Toniolo and Linder, Equation (13)

Average Error: 34.3 → 29.4

Time: 1.0m

Precision: 64

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -9.379719815826530589603530820932364731187 \cdot 10^{241}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;t \le -1.966200757559735377429830381935693182299 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;t \le 2336909186103.619140625:\\ \;\;\;\;\sqrt{\sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \cdot \sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2023726 = 2.0;
        double r2023727 = n;
        double r2023728 = r2023726 * r2023727;
        double r2023729 = U;
        double r2023730 = r2023728 * r2023729;
        double r2023731 = t;
        double r2023732 = l;
        double r2023733 = r2023732 * r2023732;
        double r2023734 = Om;
        double r2023735 = r2023733 / r2023734;
        double r2023736 = r2023726 * r2023735;
        double r2023737 = r2023731 - r2023736;
        double r2023738 = r2023732 / r2023734;
        double r2023739 = pow(r2023738, r2023726);
        double r2023740 = r2023727 * r2023739;
        double r2023741 = U_;
        double r2023742 = r2023729 - r2023741;
        double r2023743 = r2023740 * r2023742;
        double r2023744 = r2023737 - r2023743;
        double r2023745 = r2023730 * r2023744;
        double r2023746 = sqrt(r2023745);
        return r2023746;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2023747 = t;
        double r2023748 = -9.37971981582653e+241;
        bool r2023749 = r2023747 <= r2023748;
        double r2023750 = 2.0;
        double r2023751 = n;
        double r2023752 = r2023750 * r2023751;
        double r2023753 = U;
        double r2023754 = r2023752 * r2023753;
        double r2023755 = l;
        double r2023756 = Om;
        double r2023757 = r2023755 / r2023756;
        double r2023758 = r2023755 * r2023757;
        double r2023759 = r2023750 * r2023758;
        double r2023760 = r2023747 - r2023759;
        double r2023761 = r2023754 * r2023760;
        double r2023762 = sqrt(r2023761);
        double r2023763 = -1.9662007575597354e-176;
        bool r2023764 = r2023747 <= r2023763;
        double r2023765 = U_;
        double r2023766 = r2023765 - r2023753;
        double r2023767 = 2.0;
        double r2023768 = r2023750 / r2023767;
        double r2023769 = pow(r2023757, r2023768);
        double r2023770 = r2023751 * r2023769;
        double r2023771 = r2023770 * r2023769;
        double r2023772 = fma(r2023766, r2023771, r2023760);
        double r2023773 = r2023753 * r2023772;
        double r2023774 = r2023773 * r2023752;
        double r2023775 = sqrt(r2023774);
        double r2023776 = 2336909186103.619;
        bool r2023777 = r2023747 <= r2023776;
        double r2023778 = r2023772 * r2023754;
        double r2023779 = sqrt(r2023778);
        double r2023780 = sqrt(r2023779);
        double r2023781 = cbrt(r2023778);
        double r2023782 = r2023781 * r2023781;
        double r2023783 = r2023781 * r2023782;
        double r2023784 = sqrt(r2023783);
        double r2023785 = sqrt(r2023784);
        double r2023786 = r2023780 * r2023785;
        double r2023787 = sqrt(r2023754);
        double r2023788 = sqrt(r2023772);
        double r2023789 = r2023787 * r2023788;
        double r2023790 = r2023777 ? r2023786 : r2023789;
        double r2023791 = r2023764 ? r2023775 : r2023790;
        double r2023792 = r2023749 ? r2023762 : r2023791;
        return r2023792;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 4 regimes

2. if t < -9.37971981582653e+241

   1. Initial program 41.0

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 38.7

      \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}\]

   3. Taylor expanded around 0 40.3

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}}\]

   4. Simplified 37.9

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}\]

   if -9.37971981582653e+241 < t < -1.9662007575597354e-176

   1. Initial program 32.5

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 29.7

      \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}\]
   3. Using strategy rm

   4. Applied sqr-pow 29.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}\]

   5. Applied associate-*r* 29.0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}\]
   6. Using strategy rm

   7. Applied associate-*l* 28.6

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)\right)}}\]

   if -1.9662007575597354e-176 < t < 2336909186103.619

   1. Initial program 34.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 31.6

      \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}\]
   3. Using strategy rm

   4. Applied sqr-pow 31.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}\]

   5. Applied associate-*r* 30.5

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt 30.6

      \[\leadsto \color{blue}{\sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 30.7

      \[\leadsto \sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}\right) \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}}} \cdot \sqrt{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}\]

   if 2336909186103.619 < t

   1. Initial program 34.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 32.4

      \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, n \cdot {\left(\frac{\ell}{Om}\right)}^{2}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}\]
   3. Using strategy rm

   4. Applied sqr-pow 32.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}\]

   5. Applied associate-*r* 31.9

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}\]
   6. Using strategy rm

   7. Applied sqrt-prod 26.8

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 29.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.379719815826530589603530820932364731187 \cdot 10^{241}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;t \le -1.966200757559735377429830381935693182299 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;t \le 2336909186103.619140625:\\ \;\;\;\;\sqrt{\sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}} \cdot \sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))