Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.1

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -6.475205078487209 \cdot 10^{+94}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{\frac{1}{4}} \cdot {\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{\frac{1}{4}}\\ \mathbf{elif}\;n \le 2.122462322126863 \cdot 10^{-309}:\\ \;\;\;\;{\left(\left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - (\ell \cdot 2 + \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right))_* \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)} \cdot {n}^{\frac{1}{2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r16174417 = 2.0;
        double r16174418 = n;
        double r16174419 = r16174417 * r16174418;
        double r16174420 = U;
        double r16174421 = r16174419 * r16174420;
        double r16174422 = t;
        double r16174423 = l;
        double r16174424 = r16174423 * r16174423;
        double r16174425 = Om;
        double r16174426 = r16174424 / r16174425;
        double r16174427 = r16174417 * r16174426;
        double r16174428 = r16174422 - r16174427;
        double r16174429 = r16174423 / r16174425;
        double r16174430 = pow(r16174429, r16174417);
        double r16174431 = r16174418 * r16174430;
        double r16174432 = U_;
        double r16174433 = r16174420 - r16174432;
        double r16174434 = r16174431 * r16174433;
        double r16174435 = r16174428 - r16174434;
        double r16174436 = r16174421 * r16174435;
        double r16174437 = sqrt(r16174436);
        return r16174437;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r16174438 = n;
        double r16174439 = -6.475205078487209e+94;
        bool r16174440 = r16174438 <= r16174439;
        double r16174441 = U;
        double r16174442 = 2.0;
        double r16174443 = r16174441 * r16174442;
        double r16174444 = t;
        double r16174445 = l;
        double r16174446 = Om;
        double r16174447 = r16174445 / r16174446;
        double r16174448 = r16174445 * r16174442;
        double r16174449 = U_;
        double r16174450 = r16174441 - r16174449;
        double r16174451 = r16174447 * r16174450;
        double r16174452 = r16174438 * r16174451;
        double r16174453 = r16174448 + r16174452;
        double r16174454 = r16174447 * r16174453;
        double r16174455 = r16174444 - r16174454;
        double r16174456 = r16174443 * r16174455;
        double r16174457 = r16174438 * r16174456;
        double r16174458 = 0.25;
        double r16174459 = pow(r16174457, r16174458);
        double r16174460 = r16174459 * r16174459;
        double r16174461 = 2.122462322126863e-309;
        bool r16174462 = r16174438 <= r16174461;
        double r16174463 = r16174438 * r16174455;
        double r16174464 = r16174463 * r16174443;
        double r16174465 = 0.5;
        double r16174466 = pow(r16174464, r16174465);
        double r16174467 = fma(r16174445, r16174442, r16174452);
        double r16174468 = r16174467 * r16174447;
        double r16174469 = r16174444 - r16174468;
        double r16174470 = r16174469 * r16174443;
        double r16174471 = sqrt(r16174470);
        double r16174472 = pow(r16174438, r16174465);
        double r16174473 = r16174471 * r16174472;
        double r16174474 = r16174462 ? r16174466 : r16174473;
        double r16174475 = r16174440 ? r16174460 : r16174474;
        return r16174475;
}

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 < -6.475205078487209e+94

   1. Initial program 32.6

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

   3. Applied associate-/l* 31.3

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

   5. Applied unpow2 31.3

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

   6. Applied associate-*r* 29.0

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

   8. Applied pow1 29.0

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

   9. Applied sqrt-pow1 29.0

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

   10. Simplified 28.7

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

   12. Applied sqr-pow 28.9

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

   if -6.475205078487209e+94 < n < 2.122462322126863e-309

   1. Initial program 33.3

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

   3. Applied associate-/l* 29.7

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

   5. Applied unpow2 29.7

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

   6. Applied associate-*r* 29.2

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

   8. Applied pow1 29.2

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

   9. Applied sqrt-pow1 29.2

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

   10. Simplified 29.0

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

   12. Applied associate-*l* 26.9

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

   if 2.122462322126863e-309 < n

   1. Initial program 33.3

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

   3. Applied associate-/l* 30.8

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

   5. Applied unpow2 30.8

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

   6. Applied associate-*r* 29.9

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

   8. Applied pow1 29.9

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

   9. Applied sqrt-pow1 29.9

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

   10. Simplified 30.0

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

   12. Applied unpow-prod-down 22.9

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

   13. Simplified 22.9

       \[\leadsto \color{blue}{\sqrt{\left(t - (\ell \cdot 2 + \left(\left(\left(U - U*\right) \cdot \frac{\ell}{Om}\right) \cdot n\right))_* \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)}} \cdot {n}^{\left(\frac{1}{2}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 25.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -6.475205078487209 \cdot 10^{+94}:\\ \;\;\;\;{\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{\frac{1}{4}} \cdot {\left(n \cdot \left(\left(U \cdot 2\right) \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)\right)}^{\frac{1}{4}}\\ \mathbf{elif}\;n \le 2.122462322126863 \cdot 10^{-309}:\\ \;\;\;\;{\left(\left(n \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 + n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - (\ell \cdot 2 + \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right))_* \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot 2\right)} \cdot {n}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +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*))))))