Average Error: 1.8 → 0.6
Time: 28.4s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}} + \frac{\sin kx \cdot \ell}{Om} \cdot \frac{\sin kx \cdot \ell}{Om}, 1\right)}} + \frac{1}{2}}\]
\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, \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}} + \frac{\sin kx \cdot \ell}{Om} \cdot \frac{\sin kx \cdot \ell}{Om}, 1\right)}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r902563 = 1.0;
        double r902564 = 2.0;
        double r902565 = r902563 / r902564;
        double r902566 = l;
        double r902567 = r902564 * r902566;
        double r902568 = Om;
        double r902569 = r902567 / r902568;
        double r902570 = pow(r902569, r902564);
        double r902571 = kx;
        double r902572 = sin(r902571);
        double r902573 = pow(r902572, r902564);
        double r902574 = ky;
        double r902575 = sin(r902574);
        double r902576 = pow(r902575, r902564);
        double r902577 = r902573 + r902576;
        double r902578 = r902570 * r902577;
        double r902579 = r902563 + r902578;
        double r902580 = sqrt(r902579);
        double r902581 = r902563 / r902580;
        double r902582 = r902563 + r902581;
        double r902583 = r902565 * r902582;
        double r902584 = sqrt(r902583);
        return r902584;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r902585 = 0.5;
        double r902586 = 4.0;
        double r902587 = l;
        double r902588 = Om;
        double r902589 = ky;
        double r902590 = sin(r902589);
        double r902591 = r902588 / r902590;
        double r902592 = r902587 / r902591;
        double r902593 = r902592 * r902592;
        double r902594 = kx;
        double r902595 = sin(r902594);
        double r902596 = r902595 * r902587;
        double r902597 = r902596 / r902588;
        double r902598 = r902597 * r902597;
        double r902599 = r902593 + r902598;
        double r902600 = 1.0;
        double r902601 = fma(r902586, r902599, r902600);
        double r902602 = sqrt(r902601);
        double r902603 = r902585 / r902602;
        double r902604 = r902603 + r902585;
        double r902605 = sqrt(r902604);
        return r902605;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.8

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

  2. Simplified1.8

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

  3. Taylor expanded around inf 16.1

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019138 +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))))))))))