Average Error: 42.6 → 9.3
Time: 32.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.67001811380404894472223536597467413985 \cdot 10^{49}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 1.279156173644286169185104793876132292276 \cdot 10^{-253} \lor \neg \left(t \le 3.392855682492969822120603204380265318822 \cdot 10^{-169}\right) \land t \le 5.564919863052779332924405357503780952199 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]
\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 -1.67001811380404894472223536597467413985 \cdot 10^{49}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le 1.279156173644286169185104793876132292276 \cdot 10^{-253} \lor \neg \left(t \le 3.392855682492969822120603204380265318822 \cdot 10^{-169}\right) \land t \le 5.564919863052779332924405357503780952199 \cdot 10^{60}:\\
\;\;\;\;\frac{\sqrt{2} \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)}}\\

\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 r39777 = 2.0;
        double r39778 = sqrt(r39777);
        double r39779 = t;
        double r39780 = r39778 * r39779;
        double r39781 = x;
        double r39782 = 1.0;
        double r39783 = r39781 + r39782;
        double r39784 = r39781 - r39782;
        double r39785 = r39783 / r39784;
        double r39786 = l;
        double r39787 = r39786 * r39786;
        double r39788 = r39779 * r39779;
        double r39789 = r39777 * r39788;
        double r39790 = r39787 + r39789;
        double r39791 = r39785 * r39790;
        double r39792 = r39791 - r39787;
        double r39793 = sqrt(r39792);
        double r39794 = r39780 / r39793;
        return r39794;
}
↓
double f(double x, double l, double t) {
        double r39795 = t;
        double r39796 = -1.670018113804049e+49;
        bool r39797 = r39795 <= r39796;
        double r39798 = 2.0;
        double r39799 = sqrt(r39798);
        double r39800 = r39799 * r39795;
        double r39801 = x;
        double r39802 = r39799 * r39801;
        double r39803 = r39795 / r39802;
        double r39804 = r39798 * r39803;
        double r39805 = fma(r39795, r39799, r39804);
        double r39806 = -r39805;
        double r39807 = r39800 / r39806;
        double r39808 = 1.2791561736442862e-253;
        bool r39809 = r39795 <= r39808;
        double r39810 = 3.39285568249297e-169;
        bool r39811 = r39795 <= r39810;
        double r39812 = !r39811;
        double r39813 = 5.564919863052779e+60;
        bool r39814 = r39795 <= r39813;
        bool r39815 = r39812 && r39814;
        bool r39816 = r39809 || r39815;
        double r39817 = l;
        double r39818 = r39801 / r39817;
        double r39819 = r39817 / r39818;
        double r39820 = fma(r39795, r39795, r39819);
        double r39821 = 4.0;
        double r39822 = 2.0;
        double r39823 = pow(r39795, r39822);
        double r39824 = r39823 / r39801;
        double r39825 = r39821 * r39824;
        double r39826 = fma(r39798, r39820, r39825);
        double r39827 = sqrt(r39826);
        double r39828 = r39800 / r39827;
        double r39829 = r39800 / r39805;
        double r39830 = r39816 ? r39828 : r39829;
        double r39831 = r39797 ? r39807 : r39830;
        return r39831;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -1.670018113804049e+49

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around inf 42.8

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

    4. Simplified42.8

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

    5. Taylor expanded around -inf 4.0

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

    6. Simplified4.0

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

    if -1.670018113804049e+49 < t < 1.2791561736442862e-253 or 3.39285568249297e-169 < t < 5.564919863052779e+60

    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. Simplified37.7

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

    3. Taylor expanded around inf 16.8

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

    4. Simplified16.8

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

    6. Applied sqr-pow16.8

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

    7. Applied associate-/l*12.3

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

    8. Simplified12.3

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

    if 1.2791561736442862e-253 < t < 3.39285568249297e-169 or 5.564919863052779e+60 < t

    1. Initial program 49.2

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

    2. Simplified49.2

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

    3. Taylor expanded around inf 43.6

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

    4. Simplified43.6

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

    5. Taylor expanded around inf 9.1

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

    6. Simplified9.1

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.67001811380404894472223536597467413985 \cdot 10^{49}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 1.279156173644286169185104793876132292276 \cdot 10^{-253} \lor \neg \left(t \le 3.392855682492969822120603204380265318822 \cdot 10^{-169}\right) \land t \le 5.564919863052779332924405357503780952199 \cdot 10^{60}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{\ell}{\frac{x}{\ell}}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))))