Toniolo and Linder, Equation (13)

Average Error: 33.3 → 29.1

Time: 41.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 1.8088620139970252 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.5047052485579951 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 5.16907107140256 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r929162 = 2.0;
        double r929163 = n;
        double r929164 = r929162 * r929163;
        double r929165 = U;
        double r929166 = r929164 * r929165;
        double r929167 = t;
        double r929168 = l;
        double r929169 = r929168 * r929168;
        double r929170 = Om;
        double r929171 = r929169 / r929170;
        double r929172 = r929162 * r929171;
        double r929173 = r929167 - r929172;
        double r929174 = r929168 / r929170;
        double r929175 = pow(r929174, r929162);
        double r929176 = r929163 * r929175;
        double r929177 = U_;
        double r929178 = r929165 - r929177;
        double r929179 = r929176 * r929178;
        double r929180 = r929173 - r929179;
        double r929181 = r929166 * r929180;
        double r929182 = sqrt(r929181);
        return r929182;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r929183 = t;
        double r929184 = 1.8088620139970252e-254;
        bool r929185 = r929183 <= r929184;
        double r929186 = 2.0;
        double r929187 = n;
        double r929188 = r929186 * r929187;
        double r929189 = U;
        double r929190 = l;
        double r929191 = Om;
        double r929192 = r929190 / r929191;
        double r929193 = r929186 * r929192;
        double r929194 = U_;
        double r929195 = r929189 - r929194;
        double r929196 = r929192 * r929195;
        double r929197 = r929187 * r929196;
        double r929198 = r929197 * r929192;
        double r929199 = fma(r929190, r929193, r929198);
        double r929200 = r929183 - r929199;
        double r929201 = r929189 * r929200;
        double r929202 = r929188 * r929201;
        double r929203 = sqrt(r929202);
        double r929204 = 1.5047052485579951e-182;
        bool r929205 = r929183 <= r929204;
        double r929206 = r929186 * r929190;
        double r929207 = r929187 * r929192;
        double r929208 = r929192 * r929207;
        double r929209 = r929208 * r929195;
        double r929210 = fma(r929206, r929192, r929209);
        double r929211 = r929183 - r929210;
        double r929212 = sqrt(r929211);
        double r929213 = r929188 * r929189;
        double r929214 = sqrt(r929213);
        double r929215 = r929212 * r929214;
        double r929216 = 5.16907107140256e-46;
        bool r929217 = r929183 <= r929216;
        double r929218 = sqrt(r929200);
        double r929219 = r929218 * r929214;
        double r929220 = r929217 ? r929203 : r929219;
        double r929221 = r929205 ? r929215 : r929220;
        double r929222 = r929185 ? r929203 : r929221;
        return r929222;
}

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 < 1.8088620139970252e-254 or 1.5047052485579951e-182 < t < 5.16907107140256e-46

   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 *-un-lft-identity 33.2

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

   4. Applied times-frac 30.5

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

   5. Simplified 30.5

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

   7. Applied unpow2 30.5

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

   8. Applied associate-*r* 29.5

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

   10. Applied associate-*l* 29.3

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

   12. Applied associate-*l* 29.6

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

   13. Simplified 29.8

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

   if 1.8088620139970252e-254 < t < 1.5047052485579951e-182

   1. Initial program 36.8

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

   3. Applied *-un-lft-identity 36.8

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

   4. Applied times-frac 34.6

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

   5. Simplified 34.6

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

   7. Applied unpow2 34.6

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

   8. Applied associate-*r* 32.9

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

   10. Applied sqrt-prod 33.1

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

   11. Simplified 33.1

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

   if 5.16907107140256e-46 < t

   1. Initial program 32.8

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

   3. Applied *-un-lft-identity 32.8

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

   4. Applied times-frac 30.5

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

   5. Simplified 30.5

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

   7. Applied unpow2 30.5

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

   8. Applied associate-*r* 30.1

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

   10. Applied associate-*l* 30.1

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

   12. Applied sqrt-prod 26.5

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

   13. Simplified 26.7

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

4. Final simplification 29.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.8088620139970252 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.5047052485579951 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 5.16907107140256 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell, 2 \cdot \frac{\ell}{Om}, \left(n \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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*))))))