Average Error: 42.7 → 8.9
Time: 31.7s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.318745676503489222703826814331080490651 \cdot 10^{101}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 2.555309788910298182685313601676848640915 \cdot 10^{120}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]
\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 -4.318745676503489222703826814331080490651 \cdot 10^{101}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le 2.555309788910298182685313601676848640915 \cdot 10^{120}:\\
\;\;\;\;\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, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\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 r33265 = 2.0;
        double r33266 = sqrt(r33265);
        double r33267 = t;
        double r33268 = r33266 * r33267;
        double r33269 = x;
        double r33270 = 1.0;
        double r33271 = r33269 + r33270;
        double r33272 = r33269 - r33270;
        double r33273 = r33271 / r33272;
        double r33274 = l;
        double r33275 = r33274 * r33274;
        double r33276 = r33267 * r33267;
        double r33277 = r33265 * r33276;
        double r33278 = r33275 + r33277;
        double r33279 = r33273 * r33278;
        double r33280 = r33279 - r33275;
        double r33281 = sqrt(r33280);
        double r33282 = r33268 / r33281;
        return r33282;
}
↓
double f(double x, double l, double t) {
        double r33283 = t;
        double r33284 = -4.318745676503489e+101;
        bool r33285 = r33283 <= r33284;
        double r33286 = 2.0;
        double r33287 = sqrt(r33286);
        double r33288 = r33287 * r33283;
        double r33289 = x;
        double r33290 = r33287 * r33289;
        double r33291 = r33283 / r33290;
        double r33292 = r33286 * r33291;
        double r33293 = fma(r33283, r33287, r33292);
        double r33294 = -r33293;
        double r33295 = r33288 / r33294;
        double r33296 = 2.5553097889102982e+120;
        bool r33297 = r33283 <= r33296;
        double r33298 = cbrt(r33287);
        double r33299 = r33298 * r33298;
        double r33300 = r33298 * r33283;
        double r33301 = r33299 * r33300;
        double r33302 = l;
        double r33303 = fabs(r33302);
        double r33304 = r33303 / r33289;
        double r33305 = r33303 * r33304;
        double r33306 = fma(r33283, r33283, r33305);
        double r33307 = 4.0;
        double r33308 = 2.0;
        double r33309 = pow(r33283, r33308);
        double r33310 = r33309 / r33289;
        double r33311 = r33307 * r33310;
        double r33312 = fma(r33286, r33306, r33311);
        double r33313 = sqrt(r33312);
        double r33314 = r33301 / r33313;
        double r33315 = r33288 / r33293;
        double r33316 = r33297 ? r33314 : r33315;
        double r33317 = r33285 ? r33295 : r33316;
        return r33317;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -4.318745676503489e+101

    1. Initial program 50.3

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

    2. Simplified50.3

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

    3. Taylor expanded around inf 50.6

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

    4. Simplified50.6

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

    5. Taylor expanded around -inf 3.1

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

    6. Simplified3.1

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

    if -4.318745676503489e+101 < t < 2.5553097889102982e+120

    1. Initial program 36.8

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

    2. Simplified36.8

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

    3. Taylor expanded around inf 17.0

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

    4. Simplified17.0

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

    6. Applied *-un-lft-identity17.0

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

    7. Applied add-sqr-sqrt17.0

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

    8. Applied times-frac17.0

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

    9. Simplified17.0

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

    10. Simplified12.7

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

    12. Applied add-cube-cbrt12.7

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

    13. Applied associate-*l*12.7

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

    if 2.5553097889102982e+120 < t

    1. Initial program 53.8

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

    2. Simplified53.8

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

    3. Taylor expanded around inf 54.2

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

    4. Simplified54.2

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

    5. Taylor expanded around inf 2.7

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

    6. Simplified2.7

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.318745676503489222703826814331080490651 \cdot 10^{101}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{-\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 2.555309788910298182685313601676848640915 \cdot 10^{120}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right), 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{2}, 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))))