Average Error: 1.7 → 0.6
Time: 23.0s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(\frac{\sin kx \cdot \ell}{Om}, \frac{\sin kx \cdot \ell}{Om}, \frac{\ell \cdot \sin ky}{Om} \cdot \frac{\ell \cdot \sin ky}{Om}\right), 1\right)}} + \frac{1}{2}}\]
\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, \mathsf{fma}\left(\frac{\sin kx \cdot \ell}{Om}, \frac{\sin kx \cdot \ell}{Om}, \frac{\ell \cdot \sin ky}{Om} \cdot \frac{\ell \cdot \sin ky}{Om}\right), 1\right)}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r1377818 = 1.0;
        double r1377819 = 2.0;
        double r1377820 = r1377818 / r1377819;
        double r1377821 = l;
        double r1377822 = r1377819 * r1377821;
        double r1377823 = Om;
        double r1377824 = r1377822 / r1377823;
        double r1377825 = pow(r1377824, r1377819);
        double r1377826 = kx;
        double r1377827 = sin(r1377826);
        double r1377828 = pow(r1377827, r1377819);
        double r1377829 = ky;
        double r1377830 = sin(r1377829);
        double r1377831 = pow(r1377830, r1377819);
        double r1377832 = r1377828 + r1377831;
        double r1377833 = r1377825 * r1377832;
        double r1377834 = r1377818 + r1377833;
        double r1377835 = sqrt(r1377834);
        double r1377836 = r1377818 / r1377835;
        double r1377837 = r1377818 + r1377836;
        double r1377838 = r1377820 * r1377837;
        double r1377839 = sqrt(r1377838);
        return r1377839;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1377840 = 0.5;
        double r1377841 = 4.0;
        double r1377842 = kx;
        double r1377843 = sin(r1377842);
        double r1377844 = l;
        double r1377845 = r1377843 * r1377844;
        double r1377846 = Om;
        double r1377847 = r1377845 / r1377846;
        double r1377848 = ky;
        double r1377849 = sin(r1377848);
        double r1377850 = r1377844 * r1377849;
        double r1377851 = r1377850 / r1377846;
        double r1377852 = r1377851 * r1377851;
        double r1377853 = fma(r1377847, r1377847, r1377852);
        double r1377854 = 1.0;
        double r1377855 = fma(r1377841, r1377853, r1377854);
        double r1377856 = sqrt(r1377855);
        double r1377857 = r1377840 / r1377856;
        double r1377858 = r1377857 + r1377840;
        double r1377859 = sqrt(r1377858);
        return r1377859;
}

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

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

  3. Taylor expanded around inf 16.3

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019163 +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))))))))))