Average Error: 1.1 → 1.1
Time: 9.6s
Precision: binary64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
↓
\[\sqrt{0.5 + \sqrt{\frac{0.25}{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}, 1\right)}}} \]
\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 + \sqrt{\frac{0.25}{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\mathsf{hypot}\left(\sin kx, \sin ky\right)\right)}^{2}, 1\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
   (sqrt
    (/
     0.25
     (fma
      (pow (/ (* 2.0 l) Om) 2.0)
      (pow (hypot (sin kx) (sin ky)) 2.0)
      1.0))))))
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 + sqrt((0.25 / fma(pow(((2.0 * l) / Om), 2.0), pow(hypot(sin(kx), sin(ky)), 2.0), 1.0)))));
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.1

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

  2. Simplified1.1

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

  3. Applied egg-rr1.1

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2022130 
(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))))))))))