Average Error: 1.7 → 0.7
Time: 33.9s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right), 1\right)}} + \frac{1}{2}}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right), 1\right)}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r970797 = 1.0;
        double r970798 = 2.0;
        double r970799 = r970797 / r970798;
        double r970800 = l;
        double r970801 = r970798 * r970800;
        double r970802 = Om;
        double r970803 = r970801 / r970802;
        double r970804 = pow(r970803, r970798);
        double r970805 = kx;
        double r970806 = sin(r970805);
        double r970807 = pow(r970806, r970798);
        double r970808 = ky;
        double r970809 = sin(r970808);
        double r970810 = pow(r970809, r970798);
        double r970811 = r970807 + r970810;
        double r970812 = r970804 * r970811;
        double r970813 = r970797 + r970812;
        double r970814 = sqrt(r970813);
        double r970815 = r970797 / r970814;
        double r970816 = r970797 + r970815;
        double r970817 = r970799 * r970816;
        double r970818 = sqrt(r970817);
        return r970818;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r970819 = 0.5;
        double r970820 = 4.0;
        double r970821 = kx;
        double r970822 = sin(r970821);
        double r970823 = Om;
        double r970824 = l;
        double r970825 = r970823 / r970824;
        double r970826 = r970822 / r970825;
        double r970827 = r970826 * r970826;
        double r970828 = ky;
        double r970829 = sin(r970828);
        double r970830 = r970829 / r970825;
        double r970831 = r970830 * r970830;
        double r970832 = r970827 + r970831;
        double r970833 = 1.0;
        double r970834 = fma(r970820, r970832, r970833);
        double r970835 = sqrt(r970834);
        double r970836 = r970819 / r970835;
        double r970837 = r970836 + r970819;
        double r970838 = sqrt(r970837);
        return r970838;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.7

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

  2. Simplified1.7

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

  3. Taylor expanded around -inf 16.3

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))