Average Error: 43.2 → 9.6
Time: 9.9s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.24285158980977657 \cdot 10^{105}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 2.7339932236656501 \cdot 10^{113}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \frac{\ell}{\frac{x}{\ell}}} \cdot \sqrt{{t}^{2} + \frac{\ell}{\frac{x}{\ell}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot t + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{{\left(\sqrt{2}\right)}^{3}}\right)\right)}\\ \end{array}\]
\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 -2.24285158980977657 \cdot 10^{105}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 2.7339932236656501 \cdot 10^{113}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \frac{\ell}{\frac{x}{\ell}}} \cdot \sqrt{{t}^{2} + \frac{\ell}{\frac{x}{\ell}}}\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r31723 = 2.0;
        double r31724 = sqrt(r31723);
        double r31725 = t;
        double r31726 = r31724 * r31725;
        double r31727 = x;
        double r31728 = 1.0;
        double r31729 = r31727 + r31728;
        double r31730 = r31727 - r31728;
        double r31731 = r31729 / r31730;
        double r31732 = l;
        double r31733 = r31732 * r31732;
        double r31734 = r31725 * r31725;
        double r31735 = r31723 * r31734;
        double r31736 = r31733 + r31735;
        double r31737 = r31731 * r31736;
        double r31738 = r31737 - r31733;
        double r31739 = sqrt(r31738);
        double r31740 = r31726 / r31739;
        return r31740;
}
↓
double f(double x, double l, double t) {
        double r31741 = t;
        double r31742 = -2.2428515898097766e+105;
        bool r31743 = r31741 <= r31742;
        double r31744 = 2.0;
        double r31745 = sqrt(r31744);
        double r31746 = r31745 * r31741;
        double r31747 = 3.0;
        double r31748 = pow(r31745, r31747);
        double r31749 = x;
        double r31750 = 2.0;
        double r31751 = pow(r31749, r31750);
        double r31752 = r31748 * r31751;
        double r31753 = r31741 / r31752;
        double r31754 = r31745 * r31749;
        double r31755 = r31741 / r31754;
        double r31756 = r31753 - r31755;
        double r31757 = r31744 * r31756;
        double r31758 = r31757 - r31746;
        double r31759 = r31746 / r31758;
        double r31760 = 2.73399322366565e+113;
        bool r31761 = r31741 <= r31760;
        double r31762 = cbrt(r31745);
        double r31763 = r31762 * r31762;
        double r31764 = r31762 * r31741;
        double r31765 = r31763 * r31764;
        double r31766 = 4.0;
        double r31767 = pow(r31741, r31750);
        double r31768 = r31767 / r31749;
        double r31769 = r31766 * r31768;
        double r31770 = l;
        double r31771 = r31749 / r31770;
        double r31772 = r31770 / r31771;
        double r31773 = r31767 + r31772;
        double r31774 = sqrt(r31773);
        double r31775 = r31774 * r31774;
        double r31776 = r31744 * r31775;
        double r31777 = r31769 + r31776;
        double r31778 = sqrt(r31777);
        double r31779 = r31765 / r31778;
        double r31780 = r31744 * r31755;
        double r31781 = r31741 / r31751;
        double r31782 = r31744 / r31745;
        double r31783 = r31744 / r31748;
        double r31784 = r31782 - r31783;
        double r31785 = r31781 * r31784;
        double r31786 = r31780 + r31785;
        double r31787 = r31746 + r31786;
        double r31788 = r31746 / r31787;
        double r31789 = r31761 ? r31779 : r31788;
        double r31790 = r31743 ? r31759 : r31789;
        return r31790;
}

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 < -2.2428515898097766e+105

    1. Initial program 50.5

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

    2. Taylor expanded around inf 50.7

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

    3. Simplified50.7

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

    5. Applied unpow250.7

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

    6. Applied associate-/l*48.9

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

    7. Taylor expanded around -inf 2.7

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

    8. Simplified2.7

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

    if -2.2428515898097766e+105 < t < 2.73399322366565e+113

    1. Initial program 38.1

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

    2. Taylor expanded around inf 17.5

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

    3. Simplified17.5

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

    5. Applied unpow217.5

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

    6. Applied associate-/l*13.7

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

    8. Applied add-cube-cbrt13.7

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

    9. Applied associate-*l*13.7

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

    11. Applied add-sqr-sqrt13.7

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

    if 2.73399322366565e+113 < t

    1. Initial program 52.6

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

    2. Taylor expanded around inf 2.9

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

    3. Simplified2.9

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.24285158980977657 \cdot 10^{105}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 2.7339932236656501 \cdot 10^{113}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot t\right)}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \frac{\ell}{\frac{x}{\ell}}} \cdot \sqrt{{t}^{2} + \frac{\ell}{\frac{x}{\ell}}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot t + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2}} - \frac{2}{{\left(\sqrt{2}\right)}^{3}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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)))))