Average Error: 42.5 → 8.9
Time: 43.5s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.3433794278040432 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 8.264925603641206 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le 3.464075321521421 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right) + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 1.4994412213892464 \cdot 10^{+108}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right) + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right)}\\ \end{array}\]
\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 -2.3433794278040432 \cdot 10^{+67}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\

\mathbf{elif}\;t \le 8.264925603641206 \cdot 10^{-260}:\\
\;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\

\mathbf{elif}\;t \le 3.464075321521421 \cdot 10^{-166}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right) + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right)}\\

\mathbf{elif}\;t \le 1.4994412213892464 \cdot 10^{+108}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r1337320 = 2.0;
        double r1337321 = sqrt(r1337320);
        double r1337322 = t;
        double r1337323 = r1337321 * r1337322;
        double r1337324 = x;
        double r1337325 = 1.0;
        double r1337326 = r1337324 + r1337325;
        double r1337327 = r1337324 - r1337325;
        double r1337328 = r1337326 / r1337327;
        double r1337329 = l;
        double r1337330 = r1337329 * r1337329;
        double r1337331 = r1337322 * r1337322;
        double r1337332 = r1337320 * r1337331;
        double r1337333 = r1337330 + r1337332;
        double r1337334 = r1337328 * r1337333;
        double r1337335 = r1337334 - r1337330;
        double r1337336 = sqrt(r1337335);
        double r1337337 = r1337323 / r1337336;
        return r1337337;
}
↓
double f(double x, double l, double t) {
        double r1337338 = t;
        double r1337339 = -2.3433794278040432e+67;
        bool r1337340 = r1337338 <= r1337339;
        double r1337341 = 2.0;
        double r1337342 = sqrt(r1337341);
        double r1337343 = r1337342 * r1337338;
        double r1337344 = r1337341 * r1337342;
        double r1337345 = r1337338 / r1337344;
        double r1337346 = r1337338 / r1337342;
        double r1337347 = r1337345 - r1337346;
        double r1337348 = x;
        double r1337349 = r1337341 / r1337348;
        double r1337350 = r1337349 / r1337348;
        double r1337351 = r1337347 * r1337350;
        double r1337352 = r1337351 - r1337343;
        double r1337353 = r1337346 * r1337349;
        double r1337354 = r1337352 - r1337353;
        double r1337355 = r1337343 / r1337354;
        double r1337356 = 8.264925603641206e-260;
        bool r1337357 = r1337338 <= r1337356;
        double r1337358 = sqrt(r1337342);
        double r1337359 = r1337338 * r1337358;
        double r1337360 = r1337358 * r1337359;
        double r1337361 = l;
        double r1337362 = r1337348 / r1337361;
        double r1337363 = r1337361 / r1337362;
        double r1337364 = r1337338 * r1337338;
        double r1337365 = r1337363 + r1337364;
        double r1337366 = r1337365 * r1337341;
        double r1337367 = 4.0;
        double r1337368 = r1337367 / r1337348;
        double r1337369 = r1337368 * r1337364;
        double r1337370 = r1337366 + r1337369;
        double r1337371 = sqrt(r1337370);
        double r1337372 = r1337360 / r1337371;
        double r1337373 = 3.464075321521421e-166;
        bool r1337374 = r1337338 <= r1337373;
        double r1337375 = r1337353 + r1337343;
        double r1337376 = r1337346 - r1337345;
        double r1337377 = r1337350 * r1337376;
        double r1337378 = r1337375 + r1337377;
        double r1337379 = r1337343 / r1337378;
        double r1337380 = 1.4994412213892464e+108;
        bool r1337381 = r1337338 <= r1337380;
        double r1337382 = r1337343 / r1337371;
        double r1337383 = r1337381 ? r1337382 : r1337379;
        double r1337384 = r1337374 ? r1337379 : r1337383;
        double r1337385 = r1337357 ? r1337372 : r1337384;
        double r1337386 = r1337340 ? r1337355 : r1337385;
        return r1337386;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if t < -2.3433794278040432e+67

    1. Initial program 46.7

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

    2. Taylor expanded around -inf 3.5

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

    3. Simplified3.5

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

    if -2.3433794278040432e+67 < t < 8.264925603641206e-260

    1. Initial program 41.2

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

    2. Taylor expanded around -inf 18.5

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

    3. Simplified14.8

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

    5. Applied add-sqr-sqrt14.9

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

    6. Applied associate-*l*14.9

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

    if 8.264925603641206e-260 < t < 3.464075321521421e-166 or 1.4994412213892464e+108 < t

    1. Initial program 53.5

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

    2. Taylor expanded around inf 10.1

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

    3. Simplified10.1

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

    if 3.464075321521421e-166 < t < 1.4994412213892464e+108

    1. Initial program 27.4

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

    2. Taylor expanded around -inf 10.6

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

    3. Simplified5.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{4}{x} \cdot \left(t \cdot t\right) + 2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.3433794278040432 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\frac{t}{2 \cdot \sqrt{2}} - \frac{t}{\sqrt{2}}\right) \cdot \frac{\frac{2}{x}}{x} - \sqrt{2} \cdot t\right) - \frac{t}{\sqrt{2}} \cdot \frac{2}{x}}\\ \mathbf{elif}\;t \le 8.264925603641206 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(t \cdot \sqrt{\sqrt{2}}\right)}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{elif}\;t \le 3.464075321521421 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right) + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right)}\\ \mathbf{elif}\;t \le 1.4994412213892464 \cdot 10^{+108}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + \frac{4}{x} \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{\sqrt{2}} \cdot \frac{2}{x} + \sqrt{2} \cdot t\right) + \frac{\frac{2}{x}}{x} \cdot \left(\frac{t}{\sqrt{2}} - \frac{t}{2 \cdot \sqrt{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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)))))