Average Error: 1.5 → 0.6
Time: 1.2m
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \left(\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}\right), 1\right)}} + \frac{1}{2}}\]
\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, \left(\frac{\ell \cdot \sin kx}{Om} \cdot \frac{\ell \cdot \sin kx}{Om} + \frac{\sin ky \cdot \ell}{Om} \cdot \frac{\sin ky \cdot \ell}{Om}\right), 1\right)}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r5463713 = 1.0;
        double r5463714 = 2.0;
        double r5463715 = r5463713 / r5463714;
        double r5463716 = l;
        double r5463717 = r5463714 * r5463716;
        double r5463718 = Om;
        double r5463719 = r5463717 / r5463718;
        double r5463720 = pow(r5463719, r5463714);
        double r5463721 = kx;
        double r5463722 = sin(r5463721);
        double r5463723 = pow(r5463722, r5463714);
        double r5463724 = ky;
        double r5463725 = sin(r5463724);
        double r5463726 = pow(r5463725, r5463714);
        double r5463727 = r5463723 + r5463726;
        double r5463728 = r5463720 * r5463727;
        double r5463729 = r5463713 + r5463728;
        double r5463730 = sqrt(r5463729);
        double r5463731 = r5463713 / r5463730;
        double r5463732 = r5463713 + r5463731;
        double r5463733 = r5463715 * r5463732;
        double r5463734 = sqrt(r5463733);
        return r5463734;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r5463735 = 0.5;
        double r5463736 = 4.0;
        double r5463737 = l;
        double r5463738 = kx;
        double r5463739 = sin(r5463738);
        double r5463740 = r5463737 * r5463739;
        double r5463741 = Om;
        double r5463742 = r5463740 / r5463741;
        double r5463743 = r5463742 * r5463742;
        double r5463744 = ky;
        double r5463745 = sin(r5463744);
        double r5463746 = r5463745 * r5463737;
        double r5463747 = r5463746 / r5463741;
        double r5463748 = r5463747 * r5463747;
        double r5463749 = r5463743 + r5463748;
        double r5463750 = 1.0;
        double r5463751 = fma(r5463736, r5463749, r5463750);
        double r5463752 = sqrt(r5463751);
        double r5463753 = r5463735 / r5463752;
        double r5463754 = r5463753 + r5463735;
        double r5463755 = sqrt(r5463754);
        return r5463755;
}

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

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

  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

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

  5. Taylor expanded around inf 0.6

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

  6. Final simplification0.6

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

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))))))))))