Toniolo and Linder, Equation (13)

Average Error: 33.6 → 28.1

Time: 45.1s

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 r3322579 = 2.0;
        double r3322580 = n;
        double r3322581 = r3322579 * r3322580;
        double r3322582 = U;
        double r3322583 = r3322581 * r3322582;
        double r3322584 = t;
        double r3322585 = l;
        double r3322586 = r3322585 * r3322585;
        double r3322587 = Om;
        double r3322588 = r3322586 / r3322587;
        double r3322589 = r3322579 * r3322588;
        double r3322590 = r3322584 - r3322589;
        double r3322591 = r3322585 / r3322587;
        double r3322592 = pow(r3322591, r3322579);
        double r3322593 = r3322580 * r3322592;
        double r3322594 = U_;
        double r3322595 = r3322582 - r3322594;
        double r3322596 = r3322593 * r3322595;
        double r3322597 = r3322590 - r3322596;
        double r3322598 = r3322583 * r3322597;
        double r3322599 = sqrt(r3322598);
        return r3322599;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3322600 = 2.0;
        double r3322601 = n;
        double r3322602 = r3322600 * r3322601;
        double r3322603 = U;
        double r3322604 = r3322602 * r3322603;
        double r3322605 = t;
        double r3322606 = l;
        double r3322607 = r3322606 * r3322606;
        double r3322608 = Om;
        double r3322609 = r3322607 / r3322608;
        double r3322610 = r3322609 * r3322600;
        double r3322611 = r3322605 - r3322610;
        double r3322612 = r3322606 / r3322608;
        double r3322613 = pow(r3322612, r3322600);
        double r3322614 = r3322601 * r3322613;
        double r3322615 = U_;
        double r3322616 = r3322603 - r3322615;
        double r3322617 = r3322614 * r3322616;
        double r3322618 = r3322611 - r3322617;
        double r3322619 = r3322604 * r3322618;
        double r3322620 = sqrt(r3322619);
        double r3322621 = 1.0452549734454207e-138;
        bool r3322622 = r3322620 <= r3322621;
        double r3322623 = r3322600 * r3322606;
        double r3322624 = r3322615 - r3322603;
        double r3322625 = r3322601 * r3322624;
        double r3322626 = r3322612 * r3322625;
        double r3322627 = r3322623 - r3322626;
        double r3322628 = r3322627 * r3322612;
        double r3322629 = r3322605 - r3322628;
        double r3322630 = r3322601 * r3322629;
        double r3322631 = r3322630 * r3322603;
        double r3322632 = cbrt(r3322631);
        double r3322633 = r3322632 * r3322632;
        double r3322634 = r3322632 * r3322633;
        double r3322635 = r3322600 * r3322634;
        double r3322636 = sqrt(r3322635);
        double r3322637 = 1.0706139562680882e+131;
        bool r3322638 = r3322620 <= r3322637;
        double r3322639 = cbrt(r3322603);
        double r3322640 = cbrt(r3322630);
        double r3322641 = r3322639 * r3322640;
        double r3322642 = r3322641 * r3322633;
        double r3322643 = r3322600 * r3322642;
        double r3322644 = sqrt(r3322643);
        double r3322645 = r3322638 ? r3322620 : r3322644;
        double r3322646 = r3322622 ? r3322636 : r3322645;
        return r3322646;
}

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*))))))