Toniolo and Linder, Equation (13)

Average Error: 33.3 → 28.3

Time: 37.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 4.2299491368120496 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\left(\frac{\ell}{Om}\right), \left(2 \cdot \ell\right), \left(\sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;t \le 8.150193543384339 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot n\right)} \cdot \sqrt{t - \mathsf{fma}\left(\left(\frac{\ell}{Om}\right), \left(2 \cdot \ell\right), \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t \le 6.13667835067999 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\left(\frac{\ell}{Om}\right), \left(2 \cdot \ell\right), \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot n\right)} \cdot \sqrt{t - \mathsf{fma}\left(\left(\frac{\ell}{Om}\right), \left(2 \cdot \ell\right), \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1306321 = 2.0;
        double r1306322 = n;
        double r1306323 = r1306321 * r1306322;
        double r1306324 = U;
        double r1306325 = r1306323 * r1306324;
        double r1306326 = t;
        double r1306327 = l;
        double r1306328 = r1306327 * r1306327;
        double r1306329 = Om;
        double r1306330 = r1306328 / r1306329;
        double r1306331 = r1306321 * r1306330;
        double r1306332 = r1306326 - r1306331;
        double r1306333 = r1306327 / r1306329;
        double r1306334 = pow(r1306333, r1306321);
        double r1306335 = r1306322 * r1306334;
        double r1306336 = U_;
        double r1306337 = r1306324 - r1306336;
        double r1306338 = r1306335 * r1306337;
        double r1306339 = r1306332 - r1306338;
        double r1306340 = r1306325 * r1306339;
        double r1306341 = sqrt(r1306340);
        return r1306341;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1306342 = t;
        double r1306343 = 4.2299491368120496e-247;
        bool r1306344 = r1306342 <= r1306343;
        double r1306345 = l;
        double r1306346 = Om;
        double r1306347 = r1306345 / r1306346;
        double r1306348 = 2.0;
        double r1306349 = r1306348 * r1306345;
        double r1306350 = U;
        double r1306351 = U_;
        double r1306352 = r1306350 - r1306351;
        double r1306353 = cbrt(r1306352);
        double r1306354 = r1306353 * r1306353;
        double r1306355 = n;
        double r1306356 = r1306355 * r1306347;
        double r1306357 = r1306356 * r1306347;
        double r1306358 = r1306354 * r1306357;
        double r1306359 = r1306353 * r1306358;
        double r1306360 = fma(r1306347, r1306349, r1306359);
        double r1306361 = r1306342 - r1306360;
        double r1306362 = r1306348 * r1306355;
        double r1306363 = r1306350 * r1306362;
        double r1306364 = r1306361 * r1306363;
        double r1306365 = sqrt(r1306364);
        double r1306366 = 8.150193543384339e-127;
        bool r1306367 = r1306342 <= r1306366;
        double r1306368 = sqrt(r1306363);
        double r1306369 = r1306357 * r1306352;
        double r1306370 = fma(r1306347, r1306349, r1306369);
        double r1306371 = r1306342 - r1306370;
        double r1306372 = sqrt(r1306371);
        double r1306373 = r1306368 * r1306372;
        double r1306374 = 6.13667835067999e+123;
        bool r1306375 = r1306342 <= r1306374;
        double r1306376 = r1306362 * r1306371;
        double r1306377 = r1306376 * r1306350;
        double r1306378 = sqrt(r1306377);
        double r1306379 = r1306375 ? r1306378 : r1306373;
        double r1306380 = r1306367 ? r1306373 : r1306379;
        double r1306381 = r1306344 ? r1306365 : r1306380;
        return r1306381;
}

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 < 4.2299491368120496e-247

   1. Initial program 33.5

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

   2. Simplified 32.5

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

   4. Applied associate-*r* 31.5

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

   6. Applied associate-*r* 30.9

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

   8. Applied associate-*r* 29.9

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

   10. Applied add-cube-cbrt 30.0

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

   11. Applied associate-*r* 30.0

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

   if 4.2299491368120496e-247 < t < 8.150193543384339e-127 or 6.13667835067999e+123 < t

   1. Initial program 36.5

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

   2. Simplified 35.0

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

   4. Applied associate-*r* 35.0

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

   6. Applied associate-*r* 34.4

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

   8. Applied associate-*r* 33.9

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

   10. Applied sqrt-prod 27.4

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

   if 8.150193543384339e-127 < t < 6.13667835067999e+123

   1. Initial program 28.9

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

   2. Simplified 26.7

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

   4. Applied associate-*r* 26.8

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

   6. Applied associate-*r* 26.5

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

   8. Applied associate-*r* 25.7

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

   10. Applied associate-*l* 24.9

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

4. Final simplification 28.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 4.2299491368120496 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(\left(\frac{\ell}{Om}\right), \left(2 \cdot \ell\right), \left(\sqrt[3]{U - U*} \cdot \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;t \le 8.150193543384339 \cdot 10^{-127}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot n\right)} \cdot \sqrt{t - \mathsf{fma}\left(\left(\frac{\ell}{Om}\right), \left(2 \cdot \ell\right), \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t \le 6.13667835067999 \cdot 10^{+123}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\left(\frac{\ell}{Om}\right), \left(2 \cdot \ell\right), \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot U}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot n\right)} \cdot \sqrt{t - \mathsf{fma}\left(\left(\frac{\ell}{Om}\right), \left(2 \cdot \ell\right), \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +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*))))))