Average Error: 1.1 → 0.7
Time: 12.1s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
↓
\[\begin{array}{l} t_0 := {\sin ky}^{2}\\ t_1 := {\sin kx}^{2}\\ \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left(t_1 + t_0\right) \leq \infty:\\ \;\;\;\;\sqrt{0.5 + \sqrt{\frac{0.25}{\mathsf{fma}\left(4 \cdot {\left(\frac{\ell}{Om}\right)}^{2}, {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(\ell, \sqrt{4 \cdot \left(\frac{t_0}{Om \cdot Om} + \frac{t_1}{Om \cdot Om}\right)}, \log \left(1 + \mathsf{expm1}\left(\frac{0.5}{\mathsf{hypot}\left(\frac{\sin ky}{Om}, \frac{\sin kx}{Om}\right)}\right)\right) \cdot \frac{0.5}{\ell}\right)}}\\ \end{array} \]
(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 (sin ky) 2.0)) (t_1 (pow (sin kx) 2.0)))
   (if (<= (* (pow (/ (* 2.0 l) Om) 2.0) (+ t_1 t_0)) INFINITY)
     (sqrt
      (+
       0.5
       (sqrt
        (/
         0.25
         (fma
          (* 4.0 (pow (/ l Om) 2.0))
          (pow (hypot (sin kx) (sin ky)) 2.0)
          1.0)))))
     (sqrt
      (+
       0.5
       (/
        0.5
        (fma
         l
         (sqrt (* 4.0 (+ (/ t_0 (* Om Om)) (/ t_1 (* Om Om)))))
         (*
          (log (+ 1.0 (expm1 (/ 0.5 (hypot (/ (sin ky) Om) (/ (sin kx) Om))))))
          (/ 0.5 l)))))))))
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(sin(ky), 2.0);
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if ((pow(((2.0 * l) / Om), 2.0) * (t_1 + t_0)) <= ((double) INFINITY)) {
		tmp = sqrt((0.5 + sqrt((0.25 / fma((4.0 * pow((l / Om), 2.0)), pow(hypot(sin(kx), sin(ky)), 2.0), 1.0)))));
	} else {
		tmp = sqrt((0.5 + (0.5 / fma(l, sqrt((4.0 * ((t_0 / (Om * Om)) + (t_1 / (Om * Om))))), (log((1.0 + expm1((0.5 / hypot((sin(ky) / Om), (sin(kx) / Om)))))) * (0.5 / l))))));
	}
	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 = sin(ky) ^ 2.0
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64(t_1 + t_0)) <= Inf)
		tmp = sqrt(Float64(0.5 + sqrt(Float64(0.25 / fma(Float64(4.0 * (Float64(l / Om) ^ 2.0)), (hypot(sin(kx), sin(ky)) ^ 2.0), 1.0)))));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / fma(l, sqrt(Float64(4.0 * Float64(Float64(t_0 / Float64(Om * Om)) + Float64(t_1 / Float64(Om * Om))))), Float64(log(Float64(1.0 + expm1(Float64(0.5 / hypot(Float64(sin(ky) / Om), Float64(sin(kx) / Om)))))) * Float64(0.5 / l))))));
	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[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision], Infinity], N[Sqrt[N[(0.5 + N[Sqrt[N[(0.25 / N[(N[(4.0 * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[(l * N[Sqrt[N[(4.0 * N[(N[(t$95$0 / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Log[N[(1.0 + N[(Exp[N[(0.5 / N[Sqrt[N[(N[Sin[ky], $MachinePrecision] / Om), $MachinePrecision] ^ 2 + N[(N[Sin[kx], $MachinePrecision] / Om), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.5 / l), $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 := {\sin ky}^{2}\\
t_1 := {\sin kx}^{2}\\
\mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left(t_1 + t_0\right) \leq \infty:\\
\;\;\;\;\sqrt{0.5 + \sqrt{\frac{0.25}{\mathsf{fma}\left(4 \cdot {\left(\frac{\ell}{Om}\right)}^{2}, {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}, 1\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(\ell, \sqrt{4 \cdot \left(\frac{t_0}{Om \cdot Om} + \frac{t_1}{Om \cdot Om}\right)}, \log \left(1 + \mathsf{expm1}\left(\frac{0.5}{\mathsf{hypot}\left(\frac{\sin ky}{Om}, \frac{\sin kx}{Om}\right)}\right)\right) \cdot \frac{0.5}{\ell}\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

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

    3. Applied egg-rr0.0

      \[\leadsto \sqrt{0.5 + \color{blue}{\sqrt{\frac{0.25}{\mathsf{fma}\left(4 \cdot {\left(\frac{\ell}{Om}\right)}^{2}, {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}, 1\right)}}}} \]

    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

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

    3. Taylor expanded in l around inf 38.5

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

    4. Simplified38.5

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

    5. Applied egg-rr38.5

      \[\leadsto \sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(\ell, \sqrt{4 \cdot \left(\frac{{\sin ky}^{2}}{Om \cdot Om} + \frac{{\sin kx}^{2}}{Om \cdot Om}\right)}, \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{0.5}{\mathsf{hypot}\left(\frac{\sin ky}{Om}, \frac{\sin kx}{Om}\right)}\right)\right)} \cdot \frac{0.5}{\ell}\right)}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq \infty:\\ \;\;\;\;\sqrt{0.5 + \sqrt{\frac{0.25}{\mathsf{fma}\left(4 \cdot {\left(\frac{\ell}{Om}\right)}^{2}, {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}, 1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\mathsf{fma}\left(\ell, \sqrt{4 \cdot \left(\frac{{\sin ky}^{2}}{Om \cdot Om} + \frac{{\sin kx}^{2}}{Om \cdot Om}\right)}, \log \left(1 + \mathsf{expm1}\left(\frac{0.5}{\mathsf{hypot}\left(\frac{\sin ky}{Om}, \frac{\sin kx}{Om}\right)}\right)\right) \cdot \frac{0.5}{\ell}\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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))))))))))