Average Error: 1.6 → 1.6
Time: 15.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(\frac{\frac{1}{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}} + 1\right) \cdot \frac{1}{2}}\]
\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(\frac{\frac{1}{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}} + 1\right) \cdot \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r60589 = 1.0;
        double r60590 = 2.0;
        double r60591 = r60589 / r60590;
        double r60592 = l;
        double r60593 = r60590 * r60592;
        double r60594 = Om;
        double r60595 = r60593 / r60594;
        double r60596 = pow(r60595, r60590);
        double r60597 = kx;
        double r60598 = sin(r60597);
        double r60599 = pow(r60598, r60590);
        double r60600 = ky;
        double r60601 = sin(r60600);
        double r60602 = pow(r60601, r60590);
        double r60603 = r60599 + r60602;
        double r60604 = r60596 * r60603;
        double r60605 = r60589 + r60604;
        double r60606 = sqrt(r60605);
        double r60607 = r60589 / r60606;
        double r60608 = r60589 + r60607;
        double r60609 = r60591 * r60608;
        double r60610 = sqrt(r60609);
        return r60610;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r60611 = 1.0;
        double r60612 = 2.0;
        double r60613 = l;
        double r60614 = r60612 * r60613;
        double r60615 = Om;
        double r60616 = r60614 / r60615;
        double r60617 = pow(r60616, r60612);
        double r60618 = kx;
        double r60619 = sin(r60618);
        double r60620 = pow(r60619, r60612);
        double r60621 = ky;
        double r60622 = sin(r60621);
        double r60623 = pow(r60622, r60612);
        double r60624 = r60620 + r60623;
        double r60625 = fma(r60617, r60624, r60611);
        double r60626 = sqrt(r60625);
        double r60627 = cbrt(r60626);
        double r60628 = r60627 * r60627;
        double r60629 = r60611 / r60628;
        double r60630 = cbrt(r60627);
        double r60631 = r60630 * r60630;
        double r60632 = r60631 * r60630;
        double r60633 = r60629 / r60632;
        double r60634 = r60633 + r60611;
        double r60635 = r60611 / r60612;
        double r60636 = r60634 * r60635;
        double r60637 = sqrt(r60636);
        return r60637;
}

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

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

  4. Applied add-cube-cbrt1.6

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

  5. Applied associate-/r*1.6

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

  7. Applied add-cube-cbrt1.6

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

  8. Final simplification1.6

    \[\leadsto \sqrt{\left(\frac{\frac{1}{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}}{\left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2 \cdot \ell}{Om}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}} + 1\right) \cdot \frac{1}{2}}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))