Toniolo and Linder, Equation (13)

Average Error: 33.3 → 29.8

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 2.435100367453427 \cdot 10^{-204}:\\ \;\;\;\;{\left((n \cdot \left(\frac{\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}\right) + \left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 3.00502928168256 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(\ell \cdot -2 - n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;t \le 4.3432694513312935 \cdot 10^{+85}:\\ \;\;\;\;{\left((n \cdot \left(\frac{\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}\right) + \left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot 2\right)} \cdot \sqrt{\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r15180536 = 2.0;
        double r15180537 = n;
        double r15180538 = r15180536 * r15180537;
        double r15180539 = U;
        double r15180540 = r15180538 * r15180539;
        double r15180541 = t;
        double r15180542 = l;
        double r15180543 = r15180542 * r15180542;
        double r15180544 = Om;
        double r15180545 = r15180543 / r15180544;
        double r15180546 = r15180536 * r15180545;
        double r15180547 = r15180541 - r15180546;
        double r15180548 = r15180542 / r15180544;
        double r15180549 = pow(r15180548, r15180536);
        double r15180550 = r15180537 * r15180549;
        double r15180551 = U_;
        double r15180552 = r15180539 - r15180551;
        double r15180553 = r15180550 * r15180552;
        double r15180554 = r15180547 - r15180553;
        double r15180555 = r15180540 * r15180554;
        double r15180556 = sqrt(r15180555);
        return r15180556;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r15180557 = t;
        double r15180558 = 2.435100367453427e-204;
        bool r15180559 = r15180557 <= r15180558;
        double r15180560 = n;
        double r15180561 = 2.0;
        double r15180562 = l;
        double r15180563 = U;
        double r15180564 = r15180562 * r15180563;
        double r15180565 = r15180561 * r15180564;
        double r15180566 = -2.0;
        double r15180567 = r15180562 * r15180566;
        double r15180568 = Om;
        double r15180569 = r15180562 / r15180568;
        double r15180570 = r15180569 * r15180560;
        double r15180571 = U_;
        double r15180572 = r15180563 - r15180571;
        double r15180573 = r15180570 * r15180572;
        double r15180574 = r15180567 - r15180573;
        double r15180575 = r15180565 * r15180574;
        double r15180576 = r15180575 / r15180568;
        double r15180577 = r15180560 * r15180557;
        double r15180578 = r15180577 * r15180561;
        double r15180579 = r15180563 * r15180578;
        double r15180580 = fma(r15180560, r15180576, r15180579);
        double r15180581 = 0.5;
        double r15180582 = pow(r15180580, r15180581);
        double r15180583 = 3.00502928168256e-47;
        bool r15180584 = r15180557 <= r15180583;
        double r15180585 = r15180560 * r15180561;
        double r15180586 = r15180563 * r15180585;
        double r15180587 = r15180557 * r15180586;
        double r15180588 = r15180561 * r15180563;
        double r15180589 = r15180560 * r15180588;
        double r15180590 = r15180572 * r15180569;
        double r15180591 = r15180560 * r15180590;
        double r15180592 = r15180567 - r15180591;
        double r15180593 = r15180592 * r15180569;
        double r15180594 = r15180589 * r15180593;
        double r15180595 = r15180587 + r15180594;
        double r15180596 = sqrt(r15180595);
        double r15180597 = 4.3432694513312935e+85;
        bool r15180598 = r15180557 <= r15180597;
        double r15180599 = sqrt(r15180586);
        double r15180600 = r15180562 * r15180562;
        double r15180601 = r15180600 / r15180568;
        double r15180602 = r15180601 * r15180561;
        double r15180603 = r15180557 - r15180602;
        double r15180604 = pow(r15180569, r15180561);
        double r15180605 = r15180604 * r15180560;
        double r15180606 = r15180605 * r15180572;
        double r15180607 = r15180603 - r15180606;
        double r15180608 = sqrt(r15180607);
        double r15180609 = r15180599 * r15180608;
        double r15180610 = r15180598 ? r15180582 : r15180609;
        double r15180611 = r15180584 ? r15180596 : r15180610;
        double r15180612 = r15180559 ? r15180582 : r15180611;
        return r15180612;
}

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 < 2.435100367453427e-204 or 3.00502928168256e-47 < t < 4.3432694513312935e+85

   1. Initial program 33.2

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

   3. Applied sub-neg 33.2

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

   4. Applied associate--l+ 33.2

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

   5. Applied distribute-rgt-in 33.2

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

   6. Simplified 30.8

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

   8. Applied associate-*r* 30.8

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

   10. Applied associate-*r/ 31.2

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

   11. Applied associate-*l/ 30.8

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

   12. Applied associate-*l/ 31.1

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

   13. Applied associate-*r/ 31.1

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

   14. Applied associate-*l/ 32.8

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

   15. Applied sub-div 32.8

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

   16. Applied associate-*l/ 33.2

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

   17. Simplified 32.2

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

   19. Applied pow1 32.2

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

   20. Applied sqrt-pow1 32.2

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

   21. Simplified 30.9

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

   if 2.435100367453427e-204 < t < 3.00502928168256e-47

   1. Initial program 31.7

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

   3. Applied sub-neg 31.7

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

   4. Applied associate--l+ 31.7

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

   5. Applied distribute-rgt-in 31.7

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

   6. Simplified 29.1

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

   8. Applied associate-*r* 28.7

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

   10. Applied associate-*l* 29.1

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

   11. Simplified 28.4

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

   if 4.3432694513312935e+85 < t

   1. Initial program 34.6

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

   3. Applied sqrt-prod 27.0

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

4. Final simplification 29.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2.435100367453427 \cdot 10^{-204}:\\ \;\;\;\;{\left((n \cdot \left(\frac{\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}\right) + \left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 3.00502928168256 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{t \cdot \left(U \cdot \left(n \cdot 2\right)\right) + \left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(\left(\ell \cdot -2 - n \cdot \left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)}\\ \mathbf{elif}\;t \le 4.3432694513312935 \cdot 10^{+85}:\\ \;\;\;\;{\left((n \cdot \left(\frac{\left(2 \cdot \left(\ell \cdot U\right)\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}{Om}\right) + \left(U \cdot \left(\left(n \cdot t\right) \cdot 2\right)\right))_*\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot 2\right)} \cdot \sqrt{\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +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*))))))