?

Average Error: 1.0 → 0.0
Time: 18.3s
Precision: binary64
Cost: 39424

?

\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
↓
\[\sqrt{0.5 + 0.5 \cdot {\left(1 + {\left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}\right)}^{-0.5}} \]
(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))))))))))
↓
(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (+
   0.5
   (*
    0.5
    (pow
     (+ 1.0 (pow (* (* 2.0 (/ l Om)) (hypot (sin kx) (sin ky))) 2.0))
     -0.5)))))
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)))))))));
}
↓
double code(double l, double Om, double kx, double ky) {
	return sqrt((0.5 + (0.5 * pow((1.0 + pow(((2.0 * (l / Om)) * hypot(sin(kx), sin(ky))), 2.0)), -0.5))));
}

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

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

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

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

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

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

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 1.0

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

  2. Simplified1.0

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

    [Start]1.0

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

    distribute-lft-in [=>]1.0

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

    metadata-eval [=>]1.0

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

    metadata-eval [=>]1.0

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

    metadata-eval [=>]1.0

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

    associate-/l* [=>]1.0

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

  3. Applied egg-rr0.0

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

  4. Applied egg-rr0.0

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

  5. Final simplification0.0

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

Alternatives

Alternative 1
Error1.7
Cost33161
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004 \lor \neg \left(\sin ky \leq 2 \cdot 10^{-235}\right):\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\ell \cdot \frac{\sin kx}{Om}\right)\right)}}\\ \end{array} \]
Alternative 2
Error1.7
Cost33160
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin ky\right)\right)}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\ell \cdot \frac{\sin kx}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin ky\right)}\right) + -1}\\ \end{array} \]
Alternative 3
Error1.7
Cost33160
\[\begin{array}{l} \mathbf{if}\;\sin ky \leq -0.004:\\ \;\;\;\;\sqrt[3]{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin ky\right)\right)}\right)}^{1.5}}\\ \mathbf{elif}\;\sin ky \leq 2 \cdot 10^{-235}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, 2 \cdot \left(\ell \cdot \frac{\sin kx}{Om}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1.5 + \frac{0.5}{\mathsf{hypot}\left(1, \left(2 \cdot \frac{\ell}{Om}\right) \cdot \sin ky\right)}\right) + -1}\\ \end{array} \]
Alternative 4
Error0.0
Cost32832
\[\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}} \]
Alternative 5
Error3.8
Cost19968
\[\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{\ell}{Om} \cdot \left(2 \cdot \sin ky\right)\right)}} \]
Alternative 6
Error9.4
Cost13568
\[\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, \frac{2 \cdot \left(\ell \cdot ky\right)}{Om}\right)}} \]
Alternative 7
Error13.0
Cost8008
\[\begin{array}{l} \mathbf{if}\;Om \leq -3 \cdot 10^{+156}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -2.3 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{1}{1 + \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(ky \cdot ky\right)\right)}{Om \cdot Om}}}\\ \mathbf{elif}\;Om \leq 1.8 \cdot 10^{+48}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 8
Error13.3
Cost7880
\[\begin{array}{l} \mathbf{if}\;Om \leq -3.7 \cdot 10^{+183}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -2.5 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{1 + 2 \cdot \left(\left(ky \cdot ky\right) \cdot \frac{\ell}{\frac{Om \cdot Om}{\ell}}\right)}}\\ \mathbf{elif}\;Om \leq 3.15 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 9
Error14.8
Cost6992
\[\begin{array}{l} \mathbf{if}\;Om \leq -1.75 \cdot 10^{+36}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq -4.2 \cdot 10^{-78}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;Om \leq -9 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;Om \leq 4 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 10
Error24.3
Cost64
\[1 \]

Error

Reproduce?

herbie shell --seed 2023056 
(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))))))))))