Average Error: 1.1 → 0.3
Time: 14.6s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
↓
\[\begin{array}{l} t_0 := {\left(\frac{2 \cdot \ell}{Om}\right)}^{2}\\ t_1 := {\sin kx}^{2} + {\sin ky}^{2}\\ \mathbf{if}\;t_0 \cdot t_1 \leq \infty:\\ \;\;\;\;\sqrt{0.5 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(t_0, t_1, 1\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \ell \cdot \sin kx\\ \sqrt{0.5 + \frac{0.5}{2 \cdot \frac{t_2}{Om} + 0.25 \cdot \frac{Om}{t_2}}} \end{array}\\ \end{array} \]
\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 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{\sqrt{\mathsf{fma}\left(t_0, t_1, 1\right)}}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_2 := \ell \cdot \sin kx\\
\sqrt{0.5 + \frac{0.5}{2 \cdot \frac{t_2}{Om} + 0.25 \cdot \frac{Om}{t_2}}}
\end{array}\\


\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 (log1p (expm1 (/ 0.5 (sqrt (fma t_0 t_1 1.0)))))))
     (let* ((t_2 (* l (sin kx))))
       (sqrt (+ 0.5 (/ 0.5 (+ (* 2.0 (/ t_2 Om)) (* 0.25 (/ Om t_2))))))))))
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 + log1p(expm1(0.5 / sqrt(fma(t_0, t_1, 1.0)))));
	} else {
		double t_2 = l * sin(kx);
		tmp = sqrt(0.5 + (0.5 / ((2.0 * (t_2 / Om)) + (0.25 * (Om / t_2)))));
	}
	return tmp;
}

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

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

    3. Applied log1p-expm1-u_binary640.0

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

    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 ky around 0 13.7

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.5}{2 \cdot \frac{\ell \cdot \sin kx}{Om} + 0.25 \cdot \frac{Om}{\ell \cdot \sin kx}}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right) \leq \infty:\\ \;\;\;\;\sqrt{0.5 + \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{0.5}{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\sin kx}^{2} + {\sin ky}^{2}, 1\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{2 \cdot \frac{\ell \cdot \sin kx}{Om} + 0.25 \cdot \frac{Om}{\ell \cdot \sin kx}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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))))))))))