Average Error: 41.8 → 9.1
Time: 25.6s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.415525381702613 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 3.4759257316157413 \cdot 10^{-284}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{elif}\;t \le 9.831366213789788 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2 \cdot t}{\sqrt{2} \cdot x} - \frac{2}{x \cdot x} \cdot \frac{t}{2 \cdot \sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot 2 + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.0888219445242733 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2 \cdot t}{\sqrt{2} \cdot x} - \frac{2}{x \cdot x} \cdot \frac{t}{2 \cdot \sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot 2 + \sqrt{2} \cdot t\right)}\\ \end{array}\]
\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 -5.415525381702613 \cdot 10^{+62}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\

\mathbf{elif}\;t \le 3.4759257316157413 \cdot 10^{-284}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\

\mathbf{elif}\;t \le 9.831366213789788 \cdot 10^{-187}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2 \cdot t}{\sqrt{2} \cdot x} - \frac{2}{x \cdot x} \cdot \frac{t}{2 \cdot \sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot 2 + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 1.0888219445242733 \cdot 10^{+135}:\\
\;\;\;\;\frac{\left(t \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r559495 = 2.0;
        double r559496 = sqrt(r559495);
        double r559497 = t;
        double r559498 = r559496 * r559497;
        double r559499 = x;
        double r559500 = 1.0;
        double r559501 = r559499 + r559500;
        double r559502 = r559499 - r559500;
        double r559503 = r559501 / r559502;
        double r559504 = l;
        double r559505 = r559504 * r559504;
        double r559506 = r559497 * r559497;
        double r559507 = r559495 * r559506;
        double r559508 = r559505 + r559507;
        double r559509 = r559503 * r559508;
        double r559510 = r559509 - r559505;
        double r559511 = sqrt(r559510);
        double r559512 = r559498 / r559511;
        return r559512;
}
↓
double f(double x, double l, double t) {
        double r559513 = t;
        double r559514 = -5.415525381702613e+62;
        bool r559515 = r559513 <= r559514;
        double r559516 = 2.0;
        double r559517 = sqrt(r559516);
        double r559518 = r559517 * r559513;
        double r559519 = r559516 * r559517;
        double r559520 = r559513 / r559519;
        double r559521 = x;
        double r559522 = r559521 * r559521;
        double r559523 = r559520 / r559522;
        double r559524 = r559513 / r559517;
        double r559525 = r559524 / r559522;
        double r559526 = r559523 - r559525;
        double r559527 = r559516 * r559526;
        double r559528 = r559516 * r559513;
        double r559529 = r559517 * r559521;
        double r559530 = r559528 / r559529;
        double r559531 = r559518 + r559530;
        double r559532 = r559527 - r559531;
        double r559533 = r559518 / r559532;
        double r559534 = 3.4759257316157413e-284;
        bool r559535 = r559513 <= r559534;
        double r559536 = l;
        double r559537 = r559521 / r559536;
        double r559538 = r559536 / r559537;
        double r559539 = r559513 * r559513;
        double r559540 = r559538 + r559539;
        double r559541 = r559516 * r559540;
        double r559542 = 4.0;
        double r559543 = r559539 / r559521;
        double r559544 = r559542 * r559543;
        double r559545 = r559541 + r559544;
        double r559546 = sqrt(r559545);
        double r559547 = r559518 / r559546;
        double r559548 = 9.831366213789788e-187;
        bool r559549 = r559513 <= r559548;
        double r559550 = r559516 / r559522;
        double r559551 = r559550 * r559520;
        double r559552 = r559530 - r559551;
        double r559553 = r559525 * r559516;
        double r559554 = r559553 + r559518;
        double r559555 = r559552 + r559554;
        double r559556 = r559518 / r559555;
        double r559557 = 1.0888219445242733e+135;
        bool r559558 = r559513 <= r559557;
        double r559559 = cbrt(r559517);
        double r559560 = r559513 * r559559;
        double r559561 = r559559 * r559559;
        double r559562 = r559560 * r559561;
        double r559563 = r559562 / r559546;
        double r559564 = r559558 ? r559563 : r559556;
        double r559565 = r559549 ? r559556 : r559564;
        double r559566 = r559535 ? r559547 : r559565;
        double r559567 = r559515 ? r559533 : r559566;
        return r559567;
}

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 < -5.415525381702613e+62

    1. Initial program 45.3

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

    2. Taylor expanded around -inf 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

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

    if -5.415525381702613e+62 < t < 3.4759257316157413e-284

    1. Initial program 39.4

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

    2. Taylor expanded around inf 17.7

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

    3. Simplified17.7

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

    5. Applied associate-/l*14.9

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

    if 3.4759257316157413e-284 < t < 9.831366213789788e-187 or 1.0888219445242733e+135 < t

    1. Initial program 57.9

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

    2. Taylor expanded around inf 10.7

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

    3. Simplified10.8

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

    if 9.831366213789788e-187 < t < 1.0888219445242733e+135

    1. Initial program 26.0

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

    2. Taylor expanded around inf 11.7

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

    3. Simplified11.7

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

    5. Applied associate-/l*6.4

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

    7. Applied add-cube-cbrt6.4

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

    8. Applied associate-*l*6.4

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.415525381702613 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{\frac{t}{2 \cdot \sqrt{2}}}{x \cdot x} - \frac{\frac{t}{\sqrt{2}}}{x \cdot x}\right) - \left(\sqrt{2} \cdot t + \frac{2 \cdot t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \le 3.4759257316157413 \cdot 10^{-284}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{elif}\;t \le 9.831366213789788 \cdot 10^{-187}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2 \cdot t}{\sqrt{2} \cdot x} - \frac{2}{x \cdot x} \cdot \frac{t}{2 \cdot \sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot 2 + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 1.0888219445242733 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2 \cdot t}{\sqrt{2} \cdot x} - \frac{2}{x \cdot x} \cdot \frac{t}{2 \cdot \sqrt{2}}\right) + \left(\frac{\frac{t}{\sqrt{2}}}{x \cdot x} \cdot 2 + \sqrt{2} \cdot t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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)))))