Average Error: 42.4 → 9.2
Time: 40.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.9501808158464715 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 3.622605389449186 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \ell\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\ \mathbf{elif}\;t \le 5.72877141657189 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \mathbf{elif}\;t \le 3.7543841451851774 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \ell\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \end{array}\]
\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 -1.9501808158464715 \cdot 10^{+103}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\

\mathbf{elif}\;t \le 3.622605389449186 \cdot 10^{-213}:\\
\;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \ell\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\

\mathbf{elif}\;t \le 5.72877141657189 \cdot 10^{-164}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\

\mathbf{elif}\;t \le 3.7543841451851774 \cdot 10^{+122}:\\
\;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \ell\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\

\end{array}
double f(double x, double l, double t) {
        double r1658685 = 2.0;
        double r1658686 = sqrt(r1658685);
        double r1658687 = t;
        double r1658688 = r1658686 * r1658687;
        double r1658689 = x;
        double r1658690 = 1.0;
        double r1658691 = r1658689 + r1658690;
        double r1658692 = r1658689 - r1658690;
        double r1658693 = r1658691 / r1658692;
        double r1658694 = l;
        double r1658695 = r1658694 * r1658694;
        double r1658696 = r1658687 * r1658687;
        double r1658697 = r1658685 * r1658696;
        double r1658698 = r1658695 + r1658697;
        double r1658699 = r1658693 * r1658698;
        double r1658700 = r1658699 - r1658695;
        double r1658701 = sqrt(r1658700);
        double r1658702 = r1658688 / r1658701;
        return r1658702;
}
↓
double f(double x, double l, double t) {
        double r1658703 = t;
        double r1658704 = -1.9501808158464715e+103;
        bool r1658705 = r1658703 <= r1658704;
        double r1658706 = 2.0;
        double r1658707 = sqrt(r1658706);
        double r1658708 = r1658707 * r1658703;
        double r1658709 = 1.0;
        double r1658710 = r1658709 / r1658707;
        double r1658711 = r1658706 / r1658707;
        double r1658712 = r1658710 - r1658711;
        double r1658713 = x;
        double r1658714 = r1658713 * r1658713;
        double r1658715 = r1658703 / r1658714;
        double r1658716 = r1658712 * r1658715;
        double r1658717 = r1658713 * r1658707;
        double r1658718 = r1658706 / r1658717;
        double r1658719 = r1658718 * r1658703;
        double r1658720 = r1658719 + r1658708;
        double r1658721 = r1658716 - r1658720;
        double r1658722 = r1658708 / r1658721;
        double r1658723 = 3.622605389449186e-213;
        bool r1658724 = r1658703 <= r1658723;
        double r1658725 = sqrt(r1658707);
        double r1658726 = r1658703 * r1658725;
        double r1658727 = r1658726 * r1658725;
        double r1658728 = r1658703 * r1658703;
        double r1658729 = l;
        double r1658730 = r1658729 / r1658713;
        double r1658731 = r1658730 * r1658729;
        double r1658732 = r1658728 + r1658731;
        double r1658733 = r1658706 * r1658732;
        double r1658734 = r1658713 / r1658703;
        double r1658735 = r1658703 / r1658734;
        double r1658736 = 4.0;
        double r1658737 = r1658735 * r1658736;
        double r1658738 = r1658733 + r1658737;
        double r1658739 = sqrt(r1658738);
        double r1658740 = r1658727 / r1658739;
        double r1658741 = 5.72877141657189e-164;
        bool r1658742 = r1658703 <= r1658741;
        double r1658743 = r1658720 - r1658716;
        double r1658744 = r1658708 / r1658743;
        double r1658745 = 3.7543841451851774e+122;
        bool r1658746 = r1658703 <= r1658745;
        double r1658747 = r1658746 ? r1658740 : r1658744;
        double r1658748 = r1658742 ? r1658744 : r1658747;
        double r1658749 = r1658724 ? r1658740 : r1658748;
        double r1658750 = r1658705 ? r1658722 : r1658749;
        return r1658750;
}

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 3 regimes

  2. if t < -1.9501808158464715e+103

    1. Initial program 52.4

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -1.9501808158464715e+103 < t < 3.622605389449186e-213 or 5.72877141657189e-164 < t < 3.7543841451851774e+122

    1. Initial program 34.8

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

    2. Taylor expanded around -inf 16.4

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

    3. Simplified16.4

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

    5. Applied *-un-lft-identity16.4

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

    6. Applied times-frac12.0

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

    7. Simplified12.0

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

    9. Applied add-sqr-sqrt12.2

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

    10. Applied associate-*l*12.1

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

    if 3.622605389449186e-213 < t < 5.72877141657189e-164 or 3.7543841451851774e+122 < t

    1. Initial program 54.4

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

    2. Taylor expanded around inf 6.8

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

    3. Simplified6.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\sqrt{2} \cdot t + t \cdot \frac{2}{\sqrt{2} \cdot x}\right) - \frac{t}{x \cdot x} \cdot \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.9501808158464715 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x} - \left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right)}\\ \mathbf{elif}\;t \le 3.622605389449186 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \ell\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\ \mathbf{elif}\;t \le 5.72877141657189 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \mathbf{elif}\;t \le 3.7543841451851774 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(t \cdot \sqrt{\sqrt{2}}\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \ell\right) + \frac{t}{\frac{x}{t}} \cdot 4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{2}{x \cdot \sqrt{2}} \cdot t + \sqrt{2} \cdot t\right) - \left(\frac{1}{\sqrt{2}} - \frac{2}{\sqrt{2}}\right) \cdot \frac{t}{x \cdot x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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)))))