Average Error: 1.7 → 1.5
Time: 23.5s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{1}{2}}{\log \left(e^{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \frac{\ell \cdot 2}{Om} \cdot \mathsf{fma}\left(\sin ky, \sin ky, \sin kx \cdot \sin kx\right), 1\right)}}\right)} + \frac{1}{2}}\]
\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{\frac{\frac{1}{2}}{\log \left(e^{\sqrt{\mathsf{fma}\left(\frac{\ell \cdot 2}{Om}, \frac{\ell \cdot 2}{Om} \cdot \mathsf{fma}\left(\sin ky, \sin ky, \sin kx \cdot \sin kx\right), 1\right)}}\right)} + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r528485 = 1.0;
        double r528486 = 2.0;
        double r528487 = r528485 / r528486;
        double r528488 = l;
        double r528489 = r528486 * r528488;
        double r528490 = Om;
        double r528491 = r528489 / r528490;
        double r528492 = pow(r528491, r528486);
        double r528493 = kx;
        double r528494 = sin(r528493);
        double r528495 = pow(r528494, r528486);
        double r528496 = ky;
        double r528497 = sin(r528496);
        double r528498 = pow(r528497, r528486);
        double r528499 = r528495 + r528498;
        double r528500 = r528492 * r528499;
        double r528501 = r528485 + r528500;
        double r528502 = sqrt(r528501);
        double r528503 = r528485 / r528502;
        double r528504 = r528485 + r528503;
        double r528505 = r528487 * r528504;
        double r528506 = sqrt(r528505);
        return r528506;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r528507 = 0.5;
        double r528508 = l;
        double r528509 = 2.0;
        double r528510 = r528508 * r528509;
        double r528511 = Om;
        double r528512 = r528510 / r528511;
        double r528513 = ky;
        double r528514 = sin(r528513);
        double r528515 = kx;
        double r528516 = sin(r528515);
        double r528517 = r528516 * r528516;
        double r528518 = fma(r528514, r528514, r528517);
        double r528519 = r528512 * r528518;
        double r528520 = 1.0;
        double r528521 = fma(r528512, r528519, r528520);
        double r528522 = sqrt(r528521);
        double r528523 = exp(r528522);
        double r528524 = log(r528523);
        double r528525 = r528507 / r528524;
        double r528526 = r528525 + r528507;
        double r528527 = sqrt(r528526);
        return r528527;
}

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

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

  4. Applied *-un-lft-identity1.7

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

  6. Applied add-log-exp1.7

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

  7. Simplified1.5

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

  8. Final simplification1.5

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

Reproduce

herbie shell --seed 2019155 +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))))))))))