Average Error: 43.3 → 9.0
Time: 9.1s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.111000041580223529443968142716501710902 \cdot 10^{150}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le -1.071069681460846652725548995559388373135 \cdot 10^{-172}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le -3.853780726509438926031971852746276593106 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le 8.692195871820052223377296517762777409105 \cdot 10^{51}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -2.111000041580223529443968142716501710902 \cdot 10^{150}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\

\mathbf{elif}\;t \le -1.071069681460846652725548995559388373135 \cdot 10^{-172}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

\mathbf{elif}\;t \le -3.853780726509438926031971852746276593106 \cdot 10^{-213}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\

\mathbf{elif}\;t \le 8.692195871820052223377296517762777409105 \cdot 10^{51}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r38482 = 2.0;
        double r38483 = sqrt(r38482);
        double r38484 = t;
        double r38485 = r38483 * r38484;
        double r38486 = x;
        double r38487 = 1.0;
        double r38488 = r38486 + r38487;
        double r38489 = r38486 - r38487;
        double r38490 = r38488 / r38489;
        double r38491 = l;
        double r38492 = r38491 * r38491;
        double r38493 = r38484 * r38484;
        double r38494 = r38482 * r38493;
        double r38495 = r38492 + r38494;
        double r38496 = r38490 * r38495;
        double r38497 = r38496 - r38492;
        double r38498 = sqrt(r38497);
        double r38499 = r38485 / r38498;
        return r38499;
}
↓
double f(double x, double l, double t) {
        double r38500 = t;
        double r38501 = -2.1110000415802235e+150;
        bool r38502 = r38500 <= r38501;
        double r38503 = 2.0;
        double r38504 = sqrt(r38503);
        double r38505 = r38504 * r38500;
        double r38506 = 3.0;
        double r38507 = pow(r38504, r38506);
        double r38508 = x;
        double r38509 = 2.0;
        double r38510 = pow(r38508, r38509);
        double r38511 = r38507 * r38510;
        double r38512 = r38500 / r38511;
        double r38513 = r38504 * r38510;
        double r38514 = r38500 / r38513;
        double r38515 = r38504 * r38508;
        double r38516 = r38500 / r38515;
        double r38517 = r38500 * r38504;
        double r38518 = fma(r38503, r38516, r38517);
        double r38519 = fma(r38503, r38514, r38518);
        double r38520 = -r38519;
        double r38521 = fma(r38503, r38512, r38520);
        double r38522 = r38505 / r38521;
        double r38523 = -1.0710696814608467e-172;
        bool r38524 = r38500 <= r38523;
        double r38525 = cbrt(r38504);
        double r38526 = r38525 * r38525;
        double r38527 = r38525 * r38500;
        double r38528 = r38526 * r38527;
        double r38529 = pow(r38500, r38509);
        double r38530 = l;
        double r38531 = r38530 / r38508;
        double r38532 = r38530 * r38531;
        double r38533 = 4.0;
        double r38534 = r38529 / r38508;
        double r38535 = r38533 * r38534;
        double r38536 = fma(r38503, r38532, r38535);
        double r38537 = fma(r38503, r38529, r38536);
        double r38538 = sqrt(r38537);
        double r38539 = r38528 / r38538;
        double r38540 = -3.853780726509439e-213;
        bool r38541 = r38500 <= r38540;
        double r38542 = 8.692195871820052e+51;
        bool r38543 = r38500 <= r38542;
        double r38544 = r38503 * r38512;
        double r38545 = r38518 - r38544;
        double r38546 = fma(r38503, r38514, r38545);
        double r38547 = r38505 / r38546;
        double r38548 = r38543 ? r38539 : r38547;
        double r38549 = r38541 ? r38522 : r38548;
        double r38550 = r38524 ? r38539 : r38549;
        double r38551 = r38502 ? r38522 : r38550;
        return r38551;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -2.1110000415802235e+150 or -1.0710696814608467e-172 < t < -3.853780726509439e-213

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 6.8

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

    3. Simplified6.8

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

    if -2.1110000415802235e+150 < t < -1.0710696814608467e-172 or -3.853780726509439e-213 < t < 8.692195871820052e+51

    1. Initial program 36.6

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

    2. Taylor expanded around inf 16.4

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

    3. Simplified16.4

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

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

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

    6. Applied add-sqr-sqrt40.3

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

    7. Applied unpow-prod-down40.3

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

    8. Applied times-frac38.0

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

    9. Simplified38.0

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

    10. Simplified12.0

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

    12. Applied add-cube-cbrt12.0

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

    13. Applied associate-*l*11.9

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

    if 8.692195871820052e+51 < t

    1. Initial program 44.7

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

    2. Taylor expanded around inf 3.9

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

    3. Simplified3.9

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.111000041580223529443968142716501710902 \cdot 10^{150}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le -1.071069681460846652725548995559388373135 \cdot 10^{-172}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le -3.853780726509438926031971852746276593106 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le 8.692195871820052223377296517762777409105 \cdot 10^{51}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(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)))))