Toniolo and Linder, Equation (13)

Average Error: 34.3 → 27.6

Time: 1.7m

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}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} + \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) \cdot \mathsf{fma}\left(-U*, 1, U*\right)\right)\right)\right)\right) \cdot U}} \cdot \left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} + \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) \cdot \mathsf{fma}\left(-U*, 1, U*\right)\right)\right)\right)\right) \cdot U}} \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} + \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) \cdot \mathsf{fma}\left(-U*, 1, U*\right)\right)\right)\right)\right) \cdot U}}\right)\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 5.527558626767575315212582232479434470364 \cdot 10^{298}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \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)\right)\right) \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3943563 = 2.0;
        double r3943564 = n;
        double r3943565 = r3943563 * r3943564;
        double r3943566 = U;
        double r3943567 = r3943565 * r3943566;
        double r3943568 = t;
        double r3943569 = l;
        double r3943570 = r3943569 * r3943569;
        double r3943571 = Om;
        double r3943572 = r3943570 / r3943571;
        double r3943573 = r3943563 * r3943572;
        double r3943574 = r3943568 - r3943573;
        double r3943575 = r3943569 / r3943571;
        double r3943576 = pow(r3943575, r3943563);
        double r3943577 = r3943564 * r3943576;
        double r3943578 = U_;
        double r3943579 = r3943566 - r3943578;
        double r3943580 = r3943577 * r3943579;
        double r3943581 = r3943574 - r3943580;
        double r3943582 = r3943567 * r3943581;
        double r3943583 = sqrt(r3943582);
        return r3943583;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3943584 = 2.0;
        double r3943585 = n;
        double r3943586 = r3943584 * r3943585;
        double r3943587 = U;
        double r3943588 = r3943586 * r3943587;
        double r3943589 = t;
        double r3943590 = l;
        double r3943591 = r3943590 * r3943590;
        double r3943592 = Om;
        double r3943593 = r3943591 / r3943592;
        double r3943594 = r3943593 * r3943584;
        double r3943595 = r3943589 - r3943594;
        double r3943596 = r3943590 / r3943592;
        double r3943597 = pow(r3943596, r3943584);
        double r3943598 = r3943585 * r3943597;
        double r3943599 = U_;
        double r3943600 = r3943587 - r3943599;
        double r3943601 = r3943598 * r3943600;
        double r3943602 = r3943595 - r3943601;
        double r3943603 = r3943588 * r3943602;
        double r3943604 = 0.0;
        bool r3943605 = r3943603 <= r3943604;
        double r3943606 = r3943584 * r3943590;
        double r3943607 = 2.0;
        double r3943608 = r3943584 / r3943607;
        double r3943609 = pow(r3943596, r3943608);
        double r3943610 = r3943585 * r3943609;
        double r3943611 = r3943610 * r3943600;
        double r3943612 = r3943611 * r3943609;
        double r3943613 = r3943609 * r3943610;
        double r3943614 = -r3943599;
        double r3943615 = 1.0;
        double r3943616 = fma(r3943614, r3943615, r3943599);
        double r3943617 = r3943613 * r3943616;
        double r3943618 = r3943612 + r3943617;
        double r3943619 = fma(r3943596, r3943606, r3943618);
        double r3943620 = r3943589 - r3943619;
        double r3943621 = r3943585 * r3943620;
        double r3943622 = r3943584 * r3943621;
        double r3943623 = r3943622 * r3943587;
        double r3943624 = sqrt(r3943623);
        double r3943625 = cbrt(r3943624);
        double r3943626 = r3943625 * r3943625;
        double r3943627 = r3943625 * r3943626;
        double r3943628 = 5.527558626767575e+298;
        bool r3943629 = r3943603 <= r3943628;
        double r3943630 = sqrt(r3943603);
        double r3943631 = r3943600 * r3943613;
        double r3943632 = fma(r3943596, r3943606, r3943631);
        double r3943633 = r3943589 - r3943632;
        double r3943634 = r3943633 * r3943585;
        double r3943635 = r3943634 * r3943584;
        double r3943636 = sqrt(r3943635);
        double r3943637 = sqrt(r3943587);
        double r3943638 = r3943636 * r3943637;
        double r3943639 = r3943629 ? r3943630 : r3943638;
        double r3943640 = r3943605 ? r3943627 : r3943639;
        return r3943640;
}

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 (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*)))) < 0.0

   1. Initial program 56.6

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

   2. Simplified 39.7

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

   4. Applied sqr-pow 39.7

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

   5. Applied associate-*r* 37.0

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

   7. Applied *-un-lft-identity 37.0

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

   8. Applied add-sqr-sqrt 49.8

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

   9. Applied prod-diff 49.8

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

   10. Applied distribute-rgt-in 49.8

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

   11. Simplified 36.2

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

   13. Applied add-cube-cbrt 36.6

       \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{U \cdot \left(\left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right) + \mathsf{fma}\left(-U*, 1, U* \cdot 1\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right) \cdot 2\right)}} \cdot \sqrt[3]{\sqrt{U \cdot \left(\left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right) + \mathsf{fma}\left(-U*, 1, U* \cdot 1\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right) \cdot 2\right)}}\right) \cdot \sqrt[3]{\sqrt{U \cdot \left(\left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right) + \mathsf{fma}\left(-U*, 1, U* \cdot 1\right) \cdot \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right) \cdot 2\right)}}}\]

   if 0.0 < (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*)))) < 5.527558626767575e+298

   1. Initial program 1.8

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

   if 5.527558626767575e+298 < (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))

   1. Initial program 63.1

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

   2. Simplified 54.8

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

   4. Applied sqr-pow 54.8

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

   5. Applied associate-*r* 53.5

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

   7. Applied sqrt-prod 54.3

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

4. Final simplification 27.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} + \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) \cdot \mathsf{fma}\left(-U*, 1, U*\right)\right)\right)\right)\right) \cdot U}} \cdot \left(\sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} + \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) \cdot \mathsf{fma}\left(-U*, 1, U*\right)\right)\right)\right)\right) \cdot U}} \cdot \sqrt[3]{\sqrt{\left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} + \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) \cdot \mathsf{fma}\left(-U*, 1, U*\right)\right)\right)\right)\right) \cdot U}}\right)\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 5.527558626767575315212582232479434470364 \cdot 10^{298}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \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)\right)\right) \cdot n\right) \cdot 2} \cdot \sqrt{U}\\ \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*))))))