Average Error: 1.7 → 1.4
Time: 37.8s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{\frac{1}{2}}{\sqrt{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}{\sqrt{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} + \frac{1}{2}}\]
\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{\frac{1}{2}}{\sqrt{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}{\sqrt{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r1675441 = 1.0;
        double r1675442 = 2.0;
        double r1675443 = r1675441 / r1675442;
        double r1675444 = l;
        double r1675445 = r1675442 * r1675444;
        double r1675446 = Om;
        double r1675447 = r1675445 / r1675446;
        double r1675448 = pow(r1675447, r1675442);
        double r1675449 = kx;
        double r1675450 = sin(r1675449);
        double r1675451 = pow(r1675450, r1675442);
        double r1675452 = ky;
        double r1675453 = sin(r1675452);
        double r1675454 = pow(r1675453, r1675442);
        double r1675455 = r1675451 + r1675454;
        double r1675456 = r1675448 * r1675455;
        double r1675457 = r1675441 + r1675456;
        double r1675458 = sqrt(r1675457);
        double r1675459 = r1675441 / r1675458;
        double r1675460 = r1675441 + r1675459;
        double r1675461 = r1675443 * r1675460;
        double r1675462 = sqrt(r1675461);
        return r1675462;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1675463 = 0.5;
        double r1675464 = 2.0;
        double r1675465 = l;
        double r1675466 = r1675464 * r1675465;
        double r1675467 = Om;
        double r1675468 = r1675466 / r1675467;
        double r1675469 = ky;
        double r1675470 = sin(r1675469);
        double r1675471 = r1675470 * r1675470;
        double r1675472 = kx;
        double r1675473 = sin(r1675472);
        double r1675474 = r1675473 * r1675473;
        double r1675475 = r1675471 + r1675474;
        double r1675476 = r1675468 * r1675475;
        double r1675477 = r1675476 * r1675468;
        double r1675478 = 1.0;
        double r1675479 = r1675477 + r1675478;
        double r1675480 = sqrt(r1675479);
        double r1675481 = sqrt(r1675480);
        double r1675482 = r1675463 / r1675481;
        double r1675483 = r1675482 / r1675481;
        double r1675484 = r1675483 + r1675463;
        double r1675485 = sqrt(r1675484);
        return r1675485;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied associate-*r*1.4

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

  6. Applied add-sqr-sqrt1.4

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

  7. Applied sqrt-prod1.4

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

  8. Applied associate-/r*1.4

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

  9. Final simplification1.4

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

Reproduce

herbie shell --seed 2019152 
(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))))))))))