Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.3% → 100.0%

Time: 16.4s

Alternatives: 8

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% 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

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

FPCore

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

Python

def code(l, Om, kx, ky):
	return math.sqrt((0.5 + (0.5 * (1.0 / math.sqrt((1.0 + math.pow((math.hypot(math.sin(ky), math.sin(kx)) * (l * (2.0 / Om))), 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(hypot(sin(ky), sin(kx)) * Float64(l * Float64(2.0 / Om))) ^ 2.0)))))))
end

MATLAB

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. pow1 98.8%

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

   2. add-sqr-sqrt 98.8%

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

   3. pow2 98.8%

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

   4. sqrt-prod 98.8%

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

   5. unpow2 98.8%

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

   6. sqrt-prod 51.6%

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

   7. add-sqr-sqrt 98.8%

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

   8. div-inv 98.8%

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

   9. clear-num 98.8%

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

   10. unpow2 98.8%

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

   11. unpow2 98.8%

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

   12. hypot-def 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow1 100.0%

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

   2. *-commutative 100.0%

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

   3. hypot-def 98.8%

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

   4. unpow2 98.8%

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

   5. unpow2 98.8%

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

   6. +-commutative 98.8%

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

   7. unpow2 98.8%

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

   8. unpow2 98.8%

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

   9. hypot-def 100.0%

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

   10. *-commutative 100.0%

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

   11. associate-*l/ 100.0%

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

   12. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-udef 98.8%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

   3. *-commutative 100.0%

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

   4. hypot-def 98.8%

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

   5. unpow2 98.8%

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

   6. unpow2 98.8%

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

   7. +-commutative 98.8%

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

   8. unpow2 98.8%

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

   9. unpow2 98.8%

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

   10. hypot-def 100.0%

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

   11. *-commutative 100.0%

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

   12. associate-*l/ 100.0%

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

   13. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 94.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt((0.5d0 + (0.5d0 * ((1.0d0 + (((l * (sin(ky) * 2.0d0)) / om) ** 2.0d0)) ** (-0.5d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in kx around 0 76.4%

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

   1. associate-*r/ 76.4%

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

   2. *-commutative 76.4%

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

   3. associate-*r* 76.4%

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

   4. unpow2 76.4%

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

   5. unpow2 76.4%

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

6. Simplified 76.4%

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

   1. inv-pow 76.4%

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

   2. sqrt-pow2 76.4%

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

8. Applied egg-rr 94.1%

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

9. Final simplification 94.1%

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

Alternative 4: 94.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in kx around 0 76.4%

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

   1. associate-*r/ 76.4%

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

   2. *-commutative 76.4%

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

   3. associate-*r* 76.4%

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

   4. unpow2 76.4%

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

   5. unpow2 76.4%

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

6. Simplified 76.4%

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

   1. add-sqr-sqrt 76.4%

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

   2. hypot-1-def 76.4%

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

   3. sqrt-div 76.4%

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

   4. *-commutative 76.4%

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

   5. sqrt-prod 77.6%

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

   6. sqrt-prod 38.5%

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

   7. add-sqr-sqrt 86.0%

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

   8. *-commutative 86.0%

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

   9. sqrt-prod 86.0%

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

   10. unpow2 86.0%

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

   11. sqrt-prod 43.8%

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

   12. add-sqr-sqrt 91.9%

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

   13. metadata-eval 91.9%

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

   14. sqrt-prod 42.7%

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

   15. add-sqr-sqrt 94.1%

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

8. Applied egg-rr 94.1%

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

   1. expm1-log1p-u 93.4%

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

   2. expm1-udef 93.4%

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

   3. associate-*l/ 93.4%

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

   4. metadata-eval 93.4%

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

   5. associate-/l* 93.4%

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

10. Applied egg-rr 93.4%

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

    1. expm1-def 93.4%

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

    2. expm1-log1p 94.1%

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

    3. associate-/r/ 94.1%

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

    4. *-commutative 94.1%

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

12. Simplified 94.1%

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

13. Final simplification 94.1%

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

Alternative 5: 86.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(ky \cdot \ell\right)}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 1.4e+145)
   (sqrt (+ 0.5 (* 0.5 (/ 1.0 (hypot 1.0 (/ (* 2.0 (* ky l)) Om))))))
   1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.4e+145) {
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 * (ky * l)) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 1.4e+145) {
		tmp = Math.sqrt((0.5 + (0.5 * (1.0 / Math.hypot(1.0, ((2.0 * (ky * l)) / Om))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 1.4e+145:
		tmp = math.sqrt((0.5 + (0.5 * (1.0 / math.hypot(1.0, ((2.0 * (ky * l)) / Om))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 1.4e+145)
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(1.0 / hypot(1.0, Float64(Float64(2.0 * Float64(ky * l)) / Om))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if (Om <= 1.4e+145)
		tmp = sqrt((0.5 + (0.5 * (1.0 / hypot(1.0, ((2.0 * (ky * l)) / Om))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 1.4e+145], N[Sqrt[N[(0.5 + N[(0.5 * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * N[(ky * l), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 1.3999999999999999e145

   1. Initial program 98.7%

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

      1. distribute-rgt-in 98.7%

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

      2. metadata-eval 98.7%

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

      3. metadata-eval 98.7%

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

      4. associate-/l* 98.7%

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

      5. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in kx around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 77.1%

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

      4. unpow2 77.1%

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

      5. unpow2 77.1%

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

   6. Simplified 77.1%

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

      1. add-sqr-sqrt 77.1%

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

      2. hypot-1-def 77.1%

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

      3. sqrt-div 77.1%

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

      4. *-commutative 77.1%

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

      5. sqrt-prod 78.4%

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

      6. sqrt-prod 38.8%

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

      7. add-sqr-sqrt 85.3%

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

      8. *-commutative 85.3%

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

      9. sqrt-prod 85.3%

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

      10. unpow2 85.3%

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

      11. sqrt-prod 42.0%

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

      12. add-sqr-sqrt 91.8%

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

      13. metadata-eval 91.8%

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

      14. sqrt-prod 37.0%

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

      15. add-sqr-sqrt 93.5%

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

   8. Applied egg-rr 93.5%

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

   9. Taylor expanded in ky around 0 85.4%

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

   if 1.3999999999999999e145 < 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-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 69.6%

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

      1. associate-*r/ 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-*r* 69.6%

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

      4. unpow2 69.6%

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

      5. unpow2 69.6%

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

   6. Simplified 69.6%

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

      1. inv-pow 69.6%

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

      2. sqrt-pow2 69.6%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in l around 0 93.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 1.4 \cdot 10^{+145}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(ky \cdot \ell\right)}{Om}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 68.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 6 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\frac{\ell \cdot \ell}{Om \cdot Om} \cdot \left(ky \cdot ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<= Om 6e-157)
   (sqrt 0.5)
   (if (<= Om 1.4e+101)
     (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 <= 6e-157) {
		tmp = sqrt(0.5);
	} else if (Om <= 1.4e+101) {
		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 <= 6d-157) then
        tmp = sqrt(0.5d0)
    else if (om <= 1.4d+101) 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 <= 6e-157) {
		tmp = Math.sqrt(0.5);
	} else if (Om <= 1.4e+101) {
		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 <= 6e-157:
		tmp = math.sqrt(0.5)
	elif Om <= 1.4e+101:
		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 <= 6e-157)
		tmp = sqrt(0.5);
	elseif (Om <= 1.4e+101)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(1.0 + Float64(2.0 * Float64(Float64(Float64(l * l) / 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 <= 6e-157)
		tmp = sqrt(0.5);
	elseif (Om <= 1.4e+101)
		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, 6e-157], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[Om, 1.4e+101], N[Sqrt[N[(0.5 + N[(0.5 / N[(1.0 + N[(2.0 * N[(N[(N[(l * l), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if Om < 6e-157

   1. Initial program 98.3%

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

      1. distribute-rgt-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. metadata-eval 98.3%

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

      4. associate-/l* 98.3%

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

      5. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in Om around 0 54.3%

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

      1. *-commutative 54.3%

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

      2. associate-*r* 54.3%

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

      3. unpow2 54.3%

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

      4. unpow2 54.3%

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

      5. hypot-def 56.0%

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

      6. associate-*l/ 56.0%

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

      7. associate-*r/ 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in l around inf 63.7%

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

   if 6e-157 < Om < 1.39999999999999991e101

   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-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-commutative 83.9%

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

      3. associate-*r* 83.9%

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

      4. unpow2 83.9%

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

      5. unpow2 83.9%

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

   6. Simplified 83.9%

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

   7. Taylor expanded in ky around 0 73.5%

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

      1. associate-/l* 71.6%

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

      2. unpow2 71.6%

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

      3. unpow2 71.6%

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

      4. unpow2 71.6%

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

   9. Simplified 71.6%

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

       1. associate-*l/ 71.6%

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

       2. metadata-eval 71.6%

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

       3. associate-/r/ 73.1%

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

   11. Applied egg-rr 73.1%

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

   if 1.39999999999999991e101 < 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-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 71.6%

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

      1. associate-*r/ 71.6%

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

      2. *-commutative 71.6%

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

      3. associate-*r* 71.6%

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

      4. unpow2 71.6%

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

      5. unpow2 71.6%

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

   6. Simplified 71.6%

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

      1. inv-pow 71.6%

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

      2. sqrt-pow2 71.6%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in l around 0 89.7%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 6 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq 1.4 \cdot 10^{+101}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\frac{\ell \cdot \ell}{Om \cdot Om} \cdot \left(ky \cdot ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 67.1% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 6.5 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (if (<= Om 6.5e-11) (sqrt 0.5) 1.0))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if (Om <= 6.5e-11) {
		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 <= 6.5d-11) 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 <= 6.5e-11) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if Om <= 6.5e-11:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Om <= 6.5e-11)
		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 <= 6.5e-11)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[Om, 6.5e-11], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if Om < 6.49999999999999953e-11

   1. Initial program 98.5%

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

      1. distribute-rgt-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. metadata-eval 98.5%

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

      4. associate-/l* 98.5%

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

      5. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in Om around 0 55.4%

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

      1. *-commutative 55.4%

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

      2. associate-*r* 55.4%

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

      3. unpow2 55.4%

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

      4. unpow2 55.4%

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

      5. hypot-def 56.9%

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

      6. associate-*l/ 56.9%

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

      7. associate-*r/ 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in l around inf 64.1%

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

   if 6.49999999999999953e-11 < 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-rgt-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l* 100.0%

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

      5. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in kx around 0 80.0%

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

      1. associate-*r/ 80.0%

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

      2. *-commutative 80.0%

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

      3. associate-*r* 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

   6. Simplified 80.0%

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

      1. inv-pow 80.0%

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

      2. sqrt-pow2 80.0%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in l around 0 83.1%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq 6.5 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 63.0% accurate, 722.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

   1. distribute-rgt-in 98.8%

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

   2. metadata-eval 98.8%

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

   3. metadata-eval 98.8%

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

   4. associate-/l* 98.8%

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

   5. metadata-eval 98.8%

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

3. Simplified 98.8%

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

4. Taylor expanded in kx around 0 76.4%

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

   1. associate-*r/ 76.4%

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

   2. *-commutative 76.4%

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

   3. associate-*r* 76.4%

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

   4. unpow2 76.4%

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

   5. unpow2 76.4%

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

6. Simplified 76.4%

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

   1. inv-pow 76.4%

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

   2. sqrt-pow2 76.4%

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

8. Applied egg-rr 94.1%

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

9. Taylor expanded in l around 0 60.1%

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

10. Final simplification 60.1%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023215 
(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))))))))))