Toniolo and Linder, Equation (13)

Average Error: 33.6 → 28.1

Time: 48.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0452549734454207 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0706139562680882 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\sqrt[3]{U} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2387173 = 2.0;
        double r2387174 = n;
        double r2387175 = r2387173 * r2387174;
        double r2387176 = U;
        double r2387177 = r2387175 * r2387176;
        double r2387178 = t;
        double r2387179 = l;
        double r2387180 = r2387179 * r2387179;
        double r2387181 = Om;
        double r2387182 = r2387180 / r2387181;
        double r2387183 = r2387173 * r2387182;
        double r2387184 = r2387178 - r2387183;
        double r2387185 = r2387179 / r2387181;
        double r2387186 = pow(r2387185, r2387173);
        double r2387187 = r2387174 * r2387186;
        double r2387188 = U_;
        double r2387189 = r2387176 - r2387188;
        double r2387190 = r2387187 * r2387189;
        double r2387191 = r2387184 - r2387190;
        double r2387192 = r2387177 * r2387191;
        double r2387193 = sqrt(r2387192);
        return r2387193;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2387194 = 2.0;
        double r2387195 = n;
        double r2387196 = r2387194 * r2387195;
        double r2387197 = U;
        double r2387198 = r2387196 * r2387197;
        double r2387199 = t;
        double r2387200 = l;
        double r2387201 = r2387200 * r2387200;
        double r2387202 = Om;
        double r2387203 = r2387201 / r2387202;
        double r2387204 = r2387203 * r2387194;
        double r2387205 = r2387199 - r2387204;
        double r2387206 = r2387200 / r2387202;
        double r2387207 = pow(r2387206, r2387194);
        double r2387208 = r2387195 * r2387207;
        double r2387209 = U_;
        double r2387210 = r2387197 - r2387209;
        double r2387211 = r2387208 * r2387210;
        double r2387212 = r2387205 - r2387211;
        double r2387213 = r2387198 * r2387212;
        double r2387214 = sqrt(r2387213);
        double r2387215 = 1.0452549734454207e-138;
        bool r2387216 = r2387214 <= r2387215;
        double r2387217 = r2387194 * r2387200;
        double r2387218 = r2387209 - r2387197;
        double r2387219 = r2387195 * r2387218;
        double r2387220 = r2387206 * r2387219;
        double r2387221 = r2387217 - r2387220;
        double r2387222 = r2387221 * r2387206;
        double r2387223 = r2387199 - r2387222;
        double r2387224 = r2387195 * r2387223;
        double r2387225 = r2387224 * r2387197;
        double r2387226 = cbrt(r2387225);
        double r2387227 = r2387226 * r2387226;
        double r2387228 = r2387226 * r2387227;
        double r2387229 = r2387194 * r2387228;
        double r2387230 = sqrt(r2387229);
        double r2387231 = 1.0706139562680882e+131;
        bool r2387232 = r2387214 <= r2387231;
        double r2387233 = cbrt(r2387197);
        double r2387234 = cbrt(r2387224);
        double r2387235 = r2387233 * r2387234;
        double r2387236 = r2387235 * r2387227;
        double r2387237 = r2387194 * r2387236;
        double r2387238 = sqrt(r2387237);
        double r2387239 = r2387232 ? r2387214 : r2387238;
        double r2387240 = r2387216 ? r2387230 : r2387239;
        return r2387240;
}

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 (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 1.0452549734454207e-138

   1. Initial program 51.5

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

   2. Simplified 37.6

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

   4. Applied add-cube-cbrt 37.8

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

   if 1.0452549734454207e-138 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 1.0706139562680882e+131

   1. Initial program 1.3

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

   if 1.0706139562680882e+131 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

   1. Initial program 57.5

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

   2. Simplified 49.9

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

   4. Applied add-cube-cbrt 50.0

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

   6. Applied cbrt-prod 50.0

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

4. Final simplification 28.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0452549734454207 \cdot 10^{-138}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U}\right)\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.0706139562680882 \cdot 10^{+131}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\sqrt[3]{U} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right) \cdot \left(\sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U} \cdot \sqrt[3]{\left(n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)\right) \cdot U}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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*))))))