Average Error: 34.6 → 30.1
Time: 47.6s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.391153254400287088556160785530461888584 \cdot 10^{-114} \lor \neg \left(n \le 1.751535212382831785866117409411530213867 \cdot 10^{-224}\right):\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, 2, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \end{array}\]
\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.391153254400287088556160785530461888584 \cdot 10^{-114} \lor \neg \left(n \le 1.751535212382831785866117409411530213867 \cdot 10^{-224}\right):\\
\;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, 2, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r102487 = 2.0;
        double r102488 = n;
        double r102489 = r102487 * r102488;
        double r102490 = U;
        double r102491 = r102489 * r102490;
        double r102492 = t;
        double r102493 = l;
        double r102494 = r102493 * r102493;
        double r102495 = Om;
        double r102496 = r102494 / r102495;
        double r102497 = r102487 * r102496;
        double r102498 = r102492 - r102497;
        double r102499 = r102493 / r102495;
        double r102500 = pow(r102499, r102487);
        double r102501 = r102488 * r102500;
        double r102502 = U_;
        double r102503 = r102490 - r102502;
        double r102504 = r102501 * r102503;
        double r102505 = r102498 - r102504;
        double r102506 = r102491 * r102505;
        double r102507 = sqrt(r102506);
        return r102507;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r102508 = n;
        double r102509 = -2.391153254400287e-114;
        bool r102510 = r102508 <= r102509;
        double r102511 = 1.7515352123828318e-224;
        bool r102512 = r102508 <= r102511;
        double r102513 = !r102512;
        bool r102514 = r102510 || r102513;
        double r102515 = t;
        double r102516 = 2.0;
        double r102517 = l;
        double r102518 = Om;
        double r102519 = r102518 / r102517;
        double r102520 = r102517 / r102519;
        double r102521 = r102517 / r102518;
        double r102522 = 2.0;
        double r102523 = r102516 / r102522;
        double r102524 = pow(r102521, r102523);
        double r102525 = r102508 * r102524;
        double r102526 = U;
        double r102527 = U_;
        double r102528 = r102526 - r102527;
        double r102529 = r102528 * r102524;
        double r102530 = r102525 * r102529;
        double r102531 = fma(r102516, r102520, r102530);
        double r102532 = r102515 - r102531;
        double r102533 = r102516 * r102508;
        double r102534 = r102533 * r102526;
        double r102535 = r102532 * r102534;
        double r102536 = sqrt(r102535);
        double r102537 = r102522 * r102523;
        double r102538 = pow(r102521, r102537);
        double r102539 = r102538 * r102528;
        double r102540 = r102508 * r102539;
        double r102541 = fma(r102520, r102516, r102540);
        double r102542 = r102515 - r102541;
        double r102543 = r102542 * r102533;
        double r102544 = r102543 * r102526;
        double r102545 = sqrt(r102544);
        double r102546 = r102514 ? r102536 : r102545;
        return r102546;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

  1. Split input into 2 regimes

  2. if n < -2.391153254400287e-114 or 1.7515352123828318e-224 < n

    1. Initial program 33.3

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

    2. Simplified33.3

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\]
    3. Using strategy rm

    4. Applied associate-/l*30.5

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
    5. Using strategy rm

    6. Applied sqr-pow30.5

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

    7. Applied associate-*r*29.6

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
    8. Using strategy rm

    9. Applied associate-*l*28.8

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

    10. Simplified28.8

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

    if -2.391153254400287e-114 < n < 1.7515352123828318e-224

    1. Initial program 37.7

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

    2. Simplified37.7

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\]
    3. Using strategy rm

    4. Applied associate-/l*34.7

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
    5. Using strategy rm

    6. Applied sqr-pow34.7

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

    7. Applied associate-*r*33.8

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
    8. Using strategy rm

    9. Applied associate-*l*35.0

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

    10. Simplified35.0

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
    11. Using strategy rm

    12. Applied associate-*r*30.5

      \[\leadsto \sqrt{\color{blue}{\left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}}\]

    13. Simplified33.3

      \[\leadsto \sqrt{\color{blue}{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, 2, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right)} \cdot U}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification30.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.391153254400287088556160785530461888584 \cdot 10^{-114} \lor \neg \left(n \le 1.751535212382831785866117409411530213867 \cdot 10^{-224}\right):\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, 2, n \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))