Average Error: 1.7 → 2.2
Time: 41.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \left(\left(\frac{\ell}{\frac{Om}{\sin ky}}\right)\right) \cdot \left(\left(\frac{\ell}{\frac{Om}{\sin ky}}\right)\right), 1\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \left(\left(\frac{\ell}{\frac{Om}{\sin ky}}\right)\right) \cdot \left(\left(\frac{\ell}{\frac{Om}{\sin ky}}\right)\right), 1\right)}}} + \frac{1}{2}}\]
\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{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \left(\left(\frac{\ell}{\frac{Om}{\sin ky}}\right)\right) \cdot \left(\left(\frac{\ell}{\frac{Om}{\sin ky}}\right)\right), 1\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \left(\left(\frac{\ell}{\frac{Om}{\sin ky}}\right)\right) \cdot \left(\left(\frac{\ell}{\frac{Om}{\sin ky}}\right)\right), 1\right)}}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r1027788 = 1.0;
        double r1027789 = 2.0;
        double r1027790 = r1027788 / r1027789;
        double r1027791 = l;
        double r1027792 = r1027789 * r1027791;
        double r1027793 = Om;
        double r1027794 = r1027792 / r1027793;
        double r1027795 = pow(r1027794, r1027789);
        double r1027796 = kx;
        double r1027797 = sin(r1027796);
        double r1027798 = pow(r1027797, r1027789);
        double r1027799 = ky;
        double r1027800 = sin(r1027799);
        double r1027801 = pow(r1027800, r1027789);
        double r1027802 = r1027798 + r1027801;
        double r1027803 = r1027795 * r1027802;
        double r1027804 = r1027788 + r1027803;
        double r1027805 = sqrt(r1027804);
        double r1027806 = r1027788 / r1027805;
        double r1027807 = r1027788 + r1027806;
        double r1027808 = r1027790 * r1027807;
        double r1027809 = sqrt(r1027808);
        return r1027809;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1027810 = 0.5;
        double r1027811 = 4.0;
        double r1027812 = kx;
        double r1027813 = sin(r1027812);
        double r1027814 = Om;
        double r1027815 = l;
        double r1027816 = r1027814 / r1027815;
        double r1027817 = r1027813 / r1027816;
        double r1027818 = r1027817 * r1027817;
        double r1027819 = ky;
        double r1027820 = sin(r1027819);
        double r1027821 = r1027814 / r1027820;
        double r1027822 = r1027815 / r1027821;
        double r1027823 = /* ERROR: no posit support in C */;
        double r1027824 = /* ERROR: no posit support in C */;
        double r1027825 = r1027824 * r1027824;
        double r1027826 = r1027818 + r1027825;
        double r1027827 = 1.0;
        double r1027828 = fma(r1027811, r1027826, r1027827);
        double r1027829 = sqrt(r1027828);
        double r1027830 = sqrt(r1027829);
        double r1027831 = r1027830 * r1027830;
        double r1027832 = r1027810 / r1027831;
        double r1027833 = r1027832 + r1027810;
        double r1027834 = sqrt(r1027833);
        return r1027834;
}

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

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

  3. Taylor expanded around -inf 16.6

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

  4. Simplified0.6

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

  6. Applied insert-posit163.4

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

  8. Applied insert-posit162.2

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

  10. Applied add-sqr-sqrt2.2

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

  11. Final simplification2.2

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

Reproduce

herbie shell --seed 2019149 +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))))))))))