Toniolo and Linder, Equation (13)

Average Error: 32.9 → 28.0

Time: 38.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 4.583682591988209 \cdot 10^{-251}:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 6.552094729632033 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\ \mathbf{elif}\;t \le 2.539192710887033 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \frac{\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1014511 = 2.0;
        double r1014512 = n;
        double r1014513 = r1014511 * r1014512;
        double r1014514 = U;
        double r1014515 = r1014513 * r1014514;
        double r1014516 = t;
        double r1014517 = l;
        double r1014518 = r1014517 * r1014517;
        double r1014519 = Om;
        double r1014520 = r1014518 / r1014519;
        double r1014521 = r1014511 * r1014520;
        double r1014522 = r1014516 - r1014521;
        double r1014523 = r1014517 / r1014519;
        double r1014524 = pow(r1014523, r1014511);
        double r1014525 = r1014512 * r1014524;
        double r1014526 = U_;
        double r1014527 = r1014514 - r1014526;
        double r1014528 = r1014525 * r1014527;
        double r1014529 = r1014522 - r1014528;
        double r1014530 = r1014515 * r1014529;
        double r1014531 = sqrt(r1014530);
        return r1014531;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1014532 = t;
        double r1014533 = 4.583682591988209e-251;
        bool r1014534 = r1014532 <= r1014533;
        double r1014535 = l;
        double r1014536 = 2.0;
        double r1014537 = Om;
        double r1014538 = r1014535 / r1014537;
        double r1014539 = r1014536 * r1014538;
        double r1014540 = r1014535 * r1014539;
        double r1014541 = U;
        double r1014542 = U_;
        double r1014543 = r1014541 - r1014542;
        double r1014544 = r1014538 * r1014543;
        double r1014545 = n;
        double r1014546 = r1014538 * r1014545;
        double r1014547 = r1014544 * r1014546;
        double r1014548 = r1014540 + r1014547;
        double r1014549 = r1014532 - r1014548;
        double r1014550 = r1014541 * r1014536;
        double r1014551 = r1014550 * r1014545;
        double r1014552 = r1014549 * r1014551;
        double r1014553 = 0.5;
        double r1014554 = pow(r1014552, r1014553);
        double r1014555 = 6.552094729632033e-157;
        bool r1014556 = r1014532 <= r1014555;
        double r1014557 = sqrt(r1014551);
        double r1014558 = r1014532 - r1014547;
        double r1014559 = r1014536 * r1014535;
        double r1014560 = r1014538 * r1014559;
        double r1014561 = r1014558 - r1014560;
        double r1014562 = sqrt(r1014561);
        double r1014563 = r1014557 * r1014562;
        double r1014564 = 2.539192710887033e+96;
        bool r1014565 = r1014532 <= r1014564;
        double r1014566 = r1014546 * r1014535;
        double r1014567 = r1014566 * r1014543;
        double r1014568 = r1014567 / r1014537;
        double r1014569 = r1014540 + r1014568;
        double r1014570 = r1014532 - r1014569;
        double r1014571 = r1014570 * r1014545;
        double r1014572 = r1014550 * r1014571;
        double r1014573 = sqrt(r1014572);
        double r1014574 = r1014565 ? r1014573 : r1014563;
        double r1014575 = r1014556 ? r1014563 : r1014574;
        double r1014576 = r1014534 ? r1014554 : r1014575;
        return r1014576;
}

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 t < 4.583682591988209e-251

   1. Initial program 32.8

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

   2. Simplified 29.5

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

   4. Applied pow1/2 29.5

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

   6. Applied associate-*r* 29.3

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

   8. Applied associate-*l* 29.1

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

   if 4.583682591988209e-251 < t < 6.552094729632033e-157 or 2.539192710887033e+96 < t

   1. Initial program 35.6

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

   2. Simplified 30.8

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

   4. Applied pow1/2 30.8

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

   6. Applied associate-*r* 32.4

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

   8. Applied unpow-prod-down 26.3

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

   9. Simplified 26.3

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

   10. Simplified 26.6

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

   if 6.552094729632033e-157 < t < 2.539192710887033e+96

   1. Initial program 30.2

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

   2. Simplified 26.4

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

   4. Applied associate-*r/ 26.7

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

   5. Applied associate-*l/ 26.8

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

4. Final simplification 28.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 4.583682591988209 \cdot 10^{-251}:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 6.552094729632033 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\ \mathbf{elif}\;t \le 2.539192710887033 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \frac{\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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*))))))