Toniolo and Linder, Equation (13)

Average Error: 34.9 → 28.8

Time: 1.9m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -6.56714979925425613 \cdot 10^{186}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \le 3.330205482825987 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot U}\\ \mathbf{elif}\;n \le 1.2408046941130587 \cdot 10^{-213} \lor \neg \left(n \le 2.96621220568580719 \cdot 10^{43}\right):\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r268555 = 2.0;
        double r268556 = n;
        double r268557 = r268555 * r268556;
        double r268558 = U;
        double r268559 = r268557 * r268558;
        double r268560 = t;
        double r268561 = l;
        double r268562 = r268561 * r268561;
        double r268563 = Om;
        double r268564 = r268562 / r268563;
        double r268565 = r268555 * r268564;
        double r268566 = r268560 - r268565;
        double r268567 = r268561 / r268563;
        double r268568 = pow(r268567, r268555);
        double r268569 = r268556 * r268568;
        double r268570 = U_;
        double r268571 = r268558 - r268570;
        double r268572 = r268569 * r268571;
        double r268573 = r268566 - r268572;
        double r268574 = r268559 * r268573;
        double r268575 = sqrt(r268574);
        return r268575;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r268576 = n;
        double r268577 = -6.567149799254256e+186;
        bool r268578 = r268576 <= r268577;
        double r268579 = 2.0;
        double r268580 = r268579 * r268576;
        double r268581 = U;
        double r268582 = r268580 * r268581;
        double r268583 = t;
        double r268584 = l;
        double r268585 = Om;
        double r268586 = r268584 / r268585;
        double r268587 = r268584 * r268586;
        double r268588 = r268579 * r268587;
        double r268589 = r268583 - r268588;
        double r268590 = pow(r268586, r268579);
        double r268591 = r268576 * r268590;
        double r268592 = U_;
        double r268593 = r268581 - r268592;
        double r268594 = r268591 * r268593;
        double r268595 = r268589 - r268594;
        double r268596 = r268582 * r268595;
        double r268597 = sqrt(r268596);
        double r268598 = 3.330205482826e-310;
        bool r268599 = r268576 <= r268598;
        double r268600 = r268593 * r268591;
        double r268601 = fma(r268587, r268579, r268600);
        double r268602 = r268583 - r268601;
        double r268603 = r268580 * r268602;
        double r268604 = r268603 * r268581;
        double r268605 = sqrt(r268604);
        double r268606 = 1.2408046941130587e-213;
        bool r268607 = r268576 <= r268606;
        double r268608 = 2.966212205685807e+43;
        bool r268609 = r268576 <= r268608;
        double r268610 = !r268609;
        bool r268611 = r268607 || r268610;
        double r268612 = sqrt(r268580);
        double r268613 = r268602 * r268581;
        double r268614 = sqrt(r268613);
        double r268615 = r268612 * r268614;
        double r268616 = r268611 ? r268615 : r268597;
        double r268617 = r268599 ? r268605 : r268616;
        double r268618 = r268578 ? r268597 : r268617;
        return r268618;
}

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 n < -6.567149799254256e+186 or 1.2408046941130587e-213 < n < 2.966212205685807e+43

   1. Initial program 34.1

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

   3. Applied *-un-lft-identity 34.1

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

   4. Applied times-frac 31.4

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

   5. Simplified 31.4

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

   if -6.567149799254256e+186 < n < 3.330205482826e-310

   1. Initial program 34.7

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

   3. Applied *-un-lft-identity 34.7

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

   4. Applied times-frac 31.8

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

   5. Simplified 31.8

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

   7. Applied associate-*l* 31.9

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

   8. Simplified 32.8

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

   10. Applied associate-*l* 31.9

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

   12. Applied associate-*r* 30.4

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

   if 3.330205482826e-310 < n < 1.2408046941130587e-213 or 2.966212205685807e+43 < n

   1. Initial program 36.0

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

   3. Applied *-un-lft-identity 36.0

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

   4. Applied times-frac 34.1

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

   5. Simplified 34.1

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

   7. Applied associate-*l* 34.1

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

   8. Simplified 37.8

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

   10. Applied associate-*l* 34.1

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

   12. Applied sqrt-prod 23.3

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

4. Final simplification 28.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.56714979925425613 \cdot 10^{186}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;n \le 3.330205482825987 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right)\right) \cdot U}\\ \mathbf{elif}\;n \le 1.2408046941130587 \cdot 10^{-213} \lor \neg \left(n \le 2.96621220568580719 \cdot 10^{43}\right):\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))