Toniolo and Linder, Equation (13)

Average Error: 34.6 → 29.3

Time: 44.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 3.683386903089245746265762362490297308203 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r85275 = 2.0;
        double r85276 = n;
        double r85277 = r85275 * r85276;
        double r85278 = U;
        double r85279 = r85277 * r85278;
        double r85280 = t;
        double r85281 = l;
        double r85282 = r85281 * r85281;
        double r85283 = Om;
        double r85284 = r85282 / r85283;
        double r85285 = r85275 * r85284;
        double r85286 = r85280 - r85285;
        double r85287 = r85281 / r85283;
        double r85288 = pow(r85287, r85275);
        double r85289 = r85276 * r85288;
        double r85290 = U_;
        double r85291 = r85278 - r85290;
        double r85292 = r85289 * r85291;
        double r85293 = r85286 - r85292;
        double r85294 = r85279 * r85293;
        double r85295 = sqrt(r85294);
        return r85295;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r85296 = t;
        double r85297 = 3.683386903089246e+135;
        bool r85298 = r85296 <= r85297;
        double r85299 = 2.0;
        double r85300 = l;
        double r85301 = Om;
        double r85302 = r85301 / r85300;
        double r85303 = r85300 / r85302;
        double r85304 = n;
        double r85305 = cbrt(r85304);
        double r85306 = r85305 * r85305;
        double r85307 = r85300 / r85301;
        double r85308 = 2.0;
        double r85309 = r85299 / r85308;
        double r85310 = pow(r85307, r85309);
        double r85311 = r85305 * r85310;
        double r85312 = r85306 * r85311;
        double r85313 = U;
        double r85314 = U_;
        double r85315 = r85313 - r85314;
        double r85316 = r85310 * r85315;
        double r85317 = r85312 * r85316;
        double r85318 = fma(r85299, r85303, r85317);
        double r85319 = r85296 - r85318;
        double r85320 = r85299 * r85304;
        double r85321 = r85320 * r85313;
        double r85322 = r85319 * r85321;
        double r85323 = sqrt(r85322);
        double r85324 = r85299 * r85300;
        double r85325 = r85308 * r85309;
        double r85326 = pow(r85307, r85325);
        double r85327 = r85304 * r85326;
        double r85328 = r85315 * r85327;
        double r85329 = fma(r85307, r85324, r85328);
        double r85330 = r85296 - r85329;
        double r85331 = sqrt(r85330);
        double r85332 = sqrt(r85321);
        double r85333 = r85331 * r85332;
        double r85334 = r85298 ? r85323 : r85333;
        return r85334;
}

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 2 regimes

2. if t < 3.683386903089246e+135

   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. Simplified 34.1

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

   4. Applied associate-/l* 31.2

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

   6. Applied sqr-pow 31.2

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

   7. Applied associate-*r* 30.4

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

   9. Applied add-cube-cbrt 30.5

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

   10. Applied associate-*l* 30.5

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

   12. Applied associate-*l* 30.4

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

   if 3.683386903089246e+135 < t

   1. Initial program 37.8

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

   2. Simplified 37.8

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

   4. Applied associate-/l* 35.2

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

   6. Applied sqr-pow 35.2

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

   7. Applied associate-*r* 34.6

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

   9. Applied add-cube-cbrt 34.6

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

   10. Applied associate-*l* 34.6

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

   12. Applied sqrt-prod 22.6

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

   13. Simplified 23.1

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

4. Final simplification 29.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.683386903089245746265762362490297308203 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))