Average Error: 43.1 → 9.2
Time: 27.8s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.397305156304466012482107522690315080951 \cdot 10^{105}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 3.16626920683862038527006405926439879237 \cdot 10^{113}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(4, \frac{{t}^{2}}{x}, 2 \cdot \mathsf{fma}\left(t, t, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]
\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 -1.397305156304466012482107522690315080951 \cdot 10^{105}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le 3.16626920683862038527006405926439879237 \cdot 10^{113}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(4, \frac{{t}^{2}}{x}, 2 \cdot \mathsf{fma}\left(t, t, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r42585 = 2.0;
        double r42586 = sqrt(r42585);
        double r42587 = t;
        double r42588 = r42586 * r42587;
        double r42589 = x;
        double r42590 = 1.0;
        double r42591 = r42589 + r42590;
        double r42592 = r42589 - r42590;
        double r42593 = r42591 / r42592;
        double r42594 = l;
        double r42595 = r42594 * r42594;
        double r42596 = r42587 * r42587;
        double r42597 = r42585 * r42596;
        double r42598 = r42595 + r42597;
        double r42599 = r42593 * r42598;
        double r42600 = r42599 - r42595;
        double r42601 = sqrt(r42600);
        double r42602 = r42588 / r42601;
        return r42602;
}
↓
double f(double x, double l, double t) {
        double r42603 = t;
        double r42604 = -1.397305156304466e+105;
        bool r42605 = r42603 <= r42604;
        double r42606 = 2.0;
        double r42607 = sqrt(r42606);
        double r42608 = r42607 * r42603;
        double r42609 = 3.0;
        double r42610 = pow(r42607, r42609);
        double r42611 = x;
        double r42612 = 2.0;
        double r42613 = pow(r42611, r42612);
        double r42614 = r42610 * r42613;
        double r42615 = r42603 / r42614;
        double r42616 = r42607 * r42613;
        double r42617 = r42603 / r42616;
        double r42618 = r42615 - r42617;
        double r42619 = r42606 * r42618;
        double r42620 = r42607 * r42611;
        double r42621 = r42603 / r42620;
        double r42622 = r42603 * r42607;
        double r42623 = fma(r42606, r42621, r42622);
        double r42624 = r42619 - r42623;
        double r42625 = r42608 / r42624;
        double r42626 = 3.1662692068386204e+113;
        bool r42627 = r42603 <= r42626;
        double r42628 = 4.0;
        double r42629 = pow(r42603, r42612);
        double r42630 = r42629 / r42611;
        double r42631 = l;
        double r42632 = fabs(r42631);
        double r42633 = r42632 / r42611;
        double r42634 = r42632 * r42633;
        double r42635 = fma(r42603, r42603, r42634);
        double r42636 = r42606 * r42635;
        double r42637 = fma(r42628, r42630, r42636);
        double r42638 = sqrt(r42637);
        double r42639 = r42608 / r42638;
        double r42640 = r42606 * r42615;
        double r42641 = r42623 - r42640;
        double r42642 = r42608 / r42641;
        double r42643 = r42627 ? r42639 : r42642;
        double r42644 = r42605 ? r42625 : r42643;
        return r42644;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -1.397305156304466e+105

    1. Initial program 50.9

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

    2. Taylor expanded around -inf 3.0

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

    3. Simplified3.0

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

    if -1.397305156304466e+105 < t < 3.1662692068386204e+113

    1. Initial program 37.7

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

    2. Taylor expanded around inf 17.6

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

    3. Simplified17.6

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

    5. Applied *-un-lft-identity17.6

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

    6. Applied add-sqr-sqrt17.6

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

    7. Applied times-frac17.6

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

    8. Simplified17.6

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

    9. Simplified13.2

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

    if 3.1662692068386204e+113 < t

    1. Initial program 52.7

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

    2. Taylor expanded around inf 52.7

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

    3. Simplified52.7

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

    5. Applied *-un-lft-identity52.7

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

    6. Applied add-sqr-sqrt52.7

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

    7. Applied times-frac52.7

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

    8. Simplified52.7

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

    9. Simplified51.8

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

    10. Taylor expanded around inf 2.8

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

    11. Simplified2.8

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.397305156304466012482107522690315080951 \cdot 10^{105}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 3.16626920683862038527006405926439879237 \cdot 10^{113}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(4, \frac{{t}^{2}}{x}, 2 \cdot \mathsf{fma}\left(t, t, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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)))))