Toniolo and Linder, Equation (13)

Average Error: 32.9 → 28.6

Time: 46.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -4.4451804611309516 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \frac{U - U*}{\sqrt[3]{Om}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le -3.1769776407334526 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \le 3.336337607476201 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le 3.2878796761193827 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) - \left(\frac{\frac{\ell \cdot \ell}{Om}}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r927514 = 2.0;
        double r927515 = n;
        double r927516 = r927514 * r927515;
        double r927517 = U;
        double r927518 = r927516 * r927517;
        double r927519 = t;
        double r927520 = l;
        double r927521 = r927520 * r927520;
        double r927522 = Om;
        double r927523 = r927521 / r927522;
        double r927524 = r927514 * r927523;
        double r927525 = r927519 - r927524;
        double r927526 = r927520 / r927522;
        double r927527 = pow(r927526, r927514);
        double r927528 = r927515 * r927527;
        double r927529 = U_;
        double r927530 = r927517 - r927529;
        double r927531 = r927528 * r927530;
        double r927532 = r927525 - r927531;
        double r927533 = r927518 * r927532;
        double r927534 = sqrt(r927533);
        return r927534;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r927535 = U;
        double r927536 = -4.4451804611309516e+70;
        bool r927537 = r927535 <= r927536;
        double r927538 = t;
        double r927539 = 2.0;
        double r927540 = l;
        double r927541 = Om;
        double r927542 = r927540 / r927541;
        double r927543 = r927540 * r927542;
        double r927544 = r927539 * r927543;
        double r927545 = r927538 - r927544;
        double r927546 = n;
        double r927547 = r927546 * r927540;
        double r927548 = r927547 * r927542;
        double r927549 = cbrt(r927541);
        double r927550 = r927549 * r927549;
        double r927551 = r927548 / r927550;
        double r927552 = U_;
        double r927553 = r927535 - r927552;
        double r927554 = r927553 / r927549;
        double r927555 = r927551 * r927554;
        double r927556 = r927545 - r927555;
        double r927557 = r927539 * r927546;
        double r927558 = r927557 * r927535;
        double r927559 = r927556 * r927558;
        double r927560 = sqrt(r927559);
        double r927561 = -3.1769776407334526e-289;
        bool r927562 = r927535 <= r927561;
        double r927563 = r927540 * r927539;
        double r927564 = r927546 * r927542;
        double r927565 = r927542 * r927564;
        double r927566 = r927565 * r927553;
        double r927567 = fma(r927563, r927542, r927566);
        double r927568 = r927538 - r927567;
        double r927569 = r927535 * r927568;
        double r927570 = r927569 * r927557;
        double r927571 = sqrt(r927570);
        double r927572 = 3.336337607476201e-207;
        bool r927573 = r927535 <= r927572;
        double r927574 = r927553 * r927542;
        double r927575 = r927574 * r927547;
        double r927576 = r927575 / r927541;
        double r927577 = r927545 - r927576;
        double r927578 = r927577 * r927558;
        double r927579 = sqrt(r927578);
        double r927580 = 3.2878796761193827e+96;
        bool r927581 = r927535 <= r927580;
        double r927582 = r927540 * r927540;
        double r927583 = r927582 / r927541;
        double r927584 = -2.0;
        double r927585 = fma(r927583, r927584, r927538);
        double r927586 = r927583 / r927541;
        double r927587 = r927586 * r927546;
        double r927588 = r927587 * r927553;
        double r927589 = r927585 - r927588;
        double r927590 = r927558 * r927589;
        double r927591 = sqrt(r927590);
        double r927592 = r927581 ? r927571 : r927591;
        double r927593 = r927573 ? r927579 : r927592;
        double r927594 = r927562 ? r927571 : r927593;
        double r927595 = r927537 ? r927560 : r927594;
        return r927595;
}

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 4 regimes

2. if U < -4.4451804611309516e+70

   1. Initial program 27.5

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

   3. Applied *-un-lft-identity 27.5

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

   4. Applied times-frac 24.8

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

   5. Simplified 24.8

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

   7. Applied unpow2 24.8

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

   8. Applied associate-*r* 24.4

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

   10. Applied associate-*r/ 24.7

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

   11. Applied associate-*l/ 24.8

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

   12. Applied associate-*l/ 24.9

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

   14. Applied add-cube-cbrt 24.9

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

   15. Applied times-frac 25.7

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

   if -4.4451804611309516e+70 < U < -3.1769776407334526e-289 or 3.336337607476201e-207 < U < 3.2878796761193827e+96

   1. Initial program 33.1

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

   3. Applied *-un-lft-identity 33.1

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

   4. Applied times-frac 30.3

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

   5. Simplified 30.3

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

   7. Applied unpow2 30.3

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

   8. Applied associate-*r* 29.2

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

   10. Applied associate-*l* 26.9

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

   11. Simplified 26.9

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

   if -3.1769776407334526e-289 < U < 3.336337607476201e-207

   1. Initial program 42.8

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

   3. Applied *-un-lft-identity 42.8

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

   4. Applied times-frac 41.6

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

   5. Simplified 41.6

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

   7. Applied unpow2 41.6

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

   8. Applied associate-*r* 40.7

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

   10. Applied associate-*r/ 41.7

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

   11. Applied associate-*l/ 41.9

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

   12. Applied associate-*l/ 41.6

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

   14. Applied associate-*l* 41.1

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

   if 3.2878796761193827e+96 < U

   1. Initial program 26.1

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

   2. Simplified 26.6

      \[\leadsto \color{blue}{\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) - \left(n \cdot \frac{\frac{\ell \cdot \ell}{Om}}{Om}\right) \cdot \left(U - U*\right)\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 28.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -4.4451804611309516 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\left(n \cdot \ell\right) \cdot \frac{\ell}{Om}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}} \cdot \frac{U - U*}{\sqrt[3]{Om}}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le -3.1769776407334526 \cdot 10^{-289}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \le 3.336337607476201 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \frac{\left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \ell\right)}{Om}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;U \le 3.2878796761193827 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell \cdot 2, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{Om}, -2, t\right) - \left(\frac{\frac{\ell \cdot \ell}{Om}}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +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*))))))