Toniolo and Linder, Equation (13)

Average Error: 34.0 → 27.2

Time: 2.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -8.761276549602833524386767464963015900073 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(U* - U\right) \cdot \left(\left(\left(n \cdot {\ell}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{1}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;U \le 1.188289134364555924225184345803502950244 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3402684 = 2.0;
        double r3402685 = n;
        double r3402686 = r3402684 * r3402685;
        double r3402687 = U;
        double r3402688 = r3402686 * r3402687;
        double r3402689 = t;
        double r3402690 = l;
        double r3402691 = r3402690 * r3402690;
        double r3402692 = Om;
        double r3402693 = r3402691 / r3402692;
        double r3402694 = r3402684 * r3402693;
        double r3402695 = r3402689 - r3402694;
        double r3402696 = r3402690 / r3402692;
        double r3402697 = pow(r3402696, r3402684);
        double r3402698 = r3402685 * r3402697;
        double r3402699 = U_;
        double r3402700 = r3402687 - r3402699;
        double r3402701 = r3402698 * r3402700;
        double r3402702 = r3402695 - r3402701;
        double r3402703 = r3402688 * r3402702;
        double r3402704 = sqrt(r3402703);
        return r3402704;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3402705 = U;
        double r3402706 = -8.761276549602834e-54;
        bool r3402707 = r3402705 <= r3402706;
        double r3402708 = U_;
        double r3402709 = r3402708 - r3402705;
        double r3402710 = n;
        double r3402711 = l;
        double r3402712 = 2.0;
        double r3402713 = 2.0;
        double r3402714 = r3402712 / r3402713;
        double r3402715 = pow(r3402711, r3402714);
        double r3402716 = r3402710 * r3402715;
        double r3402717 = 1.0;
        double r3402718 = Om;
        double r3402719 = r3402717 / r3402718;
        double r3402720 = pow(r3402719, r3402714);
        double r3402721 = r3402716 * r3402720;
        double r3402722 = r3402711 / r3402718;
        double r3402723 = pow(r3402722, r3402714);
        double r3402724 = r3402721 * r3402723;
        double r3402725 = r3402709 * r3402724;
        double r3402726 = r3402712 * r3402711;
        double r3402727 = r3402722 * r3402726;
        double r3402728 = r3402725 - r3402727;
        double r3402729 = t;
        double r3402730 = r3402728 + r3402729;
        double r3402731 = r3402712 * r3402710;
        double r3402732 = r3402730 * r3402731;
        double r3402733 = r3402732 * r3402705;
        double r3402734 = sqrt(r3402733);
        double r3402735 = 1.188289134364556e-207;
        bool r3402736 = r3402705 <= r3402735;
        double r3402737 = r3402710 * r3402723;
        double r3402738 = r3402723 * r3402737;
        double r3402739 = r3402709 * r3402738;
        double r3402740 = r3402739 - r3402727;
        double r3402741 = r3402729 + r3402740;
        double r3402742 = r3402741 * r3402705;
        double r3402743 = r3402731 * r3402742;
        double r3402744 = sqrt(r3402743);
        double r3402745 = pow(r3402722, r3402712);
        double r3402746 = r3402710 * r3402745;
        double r3402747 = r3402746 * r3402709;
        double r3402748 = r3402747 - r3402727;
        double r3402749 = r3402729 + r3402748;
        double r3402750 = r3402749 * r3402731;
        double r3402751 = sqrt(r3402750);
        double r3402752 = sqrt(r3402705);
        double r3402753 = r3402751 * r3402752;
        double r3402754 = r3402736 ? r3402744 : r3402753;
        double r3402755 = r3402707 ? r3402734 : r3402754;
        return r3402755;
}

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 U < -8.761276549602834e-54

   1. Initial program 28.9

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

   2. Simplified 27.1

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

   4. Applied sqr-pow 27.1

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

   5. Applied associate-*r* 26.4

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

   7. Applied div-inv 26.4

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

   8. Applied unpow-prod-down 26.4

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

   9. Applied associate-*r* 26.9

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

   if -8.761276549602834e-54 < U < 1.188289134364556e-207

   1. Initial program 40.5

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

   2. Simplified 37.1

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

   4. Applied sqr-pow 37.1

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

   5. Applied associate-*r* 35.7

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

   7. Applied associate-*l* 31.4

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

   if 1.188289134364556e-207 < U

   1. Initial program 31.0

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

   2. Simplified 29.1

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

   4. Applied sqrt-prod 23.4

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

4. Final simplification 27.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -8.761276549602833524386767464963015900073 \cdot 10^{-54}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(U* - U\right) \cdot \left(\left(\left(n \cdot {\ell}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{1}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;U \le 1.188289134364555924225184345803502950244 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t + \left(\left(U* - U\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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*))))))