Average Error: 1.6 → 0.6
Time: 24.1s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}}, 1\right)}}}\right) + \frac{1}{2}}\]
\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(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}}, 1\right)}}}\right) + \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r643533 = 1.0;
        double r643534 = 2.0;
        double r643535 = r643533 / r643534;
        double r643536 = l;
        double r643537 = r643534 * r643536;
        double r643538 = Om;
        double r643539 = r643537 / r643538;
        double r643540 = pow(r643539, r643534);
        double r643541 = kx;
        double r643542 = sin(r643541);
        double r643543 = pow(r643542, r643534);
        double r643544 = ky;
        double r643545 = sin(r643544);
        double r643546 = pow(r643545, r643534);
        double r643547 = r643543 + r643546;
        double r643548 = r643540 * r643547;
        double r643549 = r643533 + r643548;
        double r643550 = sqrt(r643549);
        double r643551 = r643533 / r643550;
        double r643552 = r643533 + r643551;
        double r643553 = r643535 * r643552;
        double r643554 = sqrt(r643553);
        return r643554;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r643555 = 0.5;
        double r643556 = 4.0;
        double r643557 = kx;
        double r643558 = sin(r643557);
        double r643559 = Om;
        double r643560 = l;
        double r643561 = r643559 / r643560;
        double r643562 = r643558 / r643561;
        double r643563 = r643562 * r643562;
        double r643564 = ky;
        double r643565 = sin(r643564);
        double r643566 = r643559 / r643565;
        double r643567 = r643560 / r643566;
        double r643568 = r643567 * r643567;
        double r643569 = r643563 + r643568;
        double r643570 = 1.0;
        double r643571 = fma(r643556, r643569, r643570);
        double r643572 = sqrt(r643571);
        double r643573 = r643555 / r643572;
        double r643574 = exp(r643573);
        double r643575 = log(r643574);
        double r643576 = r643575 + r643555;
        double r643577 = sqrt(r643576);
        return r643577;
}

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. Taylor expanded around inf 16.7

    \[\leadsto \sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\color{blue}{4 \cdot \frac{{\ell}^{2} \cdot {\left(\sin ky\right)}^{2}}{{Om}^{2}} + \left(4 \cdot \frac{{\left(\sin kx\right)}^{2} \cdot {\ell}^{2}}{{Om}^{2}} + 1\right)}}}}\right) + \frac{1}{2}}\]

  6. Simplified0.6

    \[\leadsto \sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}} + \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}}, 1\right)}}}}\right) + \frac{1}{2}}\]

  7. Final simplification0.6

    \[\leadsto \sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\mathsf{fma}\left(4, \frac{\sin kx}{\frac{Om}{\ell}} \cdot \frac{\sin kx}{\frac{Om}{\ell}} + \frac{\ell}{\frac{Om}{\sin ky}} \cdot \frac{\ell}{\frac{Om}{\sin ky}}, 1\right)}}}\right) + \frac{1}{2}}\]

Reproduce

herbie shell --seed 2019153 +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))))))))))