Toniolo and Linder, Equation (13)

Average Error: 33.6 → 28.1

Time: 43.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0452549734454207 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0706139562680882 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\sqrt[3]{U} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3322582 = 2.0;
        double r3322583 = n;
        double r3322584 = r3322582 * r3322583;
        double r3322585 = U;
        double r3322586 = r3322584 * r3322585;
        double r3322587 = t;
        double r3322588 = l;
        double r3322589 = r3322588 * r3322588;
        double r3322590 = Om;
        double r3322591 = r3322589 / r3322590;
        double r3322592 = r3322582 * r3322591;
        double r3322593 = r3322587 - r3322592;
        double r3322594 = r3322588 / r3322590;
        double r3322595 = pow(r3322594, r3322582);
        double r3322596 = r3322583 * r3322595;
        double r3322597 = U_;
        double r3322598 = r3322585 - r3322597;
        double r3322599 = r3322596 * r3322598;
        double r3322600 = r3322593 - r3322599;
        double r3322601 = r3322586 * r3322600;
        double r3322602 = sqrt(r3322601);
        return r3322602;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3322603 = 2.0;
        double r3322604 = n;
        double r3322605 = r3322603 * r3322604;
        double r3322606 = U;
        double r3322607 = r3322605 * r3322606;
        double r3322608 = t;
        double r3322609 = l;
        double r3322610 = r3322609 * r3322609;
        double r3322611 = Om;
        double r3322612 = r3322610 / r3322611;
        double r3322613 = r3322612 * r3322603;
        double r3322614 = r3322608 - r3322613;
        double r3322615 = r3322609 / r3322611;
        double r3322616 = pow(r3322615, r3322603);
        double r3322617 = r3322604 * r3322616;
        double r3322618 = U_;
        double r3322619 = r3322606 - r3322618;
        double r3322620 = r3322617 * r3322619;
        double r3322621 = r3322614 - r3322620;
        double r3322622 = r3322607 * r3322621;
        double r3322623 = sqrt(r3322622);
        double r3322624 = 1.0452549734454207e-138;
        bool r3322625 = r3322623 <= r3322624;
        double r3322626 = r3322603 * r3322609;
        double r3322627 = r3322618 - r3322606;
        double r3322628 = r3322604 * r3322627;
        double r3322629 = r3322615 * r3322628;
        double r3322630 = r3322626 - r3322629;
        double r3322631 = r3322630 * r3322615;
        double r3322632 = r3322608 - r3322631;
        double r3322633 = r3322604 * r3322632;
        double r3322634 = r3322633 * r3322606;
        double r3322635 = cbrt(r3322634);
        double r3322636 = r3322635 * r3322635;
        double r3322637 = r3322635 * r3322636;
        double r3322638 = r3322603 * r3322637;
        double r3322639 = sqrt(r3322638);
        double r3322640 = 1.0706139562680882e+131;
        bool r3322641 = r3322623 <= r3322640;
        double r3322642 = cbrt(r3322606);
        double r3322643 = cbrt(r3322633);
        double r3322644 = r3322642 * r3322643;
        double r3322645 = r3322644 * r3322636;
        double r3322646 = r3322603 * r3322645;
        double r3322647 = sqrt(r3322646);
        double r3322648 = r3322641 ? r3322623 : r3322647;
        double r3322649 = r3322625 ? r3322639 : r3322648;
        return r3322649;
}

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 (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 1.0452549734454207e-138

   1. Initial program 51.5

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

   2. Simplified 37.6

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

   4. Applied add-cube-cbrt 37.8

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

   if 1.0452549734454207e-138 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 1.0706139562680882e+131

   1. Initial program 1.3

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

   if 1.0706139562680882e+131 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

   1. Initial program 57.5

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

   2. Simplified 49.9

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

   4. Applied add-cube-cbrt 50.0

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

   6. Applied cbrt-prod 50.0

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

4. Final simplification 28.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0452549734454207 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0706139562680882 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\sqrt[3]{U} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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*))))))