Average Error: 43.3 → 9.6
Time: 28.8s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.497186697554710081473314925566838223756 \cdot 10^{82}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -7.265062434046062880671280333196464651815 \cdot 10^{-166}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{elif}\;t \le -3.600112372186980511777207822247642078219 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 6.841551951533339534532177087886888745651 \cdot 10^{60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right)}\\ \end{array}\]
\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 -1.497186697554710081473314925566838223756 \cdot 10^{82}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le -7.265062434046062880671280333196464651815 \cdot 10^{-166}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}}}\\

\mathbf{elif}\;t \le -3.600112372186980511777207822247642078219 \cdot 10^{-276}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\

\mathbf{elif}\;t \le 6.841551951533339534532177087886888745651 \cdot 10^{60}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r43638 = 2.0;
        double r43639 = sqrt(r43638);
        double r43640 = t;
        double r43641 = r43639 * r43640;
        double r43642 = x;
        double r43643 = 1.0;
        double r43644 = r43642 + r43643;
        double r43645 = r43642 - r43643;
        double r43646 = r43644 / r43645;
        double r43647 = l;
        double r43648 = r43647 * r43647;
        double r43649 = r43640 * r43640;
        double r43650 = r43638 * r43649;
        double r43651 = r43648 + r43650;
        double r43652 = r43646 * r43651;
        double r43653 = r43652 - r43648;
        double r43654 = sqrt(r43653);
        double r43655 = r43641 / r43654;
        return r43655;
}
↓
double f(double x, double l, double t) {
        double r43656 = t;
        double r43657 = -1.49718669755471e+82;
        bool r43658 = r43656 <= r43657;
        double r43659 = 2.0;
        double r43660 = sqrt(r43659);
        double r43661 = r43660 * r43656;
        double r43662 = x;
        double r43663 = 2.0;
        double r43664 = pow(r43662, r43663);
        double r43665 = r43656 / r43664;
        double r43666 = r43660 * r43659;
        double r43667 = r43659 / r43666;
        double r43668 = r43659 / r43660;
        double r43669 = r43667 - r43668;
        double r43670 = r43665 * r43669;
        double r43671 = r43670 - r43661;
        double r43672 = r43660 * r43662;
        double r43673 = r43656 / r43672;
        double r43674 = r43659 * r43673;
        double r43675 = r43671 - r43674;
        double r43676 = r43661 / r43675;
        double r43677 = -7.265062434046063e-166;
        bool r43678 = r43656 <= r43677;
        double r43679 = r43656 * r43660;
        double r43680 = l;
        double r43681 = r43662 / r43680;
        double r43682 = r43680 / r43681;
        double r43683 = r43656 * r43656;
        double r43684 = r43682 + r43683;
        double r43685 = r43684 * r43659;
        double r43686 = 4.0;
        double r43687 = pow(r43656, r43663);
        double r43688 = r43687 / r43662;
        double r43689 = r43686 * r43688;
        double r43690 = r43685 + r43689;
        double r43691 = sqrt(r43690);
        double r43692 = r43679 / r43691;
        double r43693 = -3.6001123721869805e-276;
        bool r43694 = r43656 <= r43693;
        double r43695 = 6.84155195153334e+60;
        bool r43696 = r43656 <= r43695;
        double r43697 = r43674 + r43679;
        double r43698 = r43697 - r43670;
        double r43699 = r43661 / r43698;
        double r43700 = r43696 ? r43692 : r43699;
        double r43701 = r43694 ? r43676 : r43700;
        double r43702 = r43678 ? r43692 : r43701;
        double r43703 = r43658 ? r43676 : r43702;
        return r43703;
}

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 < -1.49718669755471e+82 or -7.265062434046063e-166 < t < -3.6001123721869805e-276

    1. Initial program 52.7

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

    2. Taylor expanded around -inf 11.4

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

    3. Simplified11.4

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

    if -1.49718669755471e+82 < t < -7.265062434046063e-166 or -3.6001123721869805e-276 < t < 6.84155195153334e+60

    1. Initial program 36.4

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

    2. Taylor expanded around inf 16.1

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

    3. Simplified16.1

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

    5. Applied sqr-pow16.1

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

    6. Applied associate-/l*11.6

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

    7. Simplified11.6

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

    if 6.84155195153334e+60 < t

    1. Initial program 45.6

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

    2. Taylor expanded around inf 3.4

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

    3. Simplified3.4

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.497186697554710081473314925566838223756 \cdot 10^{82}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -7.265062434046062880671280333196464651815 \cdot 10^{-166}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{elif}\;t \le -3.600112372186980511777207822247642078219 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 6.841551951533339534532177087886888745651 \cdot 10^{60}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right) \cdot 2 + 4 \cdot \frac{{t}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right) - \frac{t}{{x}^{2}} \cdot \left(\frac{2}{\sqrt{2} \cdot 2} - \frac{2}{\sqrt{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))))