Average Error: 42.7 → 9.0
Time: 18.5s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.606623675187166320307147356355623739061 \cdot 10^{75}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 4.894745026611551231238356257343815376774 \cdot 10^{-226} \lor \neg \left(t \le 5.832383717323896654554068457322403291209 \cdot 10^{-160}\right) \land t \le 4869868199152865435985673923804373450752:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]
\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 -1.606623675187166320307147356355623739061 \cdot 10^{75}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le 4.894745026611551231238356257343815376774 \cdot 10^{-226} \lor \neg \left(t \le 5.832383717323896654554068457322403291209 \cdot 10^{-160}\right) \land t \le 4869868199152865435985673923804373450752:\\
\;\;\;\;\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, \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r29617 = 2.0;
        double r29618 = sqrt(r29617);
        double r29619 = t;
        double r29620 = r29618 * r29619;
        double r29621 = x;
        double r29622 = 1.0;
        double r29623 = r29621 + r29622;
        double r29624 = r29621 - r29622;
        double r29625 = r29623 / r29624;
        double r29626 = l;
        double r29627 = r29626 * r29626;
        double r29628 = r29619 * r29619;
        double r29629 = r29617 * r29628;
        double r29630 = r29627 + r29629;
        double r29631 = r29625 * r29630;
        double r29632 = r29631 - r29627;
        double r29633 = sqrt(r29632);
        double r29634 = r29620 / r29633;
        return r29634;
}
↓
double f(double x, double l, double t) {
        double r29635 = t;
        double r29636 = -1.6066236751871663e+75;
        bool r29637 = r29635 <= r29636;
        double r29638 = 2.0;
        double r29639 = sqrt(r29638);
        double r29640 = r29639 * r29635;
        double r29641 = x;
        double r29642 = r29639 * r29641;
        double r29643 = r29635 / r29642;
        double r29644 = r29638 * r29643;
        double r29645 = fma(r29635, r29639, r29644);
        double r29646 = -r29645;
        double r29647 = r29640 / r29646;
        double r29648 = 4.894745026611551e-226;
        bool r29649 = r29635 <= r29648;
        double r29650 = 5.832383717323897e-160;
        bool r29651 = r29635 <= r29650;
        double r29652 = !r29651;
        double r29653 = 4.8698681991528654e+39;
        bool r29654 = r29635 <= r29653;
        bool r29655 = r29652 && r29654;
        bool r29656 = r29649 || r29655;
        double r29657 = cbrt(r29639);
        double r29658 = r29657 * r29657;
        double r29659 = r29657 * r29635;
        double r29660 = r29658 * r29659;
        double r29661 = l;
        double r29662 = r29641 / r29661;
        double r29663 = r29661 / r29662;
        double r29664 = fma(r29635, r29635, r29663);
        double r29665 = 4.0;
        double r29666 = 2.0;
        double r29667 = pow(r29635, r29666);
        double r29668 = r29667 / r29641;
        double r29669 = r29665 * r29668;
        double r29670 = fma(r29638, r29664, r29669);
        double r29671 = sqrt(r29670);
        double r29672 = r29660 / r29671;
        double r29673 = r29640 / r29645;
        double r29674 = r29656 ? r29672 : r29673;
        double r29675 = r29637 ? r29647 : r29674;
        return r29675;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -1.6066236751871663e+75

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around inf 45.0

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

    4. Simplified45.0

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

    5. Taylor expanded around -inf 3.0

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

    6. Simplified3.0

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

    if -1.6066236751871663e+75 < t < 4.894745026611551e-226 or 5.832383717323897e-160 < t < 4.8698681991528654e+39

    1. Initial program 38.5

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

    2. Simplified38.5

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

    3. Taylor expanded around inf 16.4

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

    4. Simplified16.4

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

    6. Applied unpow216.4

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

    7. Applied associate-/l*12.5

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

    9. Applied add-cube-cbrt12.5

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

    10. Applied associate-*l*12.5

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

    if 4.894745026611551e-226 < t < 5.832383717323897e-160 or 4.8698681991528654e+39 < t

    1. Initial program 47.3

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

    2. Simplified47.3

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

    3. Taylor expanded around inf 42.4

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

    4. Simplified42.4

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

    5. Taylor expanded around inf 8.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}\]

    6. Simplified8.0

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.606623675187166320307147356355623739061 \cdot 10^{75}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 4.894745026611551231238356257343815376774 \cdot 10^{-226} \lor \neg \left(t \le 5.832383717323896654554068457322403291209 \cdot 10^{-160}\right) \land t \le 4869868199152865435985673923804373450752:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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)))))