Average Error: 1.4 → 1.5
Time: 25.6s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(\frac{1}{\log \left(e^{\left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}\right)}\right)}\right)} + 1\right)}\]
\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{1}{2} \cdot \left(\frac{1}{\log \left(e^{\left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\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}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\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}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}} \cdot \left(\sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left({\left(\frac{2}{\frac{Om}{\ell}}\right)}^{2}, {\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}, 1\right)}}\right)}\right)}\right)} + 1\right)}
double f(double l, double Om, double kx, double ky) {
        double r1766416 = 1.0;
        double r1766417 = 2.0;
        double r1766418 = r1766416 / r1766417;
        double r1766419 = l;
        double r1766420 = r1766417 * r1766419;
        double r1766421 = Om;
        double r1766422 = r1766420 / r1766421;
        double r1766423 = pow(r1766422, r1766417);
        double r1766424 = kx;
        double r1766425 = sin(r1766424);
        double r1766426 = pow(r1766425, r1766417);
        double r1766427 = ky;
        double r1766428 = sin(r1766427);
        double r1766429 = pow(r1766428, r1766417);
        double r1766430 = r1766426 + r1766429;
        double r1766431 = r1766423 * r1766430;
        double r1766432 = r1766416 + r1766431;
        double r1766433 = sqrt(r1766432);
        double r1766434 = r1766416 / r1766433;
        double r1766435 = r1766416 + r1766434;
        double r1766436 = r1766418 * r1766435;
        double r1766437 = sqrt(r1766436);
        return r1766437;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1766438 = 1.0;
        double r1766439 = 2.0;
        double r1766440 = r1766438 / r1766439;
        double r1766441 = Om;
        double r1766442 = l;
        double r1766443 = r1766441 / r1766442;
        double r1766444 = r1766439 / r1766443;
        double r1766445 = pow(r1766444, r1766439);
        double r1766446 = kx;
        double r1766447 = sin(r1766446);
        double r1766448 = pow(r1766447, r1766439);
        double r1766449 = ky;
        double r1766450 = sin(r1766449);
        double r1766451 = pow(r1766450, r1766439);
        double r1766452 = r1766448 + r1766451;
        double r1766453 = fma(r1766445, r1766452, r1766438);
        double r1766454 = sqrt(r1766453);
        double r1766455 = cbrt(r1766454);
        double r1766456 = cbrt(r1766455);
        double r1766457 = r1766456 * r1766456;
        double r1766458 = r1766456 * r1766457;
        double r1766459 = r1766455 * r1766455;
        double r1766460 = r1766455 * r1766459;
        double r1766461 = cbrt(r1766460);
        double r1766462 = r1766455 * r1766461;
        double r1766463 = r1766458 * r1766462;
        double r1766464 = exp(r1766463);
        double r1766465 = log(r1766464);
        double r1766466 = r1766438 / r1766465;
        double r1766467 = r1766466 + r1766438;
        double r1766468 = r1766440 * r1766467;
        double r1766469 = sqrt(r1766468);
        return r1766469;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Derivation

  1. Initial program 1.4

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

  2. Simplified1.4

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

  4. Applied add-log-exp1.5

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

  6. Applied add-cube-cbrt1.5

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

  8. Applied add-cube-cbrt1.6

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

  10. Applied add-cube-cbrt1.5

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

  11. Final simplification1.5

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))