Average Error: 1.7 → 1.7
Time: 25.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(1 + \frac{1}{\sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om} \cdot 2\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right) \cdot \frac{1}{2}}\]
\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{\left(1 + \frac{1}{\sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om} \cdot 2\right)}^{2}, {\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}, 1\right)}}\right) \cdot \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r620789 = 1.0;
        double r620790 = 2.0;
        double r620791 = r620789 / r620790;
        double r620792 = l;
        double r620793 = r620790 * r620792;
        double r620794 = Om;
        double r620795 = r620793 / r620794;
        double r620796 = pow(r620795, r620790);
        double r620797 = kx;
        double r620798 = sin(r620797);
        double r620799 = pow(r620798, r620790);
        double r620800 = ky;
        double r620801 = sin(r620800);
        double r620802 = pow(r620801, r620790);
        double r620803 = r620799 + r620802;
        double r620804 = r620796 * r620803;
        double r620805 = r620789 + r620804;
        double r620806 = sqrt(r620805);
        double r620807 = r620789 / r620806;
        double r620808 = r620789 + r620807;
        double r620809 = r620791 * r620808;
        double r620810 = sqrt(r620809);
        return r620810;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r620811 = 1.0;
        double r620812 = l;
        double r620813 = Om;
        double r620814 = r620812 / r620813;
        double r620815 = 2.0;
        double r620816 = r620814 * r620815;
        double r620817 = pow(r620816, r620815);
        double r620818 = ky;
        double r620819 = sin(r620818);
        double r620820 = pow(r620819, r620815);
        double r620821 = kx;
        double r620822 = sin(r620821);
        double r620823 = pow(r620822, r620815);
        double r620824 = r620820 + r620823;
        double r620825 = fma(r620817, r620824, r620811);
        double r620826 = sqrt(r620825);
        double r620827 = r620811 / r620826;
        double r620828 = r620811 + r620827;
        double r620829 = r620811 / r620815;
        double r620830 = r620828 * r620829;
        double r620831 = sqrt(r620830);
        return r620831;
}

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

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

  3. Final simplification1.7

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))