Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 21.1s

Alternatives: 7

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. add-sqr-sqrt 97.3%

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

   2. pow2 97.3%

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

   3. sqrt-prod 97.3%

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

   4. unpow2 97.3%

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

   5. sqrt-prod 56.4%

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

   6. add-sqr-sqrt 98.4%

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

   7. associate-/l* 98.4%

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

   8. unpow2 98.4%

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

   9. unpow2 98.4%

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

   10. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. add-sqr-sqrt 97.3%

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

   2. hypot-1-def 97.3%

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

   3. sqrt-prod 97.3%

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

   4. unpow2 97.3%

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

   5. sqrt-prod 56.4%

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

   6. add-sqr-sqrt 98.3%

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

   7. associate-/l* 98.3%

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

   8. unpow2 98.3%

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

   9. unpow2 98.3%

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

   10. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. *-un-lft-identity 100.0%

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

   2. un-div-inv 100.0%

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

   3. div-inv 100.0%

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

   4. *-commutative 100.0%

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

   5. associate-*l* 100.0%

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

   6. div-inv 100.0%

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

7. Applied egg-rr 100.0%

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

   1. *-lft-identity 100.0%

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

   2. associate-*l* 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 3: 94.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * (1.0 / sqrt((1.0 + pow((((2.0 * l) / Om) * 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((0.5d0 + (0.5d0 * (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) * sin(ky)) ** 2.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. add-sqr-sqrt 97.3%

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

   2. pow2 97.3%

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

   3. sqrt-prod 97.3%

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

   4. unpow2 97.3%

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

   5. sqrt-prod 56.4%

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

   6. add-sqr-sqrt 98.4%

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

   7. associate-/l* 98.4%

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

   8. unpow2 98.4%

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

   9. unpow2 98.4%

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

   10. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 93.6%

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

7. Final simplification 93.6%

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

Alternative 4: 94.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

   1. add-sqr-sqrt 97.3%

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

   2. hypot-1-def 97.3%

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

   3. sqrt-prod 97.3%

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

   4. unpow2 97.3%

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

   5. sqrt-prod 56.4%

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

   6. add-sqr-sqrt 98.3%

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

   7. associate-/l* 98.3%

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

   8. unpow2 98.3%

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

   9. unpow2 98.3%

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

   10. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in kx around 0 93.6%

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

   1. expm1-log1p-u 93.0%

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

   2. expm1-udef 93.1%

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

   3. un-div-inv 93.1%

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

   4. *-commutative 93.1%

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

   5. *-un-lft-identity 93.1%

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

   6. times-frac 93.1%

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

   7. metadata-eval 93.1%

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

8. Applied egg-rr 93.1%

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

   1. expm1-def 93.0%

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

   2. expm1-log1p 93.6%

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

   3. *-commutative 93.6%

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

   4. associate-*r/ 93.6%

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

   5. associate-*l/ 93.6%

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

   6. associate-*r* 93.6%

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

   7. associate-/l* 93.6%

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

   8. associate-/r* 93.6%

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

10. Simplified 93.6%

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

11. Final simplification 93.6%

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

Alternative 5: 85.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 10^{+66}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, ky \cdot \frac{2}{\frac{Om}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1e+66)
   (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 (* ky (/ 2.0 (/ Om l)))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1e+66) {
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (ky * (2.0 / (Om / l)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1e+66) {
		tmp = Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, (ky * (2.0 / (Om / l)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1e+66:
		tmp = math.sqrt((0.5 + (0.5 / math.hypot(1.0, (ky * (2.0 / (Om / l)))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1e+66)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, Float64(ky * Float64(2.0 / Float64(Om / l)))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1e+66)
		tmp = sqrt((0.5 + (0.5 / hypot(1.0, (ky * (2.0 / (Om / l)))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1e+66], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(ky * N[(2.0 / N[(Om / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 9.99999999999999945e65

   1. Initial program 96.6%

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

   3. Simplified 96.6%

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

      1. add-sqr-sqrt 96.6%

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

      2. hypot-1-def 96.6%

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

      3. sqrt-prod 96.6%

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

      4. unpow2 96.6%

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

      5. sqrt-prod 54.3%

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

      6. add-sqr-sqrt 98.0%

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

      7. associate-/l* 98.0%

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

      8. unpow2 98.0%

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

      9. unpow2 98.0%

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

      10. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in kx around 0 92.6%

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

   7. Taylor expanded in ky around 0 82.8%

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

      1. expm1-log1p-u 82.8%

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

      2. expm1-udef 82.8%

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

      3. un-div-inv 82.8%

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

      4. associate-/l* 82.8%

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

   9. Applied egg-rr 82.8%

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

       1. expm1-def 82.8%

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

       2. expm1-log1p 82.8%

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

       3. associate-*r/ 82.8%

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

       4. associate-*l/ 82.8%

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

   11. Simplified 82.8%

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

   if 9.99999999999999945e65 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

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

      3. sqrt-prod 100.0%

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

      4. unpow2 100.0%

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

      5. sqrt-prod 65.3%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/l* 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. un-div-inv 100.0%

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

      3. div-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in l around 0 88.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 10^{+66}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, ky \cdot \frac{2}{\frac{Om}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 68.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.15 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 1.15e-22) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.15e-22) {
		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 <= 1.15d-22) 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 <= 1.15e-22) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.15e-22:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.15e-22)
		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 <= 1.15e-22)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.15e-22], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.1499999999999999e-22

   1. Initial program 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in Om around 0 53.6%

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

      1. unpow2 53.6%

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

      2. unpow2 53.6%

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

      3. hypot-def 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in l around inf 63.0%

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

   if 1.1499999999999999e-22 < Om

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-1-def 100.0%

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

      3. sqrt-prod 100.0%

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

      4. unpow2 100.0%

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

      5. sqrt-prod 65.6%

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

      6. add-sqr-sqrt 100.0%

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

      7. associate-/l* 100.0%

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

      8. unpow2 100.0%

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

      9. unpow2 100.0%

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

      10. hypot-def 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. un-div-inv 100.0%

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

      3. div-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

      6. div-inv 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*l* 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in l around 0 82.6%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.15 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 55.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (sqrt 0.5))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(0.5d0)
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(0.5);
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

function code(l, Om, kx, ky)
	return sqrt(0.5)
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(0.5);
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Simplified 97.3%

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

4. Taylor expanded in Om around 0 46.3%

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

   1. unpow2 46.3%

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

   2. unpow2 46.3%

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

   3. hypot-def 48.1%

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

6. Simplified 48.1%

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

7. Taylor expanded in l around inf 56.7%

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

8. Final simplification 56.7%

   \[\leadsto \sqrt{0.5} \]

Reproduce

herbie shell --seed 2023301 
(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))))))))))