Average Error: 42.9 → 8.7
Time: 8.4s
Precision: 64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.703772595808916454667436984685994375446 \cdot 10^{141}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le 2.555685633755422263184735039093424552333 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \left(\sqrt{2} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le 5.748160003020186103470224388296769238726 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 4.837673144276178631404051943017687346215 \cdot 10^{123}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \left(\sqrt{2} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\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 -8.703772595808916454667436984685994375446 \cdot 10^{141}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\

\mathbf{elif}\;t \le 2.555685633755422263184735039093424552333 \cdot 10^{-264}:\\
\;\;\;\;\frac{\sqrt{1} \cdot \left(\sqrt{2} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

\mathbf{elif}\;t \le 5.748160003020186103470224388296769238726 \cdot 10^{-158}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\mathbf{elif}\;t \le 4.837673144276178631404051943017687346215 \cdot 10^{123}:\\
\;\;\;\;\frac{\sqrt{1} \cdot \left(\sqrt{2} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\

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

\end{array}
double f(double x, double l, double t) {
        double r37723 = 2.0;
        double r37724 = sqrt(r37723);
        double r37725 = t;
        double r37726 = r37724 * r37725;
        double r37727 = x;
        double r37728 = 1.0;
        double r37729 = r37727 + r37728;
        double r37730 = r37727 - r37728;
        double r37731 = r37729 / r37730;
        double r37732 = l;
        double r37733 = r37732 * r37732;
        double r37734 = r37725 * r37725;
        double r37735 = r37723 * r37734;
        double r37736 = r37733 + r37735;
        double r37737 = r37731 * r37736;
        double r37738 = r37737 - r37733;
        double r37739 = sqrt(r37738);
        double r37740 = r37726 / r37739;
        return r37740;
}
↓
double f(double x, double l, double t) {
        double r37741 = t;
        double r37742 = -8.703772595808916e+141;
        bool r37743 = r37741 <= r37742;
        double r37744 = 2.0;
        double r37745 = sqrt(r37744);
        double r37746 = r37745 * r37741;
        double r37747 = 3.0;
        double r37748 = pow(r37745, r37747);
        double r37749 = x;
        double r37750 = 2.0;
        double r37751 = pow(r37749, r37750);
        double r37752 = r37748 * r37751;
        double r37753 = r37741 / r37752;
        double r37754 = r37745 * r37751;
        double r37755 = r37741 / r37754;
        double r37756 = r37745 * r37749;
        double r37757 = r37741 / r37756;
        double r37758 = r37741 * r37745;
        double r37759 = fma(r37744, r37757, r37758);
        double r37760 = fma(r37744, r37755, r37759);
        double r37761 = -r37760;
        double r37762 = fma(r37744, r37753, r37761);
        double r37763 = r37746 / r37762;
        double r37764 = 2.5556856337554223e-264;
        bool r37765 = r37741 <= r37764;
        double r37766 = 1.0;
        double r37767 = sqrt(r37766);
        double r37768 = r37767 * r37746;
        double r37769 = pow(r37741, r37750);
        double r37770 = l;
        double r37771 = r37749 / r37770;
        double r37772 = r37770 / r37771;
        double r37773 = 4.0;
        double r37774 = r37769 / r37749;
        double r37775 = r37773 * r37774;
        double r37776 = fma(r37744, r37772, r37775);
        double r37777 = fma(r37744, r37769, r37776);
        double r37778 = sqrt(r37777);
        double r37779 = r37768 / r37778;
        double r37780 = 5.748160003020186e-158;
        bool r37781 = r37741 <= r37780;
        double r37782 = r37744 * r37753;
        double r37783 = r37759 - r37782;
        double r37784 = fma(r37744, r37755, r37783);
        double r37785 = r37746 / r37784;
        double r37786 = 4.8376731442761786e+123;
        bool r37787 = r37741 <= r37786;
        double r37788 = r37787 ? r37779 : r37785;
        double r37789 = r37781 ? r37785 : r37788;
        double r37790 = r37765 ? r37779 : r37789;
        double r37791 = r37743 ? r37763 : r37790;
        return r37791;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 3 regimes

  2. if t < -8.703772595808916e+141

    1. Initial program 58.9

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

    2. Taylor expanded around -inf 2.1

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

    3. Simplified2.1

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

    if -8.703772595808916e+141 < t < 2.5556856337554223e-264 or 5.748160003020186e-158 < t < 4.8376731442761786e+123

    1. Initial program 32.6

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

    2. Taylor expanded around inf 14.2

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

    3. Simplified14.2

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

    5. Applied unpow214.2

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

    6. Applied associate-/l*9.6

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

    8. Applied *-un-lft-identity9.6

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

    9. Applied sqrt-prod9.6

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

    10. Applied associate-*l*9.6

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

    if 2.5556856337554223e-264 < t < 5.748160003020186e-158 or 4.8376731442761786e+123 < t

    1. Initial program 57.1

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

    2. Taylor expanded around inf 10.8

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

    3. Simplified10.8

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.703772595808916454667436984685994375446 \cdot 10^{141}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}, -\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right)\right)\right)}\\ \mathbf{elif}\;t \le 2.555685633755422263184735039093424552333 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \left(\sqrt{2} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \le 5.748160003020186103470224388296769238726 \cdot 10^{-158}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 4.837673144276178631404051943017687346215 \cdot 10^{123}:\\ \;\;\;\;\frac{\sqrt{1} \cdot \left(\sqrt{2} \cdot t\right)}{\sqrt{\mathsf{fma}\left(2, {t}^{2}, \mathsf{fma}\left(2, \frac{\ell}{\frac{x}{\ell}}, 4 \cdot \frac{{t}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot {x}^{2}}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, t \cdot \sqrt{2}\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))))