Average Error: 1.6 → 1.6
Time: 27.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om} \cdot \frac{\ell \cdot 2}{Om}, \mathsf{fma}\left(\sin kx, \sin kx, \sin ky \cdot \sin ky\right), 1\right)}}}\right) + \frac{1}{2}}\]
\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{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om} \cdot \frac{\ell \cdot 2}{Om}, \mathsf{fma}\left(\sin kx, \sin kx, \sin ky \cdot \sin ky\right), 1\right)}}}\right) + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r616318 = 1.0;
        double r616319 = 2.0;
        double r616320 = r616318 / r616319;
        double r616321 = l;
        double r616322 = r616319 * r616321;
        double r616323 = Om;
        double r616324 = r616322 / r616323;
        double r616325 = pow(r616324, r616319);
        double r616326 = kx;
        double r616327 = sin(r616326);
        double r616328 = pow(r616327, r616319);
        double r616329 = ky;
        double r616330 = sin(r616329);
        double r616331 = pow(r616330, r616319);
        double r616332 = r616328 + r616331;
        double r616333 = r616325 * r616332;
        double r616334 = r616318 + r616333;
        double r616335 = sqrt(r616334);
        double r616336 = r616318 / r616335;
        double r616337 = r616318 + r616336;
        double r616338 = r616320 * r616337;
        double r616339 = sqrt(r616338);
        return r616339;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r616340 = 0.5;
        double r616341 = l;
        double r616342 = 2.0;
        double r616343 = r616341 * r616342;
        double r616344 = Om;
        double r616345 = r616343 / r616344;
        double r616346 = r616345 * r616345;
        double r616347 = kx;
        double r616348 = sin(r616347);
        double r616349 = ky;
        double r616350 = sin(r616349);
        double r616351 = r616350 * r616350;
        double r616352 = fma(r616348, r616348, r616351);
        double r616353 = 1.0;
        double r616354 = fma(r616346, r616352, r616353);
        double r616355 = sqrt(r616354);
        double r616356 = r616340 / r616355;
        double r616357 = exp(r616356);
        double r616358 = log(r616357);
        double r616359 = r616358 + r616340;
        double r616360 = sqrt(r616359);
        return r616360;
}

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

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

  4. Applied add-log-exp1.6

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

  5. Final simplification1.6

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

Reproduce

herbie shell --seed 2019151 +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))))))))))