Toniolo and Linder, Equation (13)

Average Error: 34.3 → 27.1

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.712021468906686645219995359698905452875 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot \left(\left(\left(U* - U\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\left(\left(\left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3297176 = 2.0;
        double r3297177 = n;
        double r3297178 = r3297176 * r3297177;
        double r3297179 = U;
        double r3297180 = r3297178 * r3297179;
        double r3297181 = t;
        double r3297182 = l;
        double r3297183 = r3297182 * r3297182;
        double r3297184 = Om;
        double r3297185 = r3297183 / r3297184;
        double r3297186 = r3297176 * r3297185;
        double r3297187 = r3297181 - r3297186;
        double r3297188 = r3297182 / r3297184;
        double r3297189 = pow(r3297188, r3297176);
        double r3297190 = r3297177 * r3297189;
        double r3297191 = U_;
        double r3297192 = r3297179 - r3297191;
        double r3297193 = r3297190 * r3297192;
        double r3297194 = r3297187 - r3297193;
        double r3297195 = r3297180 * r3297194;
        double r3297196 = sqrt(r3297195);
        return r3297196;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3297197 = n;
        double r3297198 = -1.7120214689066866e-307;
        bool r3297199 = r3297197 <= r3297198;
        double r3297200 = 2.0;
        double r3297201 = r3297200 * r3297197;
        double r3297202 = U;
        double r3297203 = t;
        double r3297204 = l;
        double r3297205 = Om;
        double r3297206 = r3297204 / r3297205;
        double r3297207 = 2.0;
        double r3297208 = r3297200 / r3297207;
        double r3297209 = pow(r3297206, r3297208);
        double r3297210 = r3297209 * r3297197;
        double r3297211 = U_;
        double r3297212 = r3297211 - r3297202;
        double r3297213 = cbrt(r3297204);
        double r3297214 = cbrt(r3297205);
        double r3297215 = r3297213 / r3297214;
        double r3297216 = pow(r3297215, r3297208);
        double r3297217 = r3297212 * r3297216;
        double r3297218 = r3297213 * r3297213;
        double r3297219 = r3297214 * r3297214;
        double r3297220 = r3297218 / r3297219;
        double r3297221 = pow(r3297220, r3297208);
        double r3297222 = r3297217 * r3297221;
        double r3297223 = r3297210 * r3297222;
        double r3297224 = r3297200 * r3297204;
        double r3297225 = r3297206 * r3297224;
        double r3297226 = r3297223 - r3297225;
        double r3297227 = r3297203 + r3297226;
        double r3297228 = r3297202 * r3297227;
        double r3297229 = r3297201 * r3297228;
        double r3297230 = sqrt(r3297229);
        double r3297231 = sqrt(r3297201);
        double r3297232 = r3297212 * r3297209;
        double r3297233 = r3297232 * r3297210;
        double r3297234 = r3297233 - r3297225;
        double r3297235 = r3297234 + r3297203;
        double r3297236 = r3297235 * r3297202;
        double r3297237 = sqrt(r3297236);
        double r3297238 = r3297231 * r3297237;
        double r3297239 = r3297199 ? r3297230 : r3297238;
        return r3297239;
}

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 n < -1.7120214689066866e-307

   1. Initial program 34.3

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

   2. Simplified 32.4

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

   4. Applied associate-*l* 32.6

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

   6. Applied sqr-pow 32.6

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

   7. Applied associate-*r* 31.7

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

   9. Applied associate-*l* 31.3

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

   11. Applied add-cube-cbrt 31.3

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

   12. Applied add-cube-cbrt 31.3

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

   13. Applied times-frac 31.3

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

   14. Applied unpow-prod-down 31.3

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

   15. Applied associate-*l* 31.3

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

   if -1.7120214689066866e-307 < n

   1. Initial program 34.2

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

   2. Simplified 30.8

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

   4. Applied associate-*l* 31.1

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

   6. Applied sqr-pow 31.1

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

   7. Applied associate-*r* 30.0

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

   9. Applied associate-*l* 29.9

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

   11. Applied sqrt-prod 22.9

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

4. Final simplification 27.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.712021468906686645219995359698905452875 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot \left(\left(\left(U* - U\right) \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\left(\left(\left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))