Average Error: 42.0 → 9.1
Time: 51.9s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.236348302352089 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x} - \mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right)\right)}\\ \mathbf{elif}\;t \le 2.9520963647114176 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot t}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right) + \mathsf{fma}\left(\left(\frac{t}{\sqrt{2} \cdot x}\right), 2, \left(\sqrt{2} \cdot t\right)\right)}\\ \end{array}\]
\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 -4.236348302352089 \cdot 10^{+73}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x} - \mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right)\right)}\\

\mathbf{elif}\;t \le 2.9520963647114176 \cdot 10^{+132}:\\
\;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot t}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\\

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

\end{array}
double f(double x, double l, double t) {
        double r2626997 = 2.0;
        double r2626998 = sqrt(r2626997);
        double r2626999 = t;
        double r2627000 = r2626998 * r2626999;
        double r2627001 = x;
        double r2627002 = 1.0;
        double r2627003 = r2627001 + r2627002;
        double r2627004 = r2627001 - r2627002;
        double r2627005 = r2627003 / r2627004;
        double r2627006 = l;
        double r2627007 = r2627006 * r2627006;
        double r2627008 = r2626999 * r2626999;
        double r2627009 = r2626997 * r2627008;
        double r2627010 = r2627007 + r2627009;
        double r2627011 = r2627005 * r2627010;
        double r2627012 = r2627011 - r2627007;
        double r2627013 = sqrt(r2627012);
        double r2627014 = r2627000 / r2627013;
        return r2627014;
}
↓
double f(double x, double l, double t) {
        double r2627015 = t;
        double r2627016 = -4.236348302352089e+73;
        bool r2627017 = r2627015 <= r2627016;
        double r2627018 = 2.0;
        double r2627019 = sqrt(r2627018);
        double r2627020 = r2627019 * r2627015;
        double r2627021 = 1.0;
        double r2627022 = r2627021 / r2627019;
        double r2627023 = x;
        double r2627024 = r2627023 * r2627023;
        double r2627025 = r2627015 / r2627024;
        double r2627026 = r2627022 * r2627025;
        double r2627027 = r2627018 / r2627019;
        double r2627028 = r2627015 / r2627023;
        double r2627029 = r2627028 + r2627025;
        double r2627030 = r2627027 * r2627029;
        double r2627031 = fma(r2627015, r2627019, r2627030);
        double r2627032 = r2627026 - r2627031;
        double r2627033 = r2627020 / r2627032;
        double r2627034 = 2.9520963647114176e+132;
        bool r2627035 = r2627015 <= r2627034;
        double r2627036 = cbrt(r2627019);
        double r2627037 = r2627036 * r2627015;
        double r2627038 = l;
        double r2627039 = r2627038 / r2627023;
        double r2627040 = r2627015 * r2627015;
        double r2627041 = fma(r2627039, r2627038, r2627040);
        double r2627042 = 4.0;
        double r2627043 = r2627042 * r2627040;
        double r2627044 = r2627043 / r2627023;
        double r2627045 = fma(r2627041, r2627018, r2627044);
        double r2627046 = sqrt(r2627045);
        double r2627047 = r2627037 / r2627046;
        double r2627048 = r2627036 * r2627036;
        double r2627049 = r2627047 * r2627048;
        double r2627050 = r2627018 / r2627023;
        double r2627051 = r2627050 / r2627023;
        double r2627052 = r2627015 / r2627019;
        double r2627053 = r2627052 / r2627018;
        double r2627054 = r2627052 - r2627053;
        double r2627055 = r2627051 * r2627054;
        double r2627056 = r2627019 * r2627023;
        double r2627057 = r2627015 / r2627056;
        double r2627058 = fma(r2627057, r2627018, r2627020);
        double r2627059 = r2627055 + r2627058;
        double r2627060 = r2627020 / r2627059;
        double r2627061 = r2627035 ? r2627049 : r2627060;
        double r2627062 = r2627017 ? r2627033 : r2627061;
        return r2627062;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -4.236348302352089e+73

    1. Initial program 46.5

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -4.236348302352089e+73 < t < 2.9520963647114176e+132

    1. Initial program 36.3

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

    2. Taylor expanded around inf 17.1

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

    3. Simplified13.3

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

    5. Applied add-cube-cbrt13.3

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

    6. Applied associate-*l*13.3

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

    8. Applied *-un-lft-identity13.3

      \[\leadsto \frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\color{blue}{1 \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{\left(t \cdot t\right) \cdot 4}{x}\right)\right)}}}\]

    9. Applied sqrt-prod13.3

      \[\leadsto \frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\color{blue}{\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{\left(t \cdot t\right) \cdot 4}{x}\right)\right)}}}\]

    10. Applied times-frac13.1

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{1}} \cdot \frac{\sqrt[3]{\sqrt{2}} \cdot t}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{\left(t \cdot t\right) \cdot 4}{x}\right)\right)}}}\]

    11. Simplified13.1

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

    if 2.9520963647114176e+132 < t

    1. Initial program 56.3

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

    2. Taylor expanded around inf 2.1

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

    3. Simplified2.1

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.236348302352089 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{1}{\sqrt{2}} \cdot \frac{t}{x \cdot x} - \mathsf{fma}\left(t, \left(\sqrt{2}\right), \left(\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x} + \frac{t}{x \cdot x}\right)\right)\right)}\\ \mathbf{elif}\;t \le 2.9520963647114176 \cdot 10^{+132}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{2}} \cdot t}{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(\frac{\ell}{x}\right), \ell, \left(t \cdot t\right)\right)\right), 2, \left(\frac{4 \cdot \left(t \cdot t\right)}{x}\right)\right)}} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{\frac{t}{\sqrt{2}}}{2}\right) + \mathsf{fma}\left(\left(\frac{t}{\sqrt{2} \cdot x}\right), 2, \left(\sqrt{2} \cdot t\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))