Average Error: 43.2 → 9.4
Time: 27.9s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.940005881918408859869096974602770049363 \cdot 10^{94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)} - \frac{t}{x \cdot \sqrt{2}}\right) - \frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.839658841198905283015739574372348749582 \cdot 10^{-241} \lor \neg \left(t \le 3.247980644380315344262592988201111260205 \cdot 10^{-157}\right) \land t \le 4.214549585149478093549971775733778272653 \cdot 10^{55}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{\left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + \frac{\left(t \cdot 4\right) \cdot t}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\sqrt{2} \cdot t + \frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x}\right) + \frac{2 \cdot t}{x \cdot \sqrt{2}}\right) - \frac{2 \cdot t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}}\\ \end{array}\]
\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 -1.940005881918408859869096974602770049363 \cdot 10^{94}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)} - \frac{t}{x \cdot \sqrt{2}}\right) - \frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x}\right) - \sqrt{2} \cdot t}\\

\mathbf{elif}\;t \le 1.839658841198905283015739574372348749582 \cdot 10^{-241} \lor \neg \left(t \le 3.247980644380315344262592988201111260205 \cdot 10^{-157}\right) \land t \le 4.214549585149478093549971775733778272653 \cdot 10^{55}:\\
\;\;\;\;\frac{\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{\left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + \frac{\left(t \cdot 4\right) \cdot t}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r47712 = 2.0;
        double r47713 = sqrt(r47712);
        double r47714 = t;
        double r47715 = r47713 * r47714;
        double r47716 = x;
        double r47717 = 1.0;
        double r47718 = r47716 + r47717;
        double r47719 = r47716 - r47717;
        double r47720 = r47718 / r47719;
        double r47721 = l;
        double r47722 = r47721 * r47721;
        double r47723 = r47714 * r47714;
        double r47724 = r47712 * r47723;
        double r47725 = r47722 + r47724;
        double r47726 = r47720 * r47725;
        double r47727 = r47726 - r47722;
        double r47728 = sqrt(r47727);
        double r47729 = r47715 / r47728;
        return r47729;
}
↓
double f(double x, double l, double t) {
        double r47730 = t;
        double r47731 = -1.940005881918409e+94;
        bool r47732 = r47730 <= r47731;
        double r47733 = 2.0;
        double r47734 = sqrt(r47733);
        double r47735 = r47734 * r47730;
        double r47736 = r47733 * r47734;
        double r47737 = x;
        double r47738 = r47737 * r47737;
        double r47739 = r47736 * r47738;
        double r47740 = r47730 / r47739;
        double r47741 = r47737 * r47734;
        double r47742 = r47730 / r47741;
        double r47743 = r47740 - r47742;
        double r47744 = r47733 * r47743;
        double r47745 = r47733 * r47730;
        double r47746 = r47745 / r47734;
        double r47747 = r47746 / r47738;
        double r47748 = r47744 - r47747;
        double r47749 = r47748 - r47735;
        double r47750 = r47735 / r47749;
        double r47751 = 1.8396588411989053e-241;
        bool r47752 = r47730 <= r47751;
        double r47753 = 3.2479806443803153e-157;
        bool r47754 = r47730 <= r47753;
        double r47755 = !r47754;
        double r47756 = 4.214549585149478e+55;
        bool r47757 = r47730 <= r47756;
        bool r47758 = r47755 && r47757;
        bool r47759 = r47752 || r47758;
        double r47760 = sqrt(r47734);
        double r47761 = r47760 * r47730;
        double r47762 = r47761 * r47760;
        double r47763 = l;
        double r47764 = r47737 / r47763;
        double r47765 = r47763 / r47764;
        double r47766 = r47733 * r47765;
        double r47767 = 4.0;
        double r47768 = r47730 * r47767;
        double r47769 = r47768 * r47730;
        double r47770 = r47769 / r47737;
        double r47771 = r47766 + r47770;
        double r47772 = r47730 * r47730;
        double r47773 = r47733 * r47772;
        double r47774 = r47771 + r47773;
        double r47775 = sqrt(r47774);
        double r47776 = r47762 / r47775;
        double r47777 = r47735 + r47747;
        double r47778 = r47745 / r47741;
        double r47779 = r47777 + r47778;
        double r47780 = r47745 / r47739;
        double r47781 = r47779 - r47780;
        double r47782 = r47735 / r47781;
        double r47783 = r47759 ? r47776 : r47782;
        double r47784 = r47732 ? r47750 : r47783;
        return r47784;
}

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 < -1.940005881918409e+94

    1. Initial program 49.9

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.4

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

    if -1.940005881918409e+94 < t < 1.8396588411989053e-241 or 3.2479806443803153e-157 < t < 4.214549585149478e+55

    1. Initial program 37.0

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

    2. Taylor expanded around inf 15.8

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

    3. Simplified15.8

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

    5. Applied distribute-lft-in15.8

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

    6. Applied associate-+l+15.8

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

    7. Simplified15.8

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

    9. Applied add-sqr-sqrt15.8

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

    10. Applied sqrt-prod16.0

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

    11. Applied associate-*l*15.9

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

    12. Simplified15.9

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

    13. Taylor expanded around 0 15.9

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

    14. Simplified12.3

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

    if 1.8396588411989053e-241 < t < 3.2479806443803153e-157 or 4.214549585149478e+55 < t

    1. Initial program 48.9

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

    2. Taylor expanded around inf 8.8

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

    3. Simplified8.8

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.940005881918408859869096974602770049363 \cdot 10^{94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)} - \frac{t}{x \cdot \sqrt{2}}\right) - \frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x}\right) - \sqrt{2} \cdot t}\\ \mathbf{elif}\;t \le 1.839658841198905283015739574372348749582 \cdot 10^{-241} \lor \neg \left(t \le 3.247980644380315344262592988201111260205 \cdot 10^{-157}\right) \land t \le 4.214549585149478093549971775733778272653 \cdot 10^{55}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{2}} \cdot t\right) \cdot \sqrt{\sqrt{2}}}{\sqrt{\left(2 \cdot \frac{\ell}{\frac{x}{\ell}} + \frac{\left(t \cdot 4\right) \cdot t}{x}\right) + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\left(\sqrt{2} \cdot t + \frac{\frac{2 \cdot t}{\sqrt{2}}}{x \cdot x}\right) + \frac{2 \cdot t}{x \cdot \sqrt{2}}\right) - \frac{2 \cdot t}{\left(2 \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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)))))