Average Error: 1.1 → 0.2
Time: 9.9s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
↓
\[\begin{array}{l} t_0 := {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}\\ t_1 := {\sin kx}^{2} + {\sin ky}^{2}\\ \mathbf{if}\;t_0 \cdot t_1 \leq \infty:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(t_0, t_1, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, \frac{\ell \cdot \sin ky}{Om}, 0.25 \cdot \frac{Om}{\ell \cdot ky}\right)}}\\ \end{array} \]
(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
 (let* ((t_0 (pow (/ (* 2.0 l) Om) 2.0))
        (t_1 (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
   (if (<= (* t_0 t_1) INFINITY)
     (sqrt (+ 0.5 (/ 0.5 (sqrt (fma t_0 t_1 1.0)))))
     (sqrt
      (+
       0.5
       (/ 0.5 (fma 2.0 (/ (* l (sin ky)) Om) (* 0.25 (/ Om (* l ky))))))))))
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) {
	double t_0 = pow(((2.0 * l) / Om), 2.0);
	double t_1 = pow(sin(kx), 2.0) + pow(sin(ky), 2.0);
	double tmp;
	if ((t_0 * t_1) <= ((double) INFINITY)) {
		tmp = sqrt((0.5 + (0.5 / sqrt(fma(t_0, t_1, 1.0)))));
	} else {
		tmp = sqrt((0.5 + (0.5 / fma(2.0, ((l * sin(ky)) / Om), (0.25 * (Om / (l * ky)))))));
	}
	return tmp;
}

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)
	t_0 = Float64(Float64(2.0 * l) / Om) ^ 2.0
	t_1 = Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))
	tmp = 0.0
	if (Float64(t_0 * t_1) <= Inf)
		tmp = sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(t_0, t_1, 1.0)))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / fma(2.0, Float64(Float64(l * sin(ky)) / Om), Float64(0.25 * Float64(Om / Float64(l * ky)))))));
	end
	return tmp
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_] := Block[{t$95$0 = N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * t$95$1), $MachinePrecision], Infinity], N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(t$95$0 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[(2.0 * N[(N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision] + N[(0.25 * N[(Om / N[(l * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)}
↓
\begin{array}{l}
t_0 := {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}\\
t_1 := {\sin kx}^{2} + {\sin ky}^{2}\\
\mathbf{if}\;t_0 \cdot t_1 \leq \infty:\\
\;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(t_0, t_1, 1\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, \frac{\ell \cdot \sin ky}{Om}, 0.25 \cdot \frac{Om}{\ell \cdot ky}\right)}}\\


\end{array}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (pow.f64 (/.f64 (*.f64 2 l) Om) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))) < +inf.0

    1. Initial program 0.0

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

    2. Simplified0.0

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

    if +inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 2 l) Om) 2) (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded in Om around 0 45.6

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

    4. Simplified45.6

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\color{blue}{\mathsf{fma}\left(0.5, Om \cdot \sqrt{\frac{0.25}{\left(\ell \cdot \ell\right) \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}, \frac{\sqrt{4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)\right)}}{Om}\right)}}} \]

    5. Taylor expanded in kx around 0 12.3

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{0.25 \cdot \frac{Om}{\ell \cdot \sin ky} + 2 \cdot \frac{\ell \cdot \sin ky}{Om}}}} \]

    6. Simplified12.3

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{\mathsf{fma}\left(2, \frac{\sin ky \cdot \ell}{Om}, 0.25 \cdot \frac{Om}{\sin ky \cdot \ell}\right)}}} \]

    7. Taylor expanded in ky around 0 12.3

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, \frac{\sin ky \cdot \ell}{Om}, 0.25 \cdot \color{blue}{\frac{Om}{\ell \cdot ky}}\right)}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq \infty:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\sin kx}^{2} + {\sin ky}^{2}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(2, \frac{\ell \cdot \sin ky}{Om}, 0.25 \cdot \frac{Om}{\ell \cdot ky}\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022138 
(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))))))))))