Average Error: 0.9 → 0.0
Time: 20.5s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
↓
\[\sqrt{0.5 + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)}^{3}}\right)\right)} \]
(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
   (expm1
    (log1p
     (cbrt
      (pow
       (/ 0.5 (hypot 1.0 (* 2.0 (* (/ l Om) (hypot (sin kx) (sin ky))))))
       3.0)))))))
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 + expm1(log1p(cbrt(pow((0.5 / hypot(1.0, (2.0 * ((l / Om) * hypot(sin(kx), sin(ky)))))), 3.0))))));
}

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 + Math.expm1(Math.log1p(Math.cbrt(Math.pow((0.5 / Math.hypot(1.0, (2.0 * ((l / Om) * Math.hypot(Math.sin(kx), Math.sin(ky)))))), 3.0))))));
}

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 + expm1(log1p(cbrt((Float64(0.5 / hypot(1.0, Float64(2.0 * Float64(Float64(l / Om) * hypot(sin(kx), sin(ky)))))) ^ 3.0))))))
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[(Exp[N[Log[1 + N[Power[N[Power[N[(0.5 / N[Sqrt[1.0 ^ 2 + N[(2.0 * N[(N[(l / Om), $MachinePrecision] * N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]] - 1), $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 + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)}^{3}}\right)\right)}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.9

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

  2. Simplified0.9

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

  3. Applied egg-rr0.9

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

  4. Applied egg-rr0.0

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

  5. Applied egg-rr0.0

    \[\leadsto \sqrt{0.5 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt[3]{{\left(\frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)}^{3}}}\right)\right)} \]

  6. Final simplification0.0

    \[\leadsto \sqrt{0.5 + \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\frac{0.5}{\mathsf{hypot}\left(1, 2 \cdot \left(\frac{\ell}{Om} \cdot \mathsf{hypot}\left(\sin kx, \sin ky\right)\right)\right)}\right)}^{3}}\right)\right)} \]

Reproduce

herbie shell --seed 2022155 
(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))))))))))