Average Error: 1.6 → 0.6
Time: 30.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right), 1\right)}} + \frac{1}{2}}\]
\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 r813088 = 1.0;
        double r813089 = 2.0;
        double r813090 = r813088 / r813089;
        double r813091 = l;
        double r813092 = r813089 * r813091;
        double r813093 = Om;
        double r813094 = r813092 / r813093;
        double r813095 = pow(r813094, r813089);
        double r813096 = kx;
        double r813097 = sin(r813096);
        double r813098 = pow(r813097, r813089);
        double r813099 = ky;
        double r813100 = sin(r813099);
        double r813101 = pow(r813100, r813089);
        double r813102 = r813098 + r813101;
        double r813103 = r813095 * r813102;
        double r813104 = r813088 + r813103;
        double r813105 = sqrt(r813104);
        double r813106 = r813088 / r813105;
        double r813107 = r813088 + r813106;
        double r813108 = r813090 * r813107;
        double r813109 = sqrt(r813108);
        return r813109;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r813110 = 0.5;
        double r813111 = 4.0;
        double r813112 = kx;
        double r813113 = sin(r813112);
        double r813114 = Om;
        double r813115 = l;
        double r813116 = r813114 / r813115;
        double r813117 = r813113 / r813116;
        double r813118 = r813117 * r813117;
        double r813119 = ky;
        double r813120 = sin(r813119);
        double r813121 = r813120 / r813116;
        double r813122 = r813121 * r813121;
        double r813123 = r813118 + r813122;
        double r813124 = 1.0;
        double r813125 = fma(r813111, r813123, r813124);
        double r813126 = sqrt(r813125);
        double r813127 = r813110 / r813126;
        double r813128 = r813127 + r813110;
        double r813129 = sqrt(r813128);
        return r813129;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.6

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

  2. Simplified1.6

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

  3. Taylor expanded around inf 16.4

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019133 +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))))))))))