Toniolo and Linder, Equation (13)

Average Error: 34.9 → 29.7

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.77082420255664932142459979130796319176 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}}\\ \mathbf{elif}\;t \le 3.19389681298425902451773616259157226606 \cdot 10^{84}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r203180 = 2.0;
        double r203181 = n;
        double r203182 = r203180 * r203181;
        double r203183 = U;
        double r203184 = r203182 * r203183;
        double r203185 = t;
        double r203186 = l;
        double r203187 = r203186 * r203186;
        double r203188 = Om;
        double r203189 = r203187 / r203188;
        double r203190 = r203180 * r203189;
        double r203191 = r203185 - r203190;
        double r203192 = r203186 / r203188;
        double r203193 = pow(r203192, r203180);
        double r203194 = r203181 * r203193;
        double r203195 = U_;
        double r203196 = r203183 - r203195;
        double r203197 = r203194 * r203196;
        double r203198 = r203191 - r203197;
        double r203199 = r203184 * r203198;
        double r203200 = sqrt(r203199);
        return r203200;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r203201 = t;
        double r203202 = -1.7708242025566493e-147;
        bool r203203 = r203201 <= r203202;
        double r203204 = 2.0;
        double r203205 = n;
        double r203206 = r203204 * r203205;
        double r203207 = U;
        double r203208 = l;
        double r203209 = Om;
        double r203210 = r203208 / r203209;
        double r203211 = r203208 * r203210;
        double r203212 = 2.0;
        double r203213 = r203204 / r203212;
        double r203214 = r203212 * r203213;
        double r203215 = pow(r203210, r203214);
        double r203216 = r203215 * r203205;
        double r203217 = U_;
        double r203218 = r203207 - r203217;
        double r203219 = r203216 * r203218;
        double r203220 = fma(r203204, r203211, r203219);
        double r203221 = r203201 - r203220;
        double r203222 = cbrt(r203221);
        double r203223 = r203222 * r203222;
        double r203224 = r203207 * r203223;
        double r203225 = r203206 * r203224;
        double r203226 = r203204 * r203211;
        double r203227 = r203201 - r203226;
        double r203228 = pow(r203210, r203213);
        double r203229 = r203205 * r203228;
        double r203230 = r203218 * r203228;
        double r203231 = r203229 * r203230;
        double r203232 = r203227 - r203231;
        double r203233 = cbrt(r203232);
        double r203234 = r203225 * r203233;
        double r203235 = sqrt(r203234);
        double r203236 = 3.193896812984259e+84;
        bool r203237 = r203201 <= r203236;
        double r203238 = r203206 * r203207;
        double r203239 = r203229 * r203228;
        double r203240 = r203239 * r203218;
        double r203241 = r203227 - r203240;
        double r203242 = r203238 * r203241;
        double r203243 = sqrt(r203242);
        double r203244 = sqrt(r203238);
        double r203245 = r203210 * r203204;
        double r203246 = pow(r203210, r203204);
        double r203247 = r203205 * r203246;
        double r203248 = r203218 * r203247;
        double r203249 = fma(r203208, r203245, r203248);
        double r203250 = r203201 - r203249;
        double r203251 = sqrt(r203250);
        double r203252 = r203244 * r203251;
        double r203253 = r203237 ? r203243 : r203252;
        double r203254 = r203203 ? r203235 : r203253;
        return r203254;
}

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 < -1.7708242025566493e-147

   1. Initial program 33.8

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

   3. Applied *-un-lft-identity 33.8

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

   4. Applied times-frac 30.6

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

   5. Simplified 30.6

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

   7. Applied sqr-pow 30.6

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

   8. Applied associate-*r* 30.0

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

   10. Applied associate-*l* 30.1

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

   11. Simplified 30.1

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

   13. Applied add-cube-cbrt 30.4

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

   14. Applied associate-*r* 30.4

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

   15. Simplified 30.2

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

   if -1.7708242025566493e-147 < t < 3.193896812984259e+84

   1. Initial program 35.0

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

   3. Applied *-un-lft-identity 35.0

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

   4. Applied times-frac 32.3

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

   5. Simplified 32.3

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

   7. Applied sqr-pow 32.3

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

   8. Applied associate-*r* 31.2

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

   if 3.193896812984259e+84 < t

   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 34.2

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

   5. Simplified 34.2

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

   7. Applied sqrt-prod 25.7

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

   8. Simplified 25.7

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

4. Final simplification 29.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.77082420255664932142459979130796319176 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(U \cdot \left(\sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n\right) \cdot \left(U - U*\right)\right)}\right)\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}}\\ \mathbf{elif}\;t \le 3.19389681298425902451773616259157226606 \cdot 10^{84}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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*))))))