Toniolo and Linder, Equation (7)

Average Error: 43.1 → 10.3

Time: 10.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -9.583977798108110900162476286547585661621 \cdot 10^{138}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -6.696367244110020413430084956000329612319 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\left(\frac{\left|{\left(\sqrt[3]{\ell}\right)}^{2}\right|}{\sqrt[3]{x}} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}\right) \cdot \frac{\left|{\left(\sqrt[3]{\ell}\right)}^{2}\right| \cdot {\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{2}}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le -3.088634056538455682354250066621880548059 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 9.909309527651243252484098193013516257504 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\left|\ell\right|}{\sqrt{x}} \cdot \frac{\left|\ell\right|}{\sqrt{x}}\right)}}\\ \mathbf{elif}\;t \le 3.291216122192683495789321781028557839903 \cdot 10^{-130}:\\ \;\;\;\;\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 2.174990866497024471017537763220387342123 \cdot 10^{85}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{\frac{x}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\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}\]

C

double f(double x, double l, double t) {
        double r41459 = 2.0;
        double r41460 = sqrt(r41459);
        double r41461 = t;
        double r41462 = r41460 * r41461;
        double r41463 = x;
        double r41464 = 1.0;
        double r41465 = r41463 + r41464;
        double r41466 = r41463 - r41464;
        double r41467 = r41465 / r41466;
        double r41468 = l;
        double r41469 = r41468 * r41468;
        double r41470 = r41461 * r41461;
        double r41471 = r41459 * r41470;
        double r41472 = r41469 + r41471;
        double r41473 = r41467 * r41472;
        double r41474 = r41473 - r41469;
        double r41475 = sqrt(r41474);
        double r41476 = r41462 / r41475;
        return r41476;
}

↓

double f(double x, double l, double t) {
        double r41477 = t;
        double r41478 = -9.583977798108111e+138;
        bool r41479 = r41477 <= r41478;
        double r41480 = 2.0;
        double r41481 = sqrt(r41480);
        double r41482 = r41481 * r41477;
        double r41483 = 3.0;
        double r41484 = pow(r41481, r41483);
        double r41485 = x;
        double r41486 = 2.0;
        double r41487 = pow(r41485, r41486);
        double r41488 = r41484 * r41487;
        double r41489 = r41477 / r41488;
        double r41490 = r41481 * r41487;
        double r41491 = r41477 / r41490;
        double r41492 = r41489 - r41491;
        double r41493 = r41480 * r41492;
        double r41494 = r41493 - r41482;
        double r41495 = r41481 * r41485;
        double r41496 = r41477 / r41495;
        double r41497 = r41480 * r41496;
        double r41498 = r41494 - r41497;
        double r41499 = r41482 / r41498;
        double r41500 = -6.6963672441100204e-164;
        bool r41501 = r41477 <= r41500;
        double r41502 = 4.0;
        double r41503 = pow(r41477, r41486);
        double r41504 = r41503 / r41485;
        double r41505 = r41502 * r41504;
        double r41506 = l;
        double r41507 = cbrt(r41506);
        double r41508 = pow(r41507, r41486);
        double r41509 = fabs(r41508);
        double r41510 = cbrt(r41485);
        double r41511 = r41509 / r41510;
        double r41512 = 1.0;
        double r41513 = r41512 / r41510;
        double r41514 = r41511 * r41513;
        double r41515 = cbrt(r41507);
        double r41516 = 4.0;
        double r41517 = pow(r41515, r41516);
        double r41518 = r41514 * r41517;
        double r41519 = pow(r41515, r41486);
        double r41520 = r41509 * r41519;
        double r41521 = r41520 / r41510;
        double r41522 = r41518 * r41521;
        double r41523 = r41503 + r41522;
        double r41524 = r41480 * r41523;
        double r41525 = r41505 + r41524;
        double r41526 = sqrt(r41525);
        double r41527 = r41482 / r41526;
        double r41528 = -3.0886340565384557e-240;
        bool r41529 = r41477 <= r41528;
        double r41530 = 9.909309527651243e-270;
        bool r41531 = r41477 <= r41530;
        double r41532 = fabs(r41506);
        double r41533 = sqrt(r41485);
        double r41534 = r41532 / r41533;
        double r41535 = r41534 * r41534;
        double r41536 = r41503 + r41535;
        double r41537 = r41480 * r41536;
        double r41538 = r41505 + r41537;
        double r41539 = sqrt(r41538);
        double r41540 = r41482 / r41539;
        double r41541 = 3.2912161221926835e-130;
        bool r41542 = r41477 <= r41541;
        double r41543 = r41491 + r41496;
        double r41544 = r41480 * r41543;
        double r41545 = r41480 * r41489;
        double r41546 = r41482 - r41545;
        double r41547 = r41544 + r41546;
        double r41548 = r41482 / r41547;
        double r41549 = 2.1749908664970245e+85;
        bool r41550 = r41477 <= r41549;
        double r41551 = r41507 * r41507;
        double r41552 = pow(r41551, r41486);
        double r41553 = r41485 / r41508;
        double r41554 = r41552 / r41553;
        double r41555 = r41503 + r41554;
        double r41556 = r41480 * r41555;
        double r41557 = r41505 + r41556;
        double r41558 = sqrt(r41557);
        double r41559 = r41482 / r41558;
        double r41560 = r41550 ? r41559 : r41548;
        double r41561 = r41542 ? r41548 : r41560;
        double r41562 = r41531 ? r41540 : r41561;
        double r41563 = r41529 ? r41499 : r41562;
        double r41564 = r41501 ? r41527 : r41563;
        double r41565 = r41479 ? r41499 : r41564;
        return r41565;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 5 regimes

2. if t < -9.583977798108111e+138 or -6.6963672441100204e-164 < t < -3.0886340565384557e-240

   1. Initial program 59.1

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

   2. Taylor expanded around -inf 9.8

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

   3. Simplified 9.8

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

   if -9.583977798108111e+138 < t < -6.6963672441100204e-164

   1. Initial program 25.3

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

   2. Taylor expanded around inf 10.3

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

   3. Simplified 10.3

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

   5. Applied add-cube-cbrt 10.4

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

   6. Applied unpow-prod-down 10.4

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

   7. Applied associate-/l* 6.7

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

   9. Applied add-cube-cbrt 6.8

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

   10. Applied unpow-prod-down 6.8

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

   11. Applied add-cube-cbrt 6.8

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

   12. Applied times-frac 6.8

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

   13. Applied add-sqr-sqrt 6.8

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

   14. Applied times-frac 6.8

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

   15. Simplified 6.8

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\left(\left(\frac{\left|{\left(\sqrt[3]{\ell}\right)}^{2}\right|}{\sqrt[3]{x}} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}\right)} \cdot \frac{\sqrt{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}}{\frac{\sqrt[3]{x}}{{\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{2}}}\right)}}\]

   16. Simplified 5.6

       \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\left(\frac{\left|{\left(\sqrt[3]{\ell}\right)}^{2}\right|}{\sqrt[3]{x}} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}\right) \cdot \color{blue}{\frac{\left|{\left(\sqrt[3]{\ell}\right)}^{2}\right| \cdot {\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{2}}{\sqrt[3]{x}}}\right)}}\]

   if -3.0886340565384557e-240 < t < 9.909309527651243e-270

   1. Initial program 62.8

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

   2. Taylor expanded around inf 32.0

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

   3. Simplified 32.0

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

   5. Applied add-sqr-sqrt 32.0

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

   6. Applied add-sqr-sqrt 32.0

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

   7. Applied times-frac 32.0

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

   8. Simplified 32.0

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

   9. Simplified 31.9

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

   if 9.909309527651243e-270 < t < 3.2912161221926835e-130 or 2.1749908664970245e+85 < t

   1. Initial program 51.7

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

   2. Taylor expanded around inf 12.2

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

   3. Simplified 12.2

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

   if 3.2912161221926835e-130 < t < 2.1749908664970245e+85

   1. Initial program 25.4

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

   2. Taylor expanded around inf 9.9

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

   3. Simplified 9.9

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

   5. Applied add-cube-cbrt 10.0

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

   6. Applied unpow-prod-down 10.0

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

   7. Applied associate-/l* 6.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{\frac{x}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.583977798108110900162476286547585661621 \cdot 10^{138}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -6.696367244110020413430084956000329612319 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \left(\left(\frac{\left|{\left(\sqrt[3]{\ell}\right)}^{2}\right|}{\sqrt[3]{x}} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot {\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{4}\right) \cdot \frac{\left|{\left(\sqrt[3]{\ell}\right)}^{2}\right| \cdot {\left(\sqrt[3]{\sqrt[3]{\ell}}\right)}^{2}}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le -3.088634056538455682354250066621880548059 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 9.909309527651243252484098193013516257504 \cdot 10^{-270}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\left|\ell\right|}{\sqrt{x}} \cdot \frac{\left|\ell\right|}{\sqrt{x}}\right)}}\\ \mathbf{elif}\;t \le 3.291216122192683495789321781028557839903 \cdot 10^{-130}:\\ \;\;\;\;\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 2.174990866497024471017537763220387342123 \cdot 10^{85}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}^{2}}{\frac{x}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\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 2019346 
(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)))))