Om 1.12e+170) t_0 1.0)))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	double tmp;
	if (Om <= 10500000.0) {
		tmp = t_0;
	} else if (Om <= 8.2e+68) {
		tmp = 1.0 + (-0.5 * (pow(sin(ky), 2.0) * ((l / Om) * (l / Om))));
	} else if (Om <= 1.12e+170) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double t_0 = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (2.0 * ((l * ky) / Om))))));
	double tmp;
	if (Om <= 10500000.0) {
		tmp = t_0;
	} else if (Om <= 8.2e+68) {
		tmp = 1.0 + (-0.5 * (Math.pow(Math.sin(ky), 2.0) * ((l / Om) * (l / Om))));
	} else if (Om <= 1.12e+170) {
		tmp = t_0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	t_0 = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (2.0 * ((l * ky) / Om))))))
	tmp = 0
	if Om <= 10500000.0:
		tmp = t_0
	elif Om <= 8.2e+68:
		tmp = 1.0 + (-0.5 * (math.pow(math.sin(ky), 2.0) * ((l / Om) * (l / Om))))
	elif Om <= 1.12e+170:
		tmp = t_0
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	t_0 = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l * ky) / Om))))))
	tmp = 0.0
	if (Om <= 10500000.0)
		tmp = t_0;
	elseif (Om <= 8.2e+68)
		tmp = Float64(1.0 + Float64(-0.5 * Float64((sin(ky) ^ 2.0) * Float64(Float64(l / Om) * Float64(l / Om)))));
	elseif (Om <= 1.12e+170)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	t_0 = sqrt((0.5 + (0.5 / hypot(1.0, (2.0 * ((l * ky) / Om))))));
	tmp = 0.0;
	if (Om <= 10500000.0)
		tmp = t_0;
	elseif (Om <= 8.2e+68)
		tmp = 1.0 + (-0.5 * ((sin(ky) ^ 2.0) * ((l / Om) * (l / Om))));
	elseif (Om <= 1.12e+170)
		tmp = t_0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l * ky), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[Om, 10500000.0], t$95$0, If[LessEqual[Om, 8.2e+68], N[(1.0 + N[(-0.5 * N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[Om, 1.12e+170], t$95$0, 1.0]]]]

Derivation

1. Split input into 3 regimes

2. if Om < 1.05e7 or 8.1999999999999998e68 < Om < 1.1200000000000001e170

   1. Initial program 98.1%

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

      1. distribute-lft-in 98.1%

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

      2. metadata-eval 98.1%

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

      3. metadata-eval 98.1%

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

      4. associate-*r/ 98.1%

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

      5. metadata-eval 98.1%

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

      6. metadata-eval 98.1%

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

      7. +-commutative 98.1%

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

      8. fma-def 98.1%

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

      9. *-commutative 98.1%

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

      10. associate-*l/ 98.1%

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

      11. *-commutative 98.1%

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

   3. Simplified 98.1%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 98.7%

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

      5. unpow2 98.7%

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

      6. unpow2 98.7%

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

      7. +-commutative 98.7%

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

      8. unpow2 98.7%

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

      9. unpow2 98.7%

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

      10. hypot-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 94.2%

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

   9. Taylor expanded in ky around 0 84.4%

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

   if 1.05e7 < Om < 8.1999999999999998e68

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l/ 100.0%

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

      11. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 100.0%

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

      5. unpow2 100.0%

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

      6. unpow2 100.0%

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

      7. +-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 94.6%

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

   9. Taylor expanded in l around 0 82.1%

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

       1. associate-/l* 82.1%

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

       2. unpow2 82.1%

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

       3. unpow2 82.1%

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

   11. Simplified 82.1%

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

   12. Taylor expanded in l around 0 82.1%

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

       1. associate-*l/ 82.1%

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

       2. unpow2 82.1%

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

       3. unpow2 82.1%

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

       4. *-commutative 82.1%

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

       5. times-frac 82.2%

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

   14. Simplified 82.2%

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

   if 1.1200000000000001e170 < Om

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l/ 100.0%

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

      11. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 100.0%

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

      5. unpow2 100.0%

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

      6. unpow2 100.0%

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

      7. +-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 99.9%

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

   9. Taylor expanded in ky around 0 94.2%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 10500000:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{elif}\;Om \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;1 + -0.5 \cdot \left({\sin ky}^{2} \cdot \left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right)\right)\\ \mathbf{elif}\;Om \leq 1.12 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \frac{\ell \cdot ky}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 68.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 7.1 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 6.5 \cdot 10^{-19} \lor \neg \left(Om \leq 6.6 \cdot 10^{+81}\right) \land Om \leq 2.7 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 7.1e-159)
   (sqrt 0.5)
   (if (or (<= Om 6.5e-19) (and (not (<= Om 6.6e+81)) (<= Om 2.7e+138)))
     (sqrt (+ 0.5 (/ 0.5 (+ 1.0 (* 2.0 (/ (* l l) (/ (* Om Om) (* ky ky))))))))
     1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 7.1e-159) {
		tmp = sqrt(0.5);
	} else if ((Om <= 6.5e-19) || (!(Om <= 6.6e+81) && (Om <= 2.7e+138))) {
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * 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 (om <= 7.1d-159) then
        tmp = sqrt(0.5d0)
    else if ((om <= 6.5d-19) .or. (.not. (om <= 6.6d+81)) .and. (om <= 2.7d+138)) then
        tmp = sqrt((0.5d0 + (0.5d0 / (1.0d0 + (2.0d0 * ((l * l) / ((om * om) / (ky * 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 (Om <= 7.1e-159) {
		tmp = Math.sqrt(0.5);
	} else if ((Om <= 6.5e-19) || (!(Om <= 6.6e+81) && (Om <= 2.7e+138))) {
		tmp = Math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky))))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 7.1e-159:
		tmp = math.sqrt(0.5)
	elif (Om <= 6.5e-19) or (not (Om <= 6.6e+81) and (Om <= 2.7e+138)):
		tmp = math.sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky))))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 7.1e-159)
		tmp = sqrt(0.5);
	elseif ((Om <= 6.5e-19) || (!(Om <= 6.6e+81) && (Om <= 2.7e+138)))
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(2.0 * Float64(Float64(l * l) / Float64(Float64(Om * Om) / Float64(ky * ky))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 7.1e-159)
		tmp = sqrt(0.5);
	elseif ((Om <= 6.5e-19) || (~((Om <= 6.6e+81)) && (Om <= 2.7e+138)))
		tmp = sqrt((0.5 + (0.5 / (1.0 + (2.0 * ((l * l) / ((Om * Om) / (ky * ky))))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 7.1e-159], N[Sqrt[0.5], $MachinePrecision], If[Or[LessEqual[Om, 6.5e-19], And[N[Not[LessEqual[Om, 6.6e+81]], $MachinePrecision], LessEqual[Om, 2.7e+138]]], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(2.0 * N[(N[(l * l), $MachinePrecision] / N[(N[(Om * Om), $MachinePrecision] / N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 7.10000000000000024e-159

   1. Initial program 98.1%

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

      1. distribute-rgt-in 98.1%

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

      2. metadata-eval 98.1%

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

      3. metadata-eval 98.1%

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

      4. associate-/l* 98.1%

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

      5. metadata-eval 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in Om around 0 53.4%

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

      1. *-commutative 53.4%

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

      2. associate-*l* 53.4%

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

      3. unpow2 53.4%

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

      4. unpow2 53.4%

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

      5. hypot-def 54.7%

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

   6. Simplified 54.7%

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

   7. Taylor expanded in l around inf 61.9%

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

   if 7.10000000000000024e-159 < Om < 6.5000000000000001e-19 or 6.6e81 < Om < 2.70000000000000009e138

   1. Initial program 97.4%

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

      1. distribute-lft-in 97.4%

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

      2. metadata-eval 97.4%

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

      3. metadata-eval 97.4%

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

      4. associate-*r/ 97.4%

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

      5. metadata-eval 97.4%

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

      6. metadata-eval 97.4%

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

      7. +-commutative 97.4%

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

      8. fma-def 97.4%

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

      9. *-commutative 97.4%

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

      10. associate-*l/ 97.4%

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

      11. *-commutative 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in kx around 0 94.7%

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

      1. associate-/l* 94.7%

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

      2. unpow2 94.7%

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

      3. unpow2 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in ky around 0 89.3%

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

      1. associate-/l* 84.0%

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

      2. unpow2 84.0%

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

      3. unpow2 84.0%

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

      4. unpow2 84.0%

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

   9. Simplified 84.0%

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

   if 6.5000000000000001e-19 < Om < 6.6e81 or 2.70000000000000009e138 < Om

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l/ 100.0%

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

      11. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 100.0%

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

      5. unpow2 100.0%

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

      6. unpow2 100.0%

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

      7. +-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 95.1%

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

   9. Taylor expanded in ky around 0 85.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 7.1 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 6.5 \cdot 10^{-19} \lor \neg \left(Om \leq 6.6 \cdot 10^{+81}\right) \land Om \leq 2.7 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \frac{\ell \cdot \ell}{\frac{Om \cdot Om}{ky \cdot ky}}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 68.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 3.8 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 3.8e-19) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 3.8e-19) {
		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 <= 3.8d-19) 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 <= 3.8e-19) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 3.8e-19:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 3.8e-19)
		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 <= 3.8e-19)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 3.8e-19], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 3.8e-19

   1. Initial program 97.9%

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

      1. distribute-rgt-in 97.9%

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

      2. metadata-eval 97.9%

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

      3. metadata-eval 97.9%

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

      4. associate-/l* 97.9%

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

      5. metadata-eval 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in Om around 0 56.0%

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

      1. *-commutative 56.0%

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

      2. associate-*l* 56.0%

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

      3. unpow2 56.0%

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

      4. unpow2 56.0%

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

      5. hypot-def 57.6%

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

   6. Simplified 57.6%

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

   7. Taylor expanded in l around inf 64.3%

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

   if 3.8e-19 < Om

   1. Initial program 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. fma-def 100.0%

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

      9. *-commutative 100.0%

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

      10. associate-*l/ 100.0%

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

      11. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. *-commutative 100.0%

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

      4. hypot-def 100.0%

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

      5. unpow2 100.0%

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

      6. unpow2 100.0%

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

      7. +-commutative 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in kx around 0 94.5%

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

   9. Taylor expanded in ky around 0 80.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 3.8 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 62.3% 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 98.4%

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

   1. distribute-lft-in 98.4%

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

   2. metadata-eval 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-*r/ 98.4%

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

   5. metadata-eval 98.4%

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

   6. metadata-eval 98.4%

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

   7. +-commutative 98.4%

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

   8. fma-def 98.4%

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

   9. *-commutative 98.4%

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

   10. associate-*l/ 98.4%

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

   11. *-commutative 98.4%

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

3. Simplified 98.4%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 98.9%

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

   5. unpow2 98.9%

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

   6. unpow2 98.9%

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

   7. +-commutative 98.9%

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

   8. unpow2 98.9%

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

   9. unpow2 98.9%

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

   10. hypot-def 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in kx around 0 94.9%

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

9. Taylor expanded in ky around 0 61.0%

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

10. Final simplification 61.0%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023255 
(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))))))))))