Average Error: 1.5 → 0.6
Time: 23.9s
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 r769085 = 1.0;
        double r769086 = 2.0;
        double r769087 = r769085 / r769086;
        double r769088 = l;
        double r769089 = r769086 * r769088;
        double r769090 = Om;
        double r769091 = r769089 / r769090;
        double r769092 = pow(r769091, r769086);
        double r769093 = kx;
        double r769094 = sin(r769093);
        double r769095 = pow(r769094, r769086);
        double r769096 = ky;
        double r769097 = sin(r769096);
        double r769098 = pow(r769097, r769086);
        double r769099 = r769095 + r769098;
        double r769100 = r769092 * r769099;
        double r769101 = r769085 + r769100;
        double r769102 = sqrt(r769101);
        double r769103 = r769085 / r769102;
        double r769104 = r769085 + r769103;
        double r769105 = r769087 * r769104;
        double r769106 = sqrt(r769105);
        return r769106;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r769107 = 0.5;
        double r769108 = 4.0;
        double r769109 = kx;
        double r769110 = sin(r769109);
        double r769111 = Om;
        double r769112 = l;
        double r769113 = r769111 / r769112;
        double r769114 = r769110 / r769113;
        double r769115 = r769114 * r769114;
        double r769116 = ky;
        double r769117 = sin(r769116);
        double r769118 = r769117 / r769113;
        double r769119 = r769118 * r769118;
        double r769120 = r769115 + r769119;
        double r769121 = 1.0;
        double r769122 = fma(r769108, r769120, r769121);
        double r769123 = sqrt(r769122);
        double r769124 = r769107 / r769123;
        double r769125 = r769124 + r769107;
        double r769126 = sqrt(r769125);
        return r769126;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.5

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

  2. Simplified1.5

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

  4. Applied insert-posit162.3

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

  5. Taylor expanded around inf 16.4

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 +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))))))))))