Average Error: 42.4 → 9.5
Time: 24.0s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.526796709366070268539771028335785888924 \cdot 10^{142}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -1.100438500662438394216054691601263900469 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le 6.50243723828802950164499168405456194956 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 55644010688926858222764032:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -2.526796709366070268539771028335785888924 \cdot 10^{142}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le -1.100438500662438394216054691601263900469 \cdot 10^{-166}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\

\mathbf{elif}\;t \le 6.50243723828802950164499168405456194956 \cdot 10^{-304}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le 55644010688926858222764032:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\end{array}
double f(double x, double l, double t) {
        double r45703 = 2.0;
        double r45704 = sqrt(r45703);
        double r45705 = t;
        double r45706 = r45704 * r45705;
        double r45707 = x;
        double r45708 = 1.0;
        double r45709 = r45707 + r45708;
        double r45710 = r45707 - r45708;
        double r45711 = r45709 / r45710;
        double r45712 = l;
        double r45713 = r45712 * r45712;
        double r45714 = r45705 * r45705;
        double r45715 = r45703 * r45714;
        double r45716 = r45713 + r45715;
        double r45717 = r45711 * r45716;
        double r45718 = r45717 - r45713;
        double r45719 = sqrt(r45718);
        double r45720 = r45706 / r45719;
        return r45720;
}
↓
double f(double x, double l, double t) {
        double r45721 = t;
        double r45722 = -2.52679670936607e+142;
        bool r45723 = r45721 <= r45722;
        double r45724 = 2.0;
        double r45725 = sqrt(r45724);
        double r45726 = r45725 * r45721;
        double r45727 = 3.0;
        double r45728 = pow(r45725, r45727);
        double r45729 = x;
        double r45730 = 2.0;
        double r45731 = pow(r45729, r45730);
        double r45732 = r45728 * r45731;
        double r45733 = r45721 / r45732;
        double r45734 = r45724 * r45733;
        double r45735 = r45725 * r45729;
        double r45736 = r45721 / r45735;
        double r45737 = r45724 * r45736;
        double r45738 = r45721 * r45725;
        double r45739 = r45737 + r45738;
        double r45740 = r45734 - r45739;
        double r45741 = r45726 / r45740;
        double r45742 = -1.1004385006624384e-166;
        bool r45743 = r45721 <= r45742;
        double r45744 = 4.0;
        double r45745 = pow(r45721, r45730);
        double r45746 = r45745 / r45729;
        double r45747 = r45744 * r45746;
        double r45748 = l;
        double r45749 = fabs(r45748);
        double r45750 = cbrt(r45729);
        double r45751 = r45749 / r45750;
        double r45752 = r45751 / r45750;
        double r45753 = r45752 * r45751;
        double r45754 = r45745 + r45753;
        double r45755 = r45724 * r45754;
        double r45756 = r45747 + r45755;
        double r45757 = sqrt(r45756);
        double r45758 = r45726 / r45757;
        double r45759 = 6.5024372382880295e-304;
        bool r45760 = r45721 <= r45759;
        double r45761 = 5.564401068892686e+25;
        bool r45762 = r45721 <= r45761;
        double r45763 = r45725 * r45731;
        double r45764 = r45721 / r45763;
        double r45765 = r45764 + r45736;
        double r45766 = r45724 * r45765;
        double r45767 = r45726 - r45734;
        double r45768 = r45766 + r45767;
        double r45769 = r45726 / r45768;
        double r45770 = r45762 ? r45758 : r45769;
        double r45771 = r45760 ? r45741 : r45770;
        double r45772 = r45743 ? r45758 : r45771;
        double r45773 = r45723 ? r45741 : r45772;
        return r45773;
}

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 < -2.52679670936607e+142 or -1.1004385006624384e-166 < t < 6.5024372382880295e-304

    1. Initial program 60.2

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

    2. Taylor expanded around inf 50.5

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

    3. Simplified50.5

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

    4. Taylor expanded around -inf 15.1

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

    if -2.52679670936607e+142 < t < -1.1004385006624384e-166 or 6.5024372382880295e-304 < t < 5.564401068892686e+25

    1. Initial program 32.5

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

    2. Taylor expanded around inf 13.8

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

    3. Simplified13.8

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

    5. Applied add-cube-cbrt13.9

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

    6. Applied add-sqr-sqrt13.9

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

    7. Applied times-frac13.9

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

    8. Simplified13.9

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

    9. Simplified9.5

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

    if 5.564401068892686e+25 < t

    1. Initial program 42.9

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

    2. Taylor expanded around inf 3.9

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

    3. Simplified3.9

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.526796709366070268539771028335785888924 \cdot 10^{142}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -1.100438500662438394216054691601263900469 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le 6.50243723828802950164499168405456194956 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 55644010688926858222764032:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\frac{\left|\ell\right|}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{\left|\ell\right|}{\sqrt[3]{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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)))))