Average Error: 1.7 → 1.7
Time: 56.1s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(\sqrt[3]{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}} \cdot \sqrt[3]{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}}\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}\right) + \frac{1}{2}}\]
double f(double l, double Om, double kx, double ky) {
        double r4567383 = 1.0;
        double r4567384 = 2.0;
        double r4567385 = r4567383 / r4567384;
        double r4567386 = l;
        double r4567387 = r4567384 * r4567386;
        double r4567388 = Om;
        double r4567389 = r4567387 / r4567388;
        double r4567390 = pow(r4567389, r4567384);
        double r4567391 = kx;
        double r4567392 = sin(r4567391);
        double r4567393 = pow(r4567392, r4567384);
        double r4567394 = ky;
        double r4567395 = sin(r4567394);
        double r4567396 = pow(r4567395, r4567384);
        double r4567397 = r4567393 + r4567396;
        double r4567398 = r4567390 * r4567397;
        double r4567399 = r4567383 + r4567398;
        double r4567400 = sqrt(r4567399);
        double r4567401 = r4567383 / r4567400;
        double r4567402 = r4567383 + r4567401;
        double r4567403 = r4567385 * r4567402;
        double r4567404 = sqrt(r4567403);
        return r4567404;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r4567405 = 0.5;
        double r4567406 = sqrt(r4567405);
        double r4567407 = 2.0;
        double r4567408 = l;
        double r4567409 = r4567407 * r4567408;
        double r4567410 = Om;
        double r4567411 = r4567409 / r4567410;
        double r4567412 = r4567411 * r4567411;
        double r4567413 = ky;
        double r4567414 = sin(r4567413);
        double r4567415 = kx;
        double r4567416 = sin(r4567415);
        double r4567417 = r4567416 * r4567416;
        double r4567418 = fma(r4567414, r4567414, r4567417);
        double r4567419 = 1.0;
        double r4567420 = fma(r4567412, r4567418, r4567419);
        double r4567421 = sqrt(r4567420);
        double r4567422 = sqrt(r4567421);
        double r4567423 = r4567406 / r4567422;
        double r4567424 = cbrt(r4567423);
        double r4567425 = r4567424 * r4567424;
        double r4567426 = r4567405 / r4567421;
        double r4567427 = cbrt(r4567426);
        double r4567428 = r4567427 * r4567427;
        double r4567429 = r4567425 * r4567428;
        double r4567430 = r4567429 + r4567405;
        double r4567431 = sqrt(r4567430);
        return r4567431;
}


\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(\sqrt[3]{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}} \cdot \sqrt[3]{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}}\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}\right) + \frac{1}{2}}

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

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

  4. Applied add-cube-cbrt1.7

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

  6. Applied add-sqr-sqrt1.7

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

  7. Applied add-sqr-sqrt1.7

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

  8. Applied times-frac1.7

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

  9. Applied cbrt-prod1.7

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

  10. Final simplification1.7

    \[\leadsto \sqrt{\left(\sqrt[3]{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}} \cdot \sqrt[3]{\frac{\sqrt{\frac{1}{2}}}{\sqrt{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}}\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}\right) + \frac{1}{2}}\]

Reproduce

herbie shell --seed 2019101 +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))))))))))