Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.1

Time: 2.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -6.475205078487209 \cdot 10^{+94}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{\frac{1}{4}} \cdot {\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{\frac{1}{4}}\\ \mathbf{elif}\;n \le 2.122462322126863 \cdot 10^{-309}:\\ \;\;\;\;{\left(\left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - (\ell \cdot 2 + \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right))_* \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)} \cdot {n}^{\frac{1}{2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r16907665 = 2.0;
        double r16907666 = n;
        double r16907667 = r16907665 * r16907666;
        double r16907668 = U;
        double r16907669 = r16907667 * r16907668;
        double r16907670 = t;
        double r16907671 = l;
        double r16907672 = r16907671 * r16907671;
        double r16907673 = Om;
        double r16907674 = r16907672 / r16907673;
        double r16907675 = r16907665 * r16907674;
        double r16907676 = r16907670 - r16907675;
        double r16907677 = r16907671 / r16907673;
        double r16907678 = pow(r16907677, r16907665);
        double r16907679 = r16907666 * r16907678;
        double r16907680 = U_;
        double r16907681 = r16907668 - r16907680;
        double r16907682 = r16907679 * r16907681;
        double r16907683 = r16907676 - r16907682;
        double r16907684 = r16907669 * r16907683;
        double r16907685 = sqrt(r16907684);
        return r16907685;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r16907686 = n;
        double r16907687 = -6.475205078487209e+94;
        bool r16907688 = r16907686 <= r16907687;
        double r16907689 = U;
        double r16907690 = 2.0;
        double r16907691 = r16907689 * r16907690;
        double r16907692 = t;
        double r16907693 = l;
        double r16907694 = Om;
        double r16907695 = r16907693 / r16907694;
        double r16907696 = r16907693 * r16907690;
        double r16907697 = U_;
        double r16907698 = r16907689 - r16907697;
        double r16907699 = r16907695 * r16907698;
        double r16907700 = r16907686 * r16907699;
        double r16907701 = r16907696 + r16907700;
        double r16907702 = r16907695 * r16907701;
        double r16907703 = r16907692 - r16907702;
        double r16907704 = r16907691 * r16907703;
        double r16907705 = r16907686 * r16907704;
        double r16907706 = 0.25;
        double r16907707 = pow(r16907705, r16907706);
        double r16907708 = r16907707 * r16907707;
        double r16907709 = 2.122462322126863e-309;
        bool r16907710 = r16907686 <= r16907709;
        double r16907711 = r16907686 * r16907703;
        double r16907712 = r16907711 * r16907691;
        double r16907713 = 0.5;
        double r16907714 = pow(r16907712, r16907713);
        double r16907715 = fma(r16907693, r16907690, r16907700);
        double r16907716 = r16907715 * r16907695;
        double r16907717 = r16907692 - r16907716;
        double r16907718 = r16907717 * r16907691;
        double r16907719 = sqrt(r16907718);
        double r16907720 = pow(r16907686, r16907713);
        double r16907721 = r16907719 * r16907720;
        double r16907722 = r16907710 ? r16907714 : r16907721;
        double r16907723 = r16907688 ? r16907708 : r16907722;
        return r16907723;
}

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 < -6.475205078487209e+94

   1. Initial program 32.6

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

   3. Applied associate-/l* 31.3

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

   5. Applied unpow2 31.3

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

   6. Applied associate-*r* 29.0

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

   8. Applied pow1 29.0

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

   9. Applied pow1 29.0

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

   10. Applied pow1 29.0

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

   11. Applied pow-prod-down 29.0

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

   12. Applied pow-prod-down 29.0

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

   13. Applied sqrt-pow1 29.0

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

   14. Simplified 28.7

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

   16. Applied sqr-pow 28.9

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

   if -6.475205078487209e+94 < n < 2.122462322126863e-309

   1. Initial program 33.3

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

   3. Applied associate-/l* 29.7

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

   5. Applied unpow2 29.7

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

   6. Applied associate-*r* 29.2

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

   8. Applied pow1 29.2

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

   9. Applied pow1 29.2

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

   10. Applied pow1 29.2

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

   11. Applied pow-prod-down 29.2

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

   12. Applied pow-prod-down 29.2

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

   13. Applied sqrt-pow1 29.2

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

   14. Simplified 29.0

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

   16. Applied associate-*l* 26.9

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

   if 2.122462322126863e-309 < n

   1. Initial program 33.3

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

   3. Applied associate-/l* 30.8

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

   5. Applied unpow2 30.8

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

   6. Applied associate-*r* 29.9

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

   8. Applied pow1 29.9

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

   9. Applied pow1 29.9

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

   10. Applied pow1 29.9

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

   11. Applied pow-prod-down 29.9

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

   12. Applied pow-prod-down 29.9

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

   13. Applied sqrt-pow1 29.9

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

   14. Simplified 30.0

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

   16. Applied unpow-prod-down 22.9

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

   17. Simplified 22.9

       \[\leadsto \color{blue}{\sqrt{\left(t - (\ell \cdot 2 + \left(\left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot n\right))_* \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)}} \cdot {n}^{\left(\frac{1}{2}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 25.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.475205078487209 \cdot 10^{+94}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{\frac{1}{4}} \cdot {\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{\frac{1}{4}}\\ \mathbf{elif}\;n \le 2.122462322126863 \cdot 10^{-309}:\\ \;\;\;\;{\left(\left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - (\ell \cdot 2 + \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right))_* \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)} \cdot {n}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))