Toniolo and Linder, Equation (13)

Average Error: 34.2 → 30.0

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 3.292637727838382735903492207891209195801 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \le 3.133024137349082252862067310328422033991 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(U - U*\right) \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\\ \mathbf{elif}\;t \le 1.321746129353034672470618178877471414334 \cdot 10^{126}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(\left(-2\right) + 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r206661 = 2.0;
        double r206662 = n;
        double r206663 = r206661 * r206662;
        double r206664 = U;
        double r206665 = r206663 * r206664;
        double r206666 = t;
        double r206667 = l;
        double r206668 = r206667 * r206667;
        double r206669 = Om;
        double r206670 = r206668 / r206669;
        double r206671 = r206661 * r206670;
        double r206672 = r206666 - r206671;
        double r206673 = r206667 / r206669;
        double r206674 = pow(r206673, r206661);
        double r206675 = r206662 * r206674;
        double r206676 = U_;
        double r206677 = r206664 - r206676;
        double r206678 = r206675 * r206677;
        double r206679 = r206672 - r206678;
        double r206680 = r206665 * r206679;
        double r206681 = sqrt(r206680);
        return r206681;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r206682 = t;
        double r206683 = 3.2926377278383827e-149;
        bool r206684 = r206682 <= r206683;
        double r206685 = 2.0;
        double r206686 = n;
        double r206687 = r206685 * r206686;
        double r206688 = U;
        double r206689 = r206687 * r206688;
        double r206690 = l;
        double r206691 = Om;
        double r206692 = r206691 / r206690;
        double r206693 = r206690 / r206692;
        double r206694 = r206685 * r206693;
        double r206695 = r206682 - r206694;
        double r206696 = r206690 / r206691;
        double r206697 = 2.0;
        double r206698 = r206685 / r206697;
        double r206699 = pow(r206696, r206698);
        double r206700 = r206686 * r206699;
        double r206701 = U_;
        double r206702 = r206688 - r206701;
        double r206703 = r206702 * r206699;
        double r206704 = r206700 * r206703;
        double r206705 = r206695 - r206704;
        double r206706 = r206689 * r206705;
        double r206707 = sqrt(r206706);
        double r206708 = 3.133024137349082e-104;
        bool r206709 = r206682 <= r206708;
        double r206710 = sqrt(r206689);
        double r206711 = r206702 * r206686;
        double r206712 = pow(r206696, r206685);
        double r206713 = r206711 * r206712;
        double r206714 = fma(r206685, r206693, r206713);
        double r206715 = r206682 - r206714;
        double r206716 = sqrt(r206715);
        double r206717 = r206710 * r206716;
        double r206718 = 1.3217461293530347e+126;
        bool r206719 = r206682 <= r206718;
        double r206720 = r206689 * r206695;
        double r206721 = -r206685;
        double r206722 = r206721 + r206685;
        double r206723 = r206693 * r206722;
        double r206724 = r206697 * r206698;
        double r206725 = pow(r206696, r206724);
        double r206726 = r206686 * r206725;
        double r206727 = r206726 * r206702;
        double r206728 = r206723 - r206727;
        double r206729 = r206688 * r206728;
        double r206730 = r206687 * r206729;
        double r206731 = r206720 + r206730;
        double r206732 = sqrt(r206731);
        double r206733 = r206690 * r206690;
        double r206734 = r206733 / r206691;
        double r206735 = r206686 * r206712;
        double r206736 = r206702 * r206735;
        double r206737 = fma(r206685, r206734, r206736);
        double r206738 = r206682 - r206737;
        double r206739 = sqrt(r206738);
        double r206740 = r206710 * r206739;
        double r206741 = r206719 ? r206732 : r206740;
        double r206742 = r206709 ? r206717 : r206741;
        double r206743 = r206684 ? r206707 : r206742;
        return r206743;
}

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 < 3.2926377278383827e-149

   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 associate-/l* 32.0

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

   5. Applied sqr-pow 32.0

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

   6. Applied associate-*r* 31.2

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

   8. Applied associate-*l* 30.9

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

   9. Simplified 30.9

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

   if 3.2926377278383827e-149 < t < 3.133024137349082e-104

   1. Initial program 32.2

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

   3. Applied associate-/l* 30.2

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

   5. Applied sqrt-prod 33.9

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

   6. Simplified 34.3

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

   if 3.133024137349082e-104 < t < 1.3217461293530347e+126

   1. Initial program 32.1

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

   3. Applied associate-/l* 29.4

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

   5. Applied sqr-pow 29.4

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

   6. Applied associate-*r* 28.9

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

   8. Applied add-sqr-sqrt 28.9

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

   9. Applied prod-diff 28.9

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

   10. Applied associate--l+ 28.9

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

   11. Applied distribute-lft-in 28.9

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

   12. Simplified 28.9

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

   13. Simplified 29.4

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

   if 1.3217461293530347e+126 < t

   1. Initial program 35.2

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

   3. Applied sqrt-prod 26.1

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

   4. Simplified 26.1

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

4. Final simplification 30.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.292637727838382735903492207891209195801 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)}\\ \mathbf{elif}\;t \le 3.133024137349082252862067310328422033991 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(U - U*\right) \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\\ \mathbf{elif}\;t \le 1.321746129353034672470618178877471414334 \cdot 10^{126}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) + \left(2 \cdot n\right) \cdot \left(U \cdot \left(\frac{\ell}{\frac{Om}{\ell}} \cdot \left(\left(-2\right) + 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))