Average Error: 43.1 → 9.6
Time: 8.5s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.08866278392021945 \cdot 10^{94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 3.92596229546232498 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \le 6.13027204388918591 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 1.4317046531863695 \cdot 10^{69}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -8.08866278392021945 \cdot 10^{94}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le 3.92596229546232498 \cdot 10^{-260}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\

\mathbf{elif}\;t \le 6.13027204388918591 \cdot 10^{-184}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\mathbf{elif}\;t \le 1.4317046531863695 \cdot 10^{69}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\end{array}
double f(double x, double l, double t) {
        double r35455 = 2.0;
        double r35456 = sqrt(r35455);
        double r35457 = t;
        double r35458 = r35456 * r35457;
        double r35459 = x;
        double r35460 = 1.0;
        double r35461 = r35459 + r35460;
        double r35462 = r35459 - r35460;
        double r35463 = r35461 / r35462;
        double r35464 = l;
        double r35465 = r35464 * r35464;
        double r35466 = r35457 * r35457;
        double r35467 = r35455 * r35466;
        double r35468 = r35465 + r35467;
        double r35469 = r35463 * r35468;
        double r35470 = r35469 - r35465;
        double r35471 = sqrt(r35470);
        double r35472 = r35458 / r35471;
        return r35472;
}
↓
double f(double x, double l, double t) {
        double r35473 = t;
        double r35474 = -8.08866278392022e+94;
        bool r35475 = r35473 <= r35474;
        double r35476 = 2.0;
        double r35477 = sqrt(r35476);
        double r35478 = r35477 * r35473;
        double r35479 = 3.0;
        double r35480 = pow(r35477, r35479);
        double r35481 = x;
        double r35482 = 2.0;
        double r35483 = pow(r35481, r35482);
        double r35484 = r35480 * r35483;
        double r35485 = r35473 / r35484;
        double r35486 = r35476 * r35485;
        double r35487 = r35477 * r35481;
        double r35488 = r35473 / r35487;
        double r35489 = r35476 * r35488;
        double r35490 = r35473 * r35477;
        double r35491 = r35489 + r35490;
        double r35492 = r35486 - r35491;
        double r35493 = r35478 / r35492;
        double r35494 = 3.925962295462325e-260;
        bool r35495 = r35473 <= r35494;
        double r35496 = 4.0;
        double r35497 = pow(r35473, r35482);
        double r35498 = r35497 / r35481;
        double r35499 = r35496 * r35498;
        double r35500 = l;
        double r35501 = r35500 / r35481;
        double r35502 = r35500 * r35501;
        double r35503 = r35497 + r35502;
        double r35504 = r35476 * r35503;
        double r35505 = r35499 + r35504;
        double r35506 = sqrt(r35505);
        double r35507 = r35478 / r35506;
        double r35508 = 6.130272043889186e-184;
        bool r35509 = r35473 <= r35508;
        double r35510 = r35477 * r35483;
        double r35511 = r35473 / r35510;
        double r35512 = r35511 + r35488;
        double r35513 = r35476 * r35512;
        double r35514 = r35478 - r35486;
        double r35515 = r35513 + r35514;
        double r35516 = r35478 / r35515;
        double r35517 = 1.4317046531863695e+69;
        bool r35518 = r35473 <= r35517;
        double r35519 = r35518 ? r35507 : r35516;
        double r35520 = r35509 ? r35516 : r35519;
        double r35521 = r35495 ? r35507 : r35520;
        double r35522 = r35475 ? r35493 : r35521;
        return r35522;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -8.08866278392022e+94

    1. Initial program 50.3

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 50.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified50.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{x}\right)}}}\]

    4. Taylor expanded around -inf 3.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}}\]

    if -8.08866278392022e+94 < t < 3.925962295462325e-260 or 6.130272043889186e-184 < t < 1.4317046531863695e+69

    1. Initial program 37.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 16.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 4 \cdot \frac{{t}^{2}}{x}\right)}}}\]

    3. Simplified16.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{x}\right)}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity16.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{\color{blue}{1 \cdot x}}\right)}}\]

    6. Applied add-sqr-sqrt40.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\color{blue}{\left(\sqrt{\ell} \cdot \sqrt{\ell}\right)}}^{2}}{1 \cdot x}\right)}}\]

    7. Applied unpow-prod-down40.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\color{blue}{{\left(\sqrt{\ell}\right)}^{2} \cdot {\left(\sqrt{\ell}\right)}^{2}}}{1 \cdot x}\right)}}\]

    8. Applied times-frac38.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{{\left(\sqrt{\ell}\right)}^{2}}{1} \cdot \frac{{\left(\sqrt{\ell}\right)}^{2}}{x}}\right)}}\]

    9. Simplified38.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\ell} \cdot \frac{{\left(\sqrt{\ell}\right)}^{2}}{x}\right)}}\]

    10. Simplified12.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \color{blue}{\frac{\ell}{x}}\right)}}\]

    if 3.925962295462325e-260 < t < 6.130272043889186e-184 or 1.4317046531863695e+69 < t

    1. Initial program 49.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 8.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]

    3. Simplified8.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.08866278392021945 \cdot 10^{94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 3.92596229546232498 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \le 6.13027204388918591 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 1.4317046531863695 \cdot 10^{69}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))