Toniolo and Linder, Equation (13)

Average Error: 34.9 → 28.8

Time: 1.6m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -6.56714979925425613 \cdot 10^{186}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \le 3.330205482825987 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot U}\\ \mathbf{elif}\;n \le 1.2408046941130587 \cdot 10^{-213} \lor \neg \left(n \le 2.96621220568580719 \cdot 10^{43}\right):\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r247717 = 2.0;
        double r247718 = n;
        double r247719 = r247717 * r247718;
        double r247720 = U;
        double r247721 = r247719 * r247720;
        double r247722 = t;
        double r247723 = l;
        double r247724 = r247723 * r247723;
        double r247725 = Om;
        double r247726 = r247724 / r247725;
        double r247727 = r247717 * r247726;
        double r247728 = r247722 - r247727;
        double r247729 = r247723 / r247725;
        double r247730 = pow(r247729, r247717);
        double r247731 = r247718 * r247730;
        double r247732 = U_;
        double r247733 = r247720 - r247732;
        double r247734 = r247731 * r247733;
        double r247735 = r247728 - r247734;
        double r247736 = r247721 * r247735;
        double r247737 = sqrt(r247736);
        return r247737;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r247738 = n;
        double r247739 = -6.567149799254256e+186;
        bool r247740 = r247738 <= r247739;
        double r247741 = 2.0;
        double r247742 = r247741 * r247738;
        double r247743 = U;
        double r247744 = r247742 * r247743;
        double r247745 = t;
        double r247746 = l;
        double r247747 = Om;
        double r247748 = r247746 / r247747;
        double r247749 = r247746 * r247748;
        double r247750 = r247741 * r247749;
        double r247751 = r247745 - r247750;
        double r247752 = pow(r247748, r247741);
        double r247753 = r247738 * r247752;
        double r247754 = U_;
        double r247755 = r247743 - r247754;
        double r247756 = r247753 * r247755;
        double r247757 = r247751 - r247756;
        double r247758 = r247744 * r247757;
        double r247759 = sqrt(r247758);
        double r247760 = 3.330205482826e-310;
        bool r247761 = r247738 <= r247760;
        double r247762 = r247755 * r247753;
        double r247763 = fma(r247749, r247741, r247762);
        double r247764 = r247745 - r247763;
        double r247765 = r247742 * r247764;
        double r247766 = r247765 * r247743;
        double r247767 = sqrt(r247766);
        double r247768 = 1.2408046941130587e-213;
        bool r247769 = r247738 <= r247768;
        double r247770 = 2.966212205685807e+43;
        bool r247771 = r247738 <= r247770;
        double r247772 = !r247771;
        bool r247773 = r247769 || r247772;
        double r247774 = sqrt(r247742);
        double r247775 = r247764 * r247743;
        double r247776 = sqrt(r247775);
        double r247777 = r247774 * r247776;
        double r247778 = r247773 ? r247777 : r247759;
        double r247779 = r247761 ? r247767 : r247778;
        double r247780 = r247740 ? r247759 : r247779;
        return r247780;
}

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 3 regimes

2. if n < -6.567149799254256e+186 or 1.2408046941130587e-213 < n < 2.966212205685807e+43

   1. Initial program 34.1

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

   3. Applied *-un-lft-identity 34.1

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied times-frac 31.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Simplified 31.4

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

   if -6.567149799254256e+186 < n < 3.330205482826e-310

   1. Initial program 34.7

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

   3. Applied *-un-lft-identity 34.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied times-frac 31.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Simplified 31.8

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

   7. Applied associate-*l* 31.9

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

   8. Simplified 32.8

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

   10. Applied associate-*l* 31.9

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

   12. Applied associate-*r* 30.4

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

   if 3.330205482826e-310 < n < 1.2408046941130587e-213 or 2.966212205685807e+43 < n

   1. Initial program 36.0

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

   3. Applied *-un-lft-identity 36.0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied times-frac 34.1

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{Om}\right)}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Simplified 34.1

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

   7. Applied associate-*l* 34.1

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

   8. Simplified 37.8

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

   10. Applied associate-*l* 34.1

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

   12. Applied sqrt-prod 23.3

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

4. Final simplification 28.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.56714979925425613 \cdot 10^{186}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \le 3.330205482825987 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot U}\\ \mathbf{elif}\;n \le 1.2408046941130587 \cdot 10^{-213} \lor \neg \left(n \le 2.96621220568580719 \cdot 10^{43}\right):\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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*))))))