Average Error: 41.8 → 9.1
Time: 28.1s
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 r569667 = 2.0;
        double r569668 = sqrt(r569667);
        double r569669 = t;
        double r569670 = r569668 * r569669;
        double r569671 = x;
        double r569672 = 1.0;
        double r569673 = r569671 + r569672;
        double r569674 = r569671 - r569672;
        double r569675 = r569673 / r569674;
        double r569676 = l;
        double r569677 = r569676 * r569676;
        double r569678 = r569669 * r569669;
        double r569679 = r569667 * r569678;
        double r569680 = r569677 + r569679;
        double r569681 = r569675 * r569680;
        double r569682 = r569681 - r569677;
        double r569683 = sqrt(r569682);
        double r569684 = r569670 / r569683;
        return r569684;
}
↓
double f(double x, double l, double t) {
        double r569685 = t;
        double r569686 = -5.415525381702613e+62;
        bool r569687 = r569685 <= r569686;
        double r569688 = 2.0;
        double r569689 = sqrt(r569688);
        double r569690 = r569689 * r569685;
        double r569691 = r569688 * r569689;
        double r569692 = r569685 / r569691;
        double r569693 = x;
        double r569694 = r569693 * r569693;
        double r569695 = r569692 / r569694;
        double r569696 = r569685 / r569689;
        double r569697 = r569696 / r569694;
        double r569698 = r569695 - r569697;
        double r569699 = r569688 * r569698;
        double r569700 = r569688 * r569685;
        double r569701 = r569689 * r569693;
        double r569702 = r569700 / r569701;
        double r569703 = r569690 + r569702;
        double r569704 = r569699 - r569703;
        double r569705 = r569690 / r569704;
        double r569706 = 3.4759257316157413e-284;
        bool r569707 = r569685 <= r569706;
        double r569708 = l;
        double r569709 = r569693 / r569708;
        double r569710 = r569708 / r569709;
        double r569711 = r569685 * r569685;
        double r569712 = r569710 + r569711;
        double r569713 = r569688 * r569712;
        double r569714 = 4.0;
        double r569715 = r569711 / r569693;
        double r569716 = r569714 * r569715;
        double r569717 = r569713 + r569716;
        double r569718 = sqrt(r569717);
        double r569719 = r569690 / r569718;
        double r569720 = 9.831366213789788e-187;
        bool r569721 = r569685 <= r569720;
        double r569722 = r569688 / r569694;
        double r569723 = r569722 * r569692;
        double r569724 = r569702 - r569723;
        double r569725 = r569697 * r569688;
        double r569726 = r569725 + r569690;
        double r569727 = r569724 + r569726;
        double r569728 = r569690 / r569727;
        double r569729 = 1.0888219445242733e+135;
        bool r569730 = r569685 <= r569729;
        double r569731 = cbrt(r569689);
        double r569732 = r569685 * r569731;
        double r569733 = r569731 * r569731;
        double r569734 = r569732 * r569733;
        double r569735 = r569734 / r569718;
        double r569736 = r569730 ? r569735 : r569728;
        double r569737 = r569721 ? r569728 : r569736;
        double r569738 = r569707 ? r569719 : r569737;
        double r569739 = r569687 ? r569705 : r569738;
        return r569739;
}

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)))))