Average Error: 43.4 → 10.1
Time: 32.5s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.653947709277397100618091830748612157853 \cdot 10^{151}:\\ \;\;\;\;\frac{\sqrt{2}}{\left(\left(-\sqrt{2}\right) \cdot t - \frac{\frac{2 \cdot t}{x}}{\sqrt{2}}\right) \cdot \frac{1}{t}}\\ \mathbf{elif}\;t \le 2.888100493419352308068723480891983028538 \cdot 10^{-285}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{\ell \cdot 2}{\frac{x}{\ell}} + t \cdot \left(2 \cdot t\right)\right) + \frac{t}{\frac{x}{t}} \cdot 4}}{t}}\\ \mathbf{elif}\;t \le 2.813385033315984081750629552156790319468 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2}}\\ \mathbf{elif}\;t \le 1913716763334430514281189625298944:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{\ell \cdot 2}{\frac{x}{\ell}} + t \cdot \left(2 \cdot t\right)\right) + \frac{t}{\frac{x}{t}} \cdot 4}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2 \cdot t}{\sqrt{2} \cdot x} + \sqrt{2} \cdot t}{t}}\\ \end{array}\]
\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 -1.653947709277397100618091830748612157853 \cdot 10^{151}:\\
\;\;\;\;\frac{\sqrt{2}}{\left(\left(-\sqrt{2}\right) \cdot t - \frac{\frac{2 \cdot t}{x}}{\sqrt{2}}\right) \cdot \frac{1}{t}}\\

\mathbf{elif}\;t \le 2.888100493419352308068723480891983028538 \cdot 10^{-285}:\\
\;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{\ell \cdot 2}{\frac{x}{\ell}} + t \cdot \left(2 \cdot t\right)\right) + \frac{t}{\frac{x}{t}} \cdot 4}}{t}}\\

\mathbf{elif}\;t \le 2.813385033315984081750629552156790319468 \cdot 10^{-161}:\\
\;\;\;\;\frac{\sqrt{2}}{\sqrt{2}}\\

\mathbf{elif}\;t \le 1913716763334430514281189625298944:\\
\;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{\ell \cdot 2}{\frac{x}{\ell}} + t \cdot \left(2 \cdot t\right)\right) + \frac{t}{\frac{x}{t}} \cdot 4}}{t}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2 \cdot t}{\sqrt{2} \cdot x} + \sqrt{2} \cdot t}{t}}\\

\end{array}
double f(double x, double l, double t) {
        double r46747 = 2.0;
        double r46748 = sqrt(r46747);
        double r46749 = t;
        double r46750 = r46748 * r46749;
        double r46751 = x;
        double r46752 = 1.0;
        double r46753 = r46751 + r46752;
        double r46754 = r46751 - r46752;
        double r46755 = r46753 / r46754;
        double r46756 = l;
        double r46757 = r46756 * r46756;
        double r46758 = r46749 * r46749;
        double r46759 = r46747 * r46758;
        double r46760 = r46757 + r46759;
        double r46761 = r46755 * r46760;
        double r46762 = r46761 - r46757;
        double r46763 = sqrt(r46762);
        double r46764 = r46750 / r46763;
        return r46764;
}
↓
double f(double x, double l, double t) {
        double r46765 = t;
        double r46766 = -1.653947709277397e+151;
        bool r46767 = r46765 <= r46766;
        double r46768 = 2.0;
        double r46769 = sqrt(r46768);
        double r46770 = -r46769;
        double r46771 = r46770 * r46765;
        double r46772 = r46768 * r46765;
        double r46773 = x;
        double r46774 = r46772 / r46773;
        double r46775 = r46774 / r46769;
        double r46776 = r46771 - r46775;
        double r46777 = 1.0;
        double r46778 = r46777 / r46765;
        double r46779 = r46776 * r46778;
        double r46780 = r46769 / r46779;
        double r46781 = 2.8881004934193523e-285;
        bool r46782 = r46765 <= r46781;
        double r46783 = l;
        double r46784 = r46783 * r46768;
        double r46785 = r46773 / r46783;
        double r46786 = r46784 / r46785;
        double r46787 = r46765 * r46772;
        double r46788 = r46786 + r46787;
        double r46789 = r46773 / r46765;
        double r46790 = r46765 / r46789;
        double r46791 = 4.0;
        double r46792 = r46790 * r46791;
        double r46793 = r46788 + r46792;
        double r46794 = sqrt(r46793);
        double r46795 = r46794 / r46765;
        double r46796 = r46769 / r46795;
        double r46797 = 2.813385033315984e-161;
        bool r46798 = r46765 <= r46797;
        double r46799 = r46769 / r46769;
        double r46800 = 1.9137167633344305e+33;
        bool r46801 = r46765 <= r46800;
        double r46802 = r46769 * r46773;
        double r46803 = r46772 / r46802;
        double r46804 = r46769 * r46765;
        double r46805 = r46803 + r46804;
        double r46806 = r46805 / r46765;
        double r46807 = r46769 / r46806;
        double r46808 = r46801 ? r46796 : r46807;
        double r46809 = r46798 ? r46799 : r46808;
        double r46810 = r46782 ? r46796 : r46809;
        double r46811 = r46767 ? r46780 : r46810;
        return r46811;
}

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 < -1.653947709277397e+151

    1. Initial program 61.9

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

    2. Simplified61.9

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

    3. Taylor expanded around inf 62.3

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

    4. Simplified62.3

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

    5. Taylor expanded around 0 62.3

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

    6. Simplified61.8

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

    8. Applied div-inv61.8

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

    9. Taylor expanded around -inf 2.4

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

    10. Simplified2.5

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

    if -1.653947709277397e+151 < t < 2.8881004934193523e-285 or 2.813385033315984e-161 < t < 1.9137167633344305e+33

    1. Initial program 34.1

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

    2. Simplified34.1

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

    3. Taylor expanded around inf 15.0

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

    4. Simplified15.0

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

    5. Taylor expanded around 0 15.0

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

    6. Simplified10.9

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

    8. Applied div-inv11.0

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

    10. Applied un-div-inv10.9

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

    if 2.8881004934193523e-285 < t < 2.813385033315984e-161

    1. Initial program 62.9

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

    2. Simplified62.9

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

    3. Taylor expanded around inf 36.8

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2}}}\]

    if 1.9137167633344305e+33 < t

    1. Initial program 44.0

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

    2. Simplified44.0

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

    3. Taylor expanded around inf 4.8

      \[\leadsto \frac{\sqrt{2}}{\frac{\color{blue}{\sqrt{2} \cdot t + 2 \cdot \frac{t}{\sqrt{2} \cdot x}}}{t}}\]

    4. Simplified4.9

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.653947709277397100618091830748612157853 \cdot 10^{151}:\\ \;\;\;\;\frac{\sqrt{2}}{\left(\left(-\sqrt{2}\right) \cdot t - \frac{\frac{2 \cdot t}{x}}{\sqrt{2}}\right) \cdot \frac{1}{t}}\\ \mathbf{elif}\;t \le 2.888100493419352308068723480891983028538 \cdot 10^{-285}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{\ell \cdot 2}{\frac{x}{\ell}} + t \cdot \left(2 \cdot t\right)\right) + \frac{t}{\frac{x}{t}} \cdot 4}}{t}}\\ \mathbf{elif}\;t \le 2.813385033315984081750629552156790319468 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2}}\\ \mathbf{elif}\;t \le 1913716763334430514281189625298944:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{\ell \cdot 2}{\frac{x}{\ell}} + t \cdot \left(2 \cdot t\right)\right) + \frac{t}{\frac{x}{t}} \cdot 4}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\frac{2 \cdot t}{\sqrt{2} \cdot x} + \sqrt{2} \cdot t}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))