Average Error: 43.8 → 9.8
Time: 14.0s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.1715607139623251 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -1.177683054019724 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \end{array}\]
\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 -8.1715607139623251 \cdot 10^{60}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le -1.177683054019724 \cdot 10^{-181}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\

\mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\end{array}
double f(double x, double l, double t) {
        double r40389 = 2.0;
        double r40390 = sqrt(r40389);
        double r40391 = t;
        double r40392 = r40390 * r40391;
        double r40393 = x;
        double r40394 = 1.0;
        double r40395 = r40393 + r40394;
        double r40396 = r40393 - r40394;
        double r40397 = r40395 / r40396;
        double r40398 = l;
        double r40399 = r40398 * r40398;
        double r40400 = r40391 * r40391;
        double r40401 = r40389 * r40400;
        double r40402 = r40399 + r40401;
        double r40403 = r40397 * r40402;
        double r40404 = r40403 - r40399;
        double r40405 = sqrt(r40404);
        double r40406 = r40392 / r40405;
        return r40406;
}
↓
double f(double x, double l, double t) {
        double r40407 = t;
        double r40408 = -8.171560713962325e+60;
        bool r40409 = r40407 <= r40408;
        double r40410 = 2.0;
        double r40411 = sqrt(r40410);
        double r40412 = r40411 * r40407;
        double r40413 = 3.0;
        double r40414 = pow(r40411, r40413);
        double r40415 = x;
        double r40416 = 2.0;
        double r40417 = pow(r40415, r40416);
        double r40418 = r40414 * r40417;
        double r40419 = r40407 / r40418;
        double r40420 = r40410 * r40419;
        double r40421 = r40411 * r40417;
        double r40422 = r40407 / r40421;
        double r40423 = r40411 * r40415;
        double r40424 = r40407 / r40423;
        double r40425 = r40422 + r40424;
        double r40426 = r40410 * r40425;
        double r40427 = r40420 - r40426;
        double r40428 = r40427 - r40412;
        double r40429 = r40412 / r40428;
        double r40430 = -1.1776830540197237e-181;
        bool r40431 = r40407 <= r40430;
        double r40432 = pow(r40407, r40416);
        double r40433 = r40410 * r40432;
        double r40434 = l;
        double r40435 = r40434 / r40415;
        double r40436 = r40434 * r40435;
        double r40437 = r40410 * r40436;
        double r40438 = 4.0;
        double r40439 = r40432 / r40415;
        double r40440 = r40438 * r40439;
        double r40441 = r40437 + r40440;
        double r40442 = r40433 + r40441;
        double r40443 = sqrt(r40442);
        double r40444 = r40412 / r40443;
        double r40445 = -5.07338813022938e-287;
        bool r40446 = r40407 <= r40445;
        double r40447 = 6.769189022626266e+24;
        bool r40448 = r40407 <= r40447;
        double r40449 = r40407 * r40411;
        double r40450 = r40410 * r40424;
        double r40451 = r40449 + r40450;
        double r40452 = r40412 / r40451;
        double r40453 = r40448 ? r40444 : r40452;
        double r40454 = r40446 ? r40429 : r40453;
        double r40455 = r40431 ? r40444 : r40454;
        double r40456 = r40409 ? r40429 : r40455;
        return r40456;
}

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 < -8.171560713962325e+60 or -1.1776830540197237e-181 < t < -5.07338813022938e-287

    1. Initial program 51.2

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

    2. Taylor expanded around -inf 11.5

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

    3. Simplified11.5

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

    if -8.171560713962325e+60 < t < -1.1776830540197237e-181 or -5.07338813022938e-287 < t < 6.769189022626266e+24

    1. Initial program 39.5

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

    2. Taylor expanded around inf 16.3

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

    4. Applied *-un-lft-identity16.3

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

    5. Applied add-sqr-sqrt40.4

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

    6. Applied unpow-prod-down40.4

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

    7. Applied times-frac38.3

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

    8. Simplified38.3

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

    9. Simplified12.0

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

    if 6.769189022626266e+24 < t

    1. Initial program 41.9

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

    2. Taylor expanded around inf 39.8

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

    3. Taylor expanded around inf 4.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.1715607139623251 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le -1.177683054019724 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{elif}\;t \le -5.0733881302293798 \cdot 10^{-287}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - 2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right)\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 6.76918902262626632 \cdot 10^{24}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))