Toniolo and Linder, Equation (13)

Average Error: 32.9 → 26.7

Time: 34.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.6344908840558971 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + 2 \cdot \ell}{\frac{Om}{\ell}}\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 5.316465484859485 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + 2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1437156 = 2.0;
        double r1437157 = n;
        double r1437158 = r1437156 * r1437157;
        double r1437159 = U;
        double r1437160 = r1437158 * r1437159;
        double r1437161 = t;
        double r1437162 = l;
        double r1437163 = r1437162 * r1437162;
        double r1437164 = Om;
        double r1437165 = r1437163 / r1437164;
        double r1437166 = r1437156 * r1437165;
        double r1437167 = r1437161 - r1437166;
        double r1437168 = r1437162 / r1437164;
        double r1437169 = pow(r1437168, r1437156);
        double r1437170 = r1437157 * r1437169;
        double r1437171 = U_;
        double r1437172 = r1437159 - r1437171;
        double r1437173 = r1437170 * r1437172;
        double r1437174 = r1437167 - r1437173;
        double r1437175 = r1437160 * r1437174;
        double r1437176 = sqrt(r1437175);
        return r1437176;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1437177 = 2.0;
        double r1437178 = n;
        double r1437179 = r1437177 * r1437178;
        double r1437180 = U;
        double r1437181 = r1437179 * r1437180;
        double r1437182 = t;
        double r1437183 = l;
        double r1437184 = r1437183 * r1437183;
        double r1437185 = Om;
        double r1437186 = r1437184 / r1437185;
        double r1437187 = r1437186 * r1437177;
        double r1437188 = r1437182 - r1437187;
        double r1437189 = r1437183 / r1437185;
        double r1437190 = pow(r1437189, r1437177);
        double r1437191 = r1437178 * r1437190;
        double r1437192 = U_;
        double r1437193 = r1437180 - r1437192;
        double r1437194 = r1437191 * r1437193;
        double r1437195 = r1437188 - r1437194;
        double r1437196 = r1437181 * r1437195;
        double r1437197 = sqrt(r1437196);
        double r1437198 = 1.6344908840558971e-152;
        bool r1437199 = r1437197 <= r1437198;
        double r1437200 = r1437185 / r1437183;
        double r1437201 = r1437178 / r1437200;
        double r1437202 = r1437201 * r1437193;
        double r1437203 = r1437177 * r1437183;
        double r1437204 = r1437202 + r1437203;
        double r1437205 = r1437204 / r1437200;
        double r1437206 = r1437182 - r1437205;
        double r1437207 = r1437180 * r1437206;
        double r1437208 = sqrt(r1437207);
        double r1437209 = sqrt(r1437179);
        double r1437210 = r1437208 * r1437209;
        double r1437211 = 5.316465484859485e+126;
        bool r1437212 = r1437197 <= r1437211;
        double r1437213 = r1437179 * r1437207;
        double r1437214 = sqrt(r1437213);
        double r1437215 = r1437212 ? r1437197 : r1437214;
        double r1437216 = r1437199 ? r1437210 : r1437215;
        return r1437216;
}

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*))))) < 1.6344908840558971e-152

   1. Initial program 53.7

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

   2. Simplified 37.6

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

   4. Applied associate-*r* 53.5

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

   6. Applied pow1 53.5

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

   7. Applied pow1 53.5

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

   8. Applied pow-prod-down 53.5

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

   9. Simplified 38.3

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

   11. Applied unpow-prod-down 38.3

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

   12. Applied sqrt-prod 38.0

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

   13. Simplified 37.5

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

   14. Simplified 37.5

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

   if 1.6344908840558971e-152 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 5.316465484859485e+126

   1. Initial program 1.6

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

   if 5.316465484859485e+126 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

   1. Initial program 56.3

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

   2. Simplified 48.8

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

   4. Applied associate-*r* 48.9

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

   6. Applied pow1 48.9

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

   7. Applied pow1 48.9

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

   8. Applied pow-prod-down 48.9

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

   9. Simplified 47.6

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

   11. Applied *-un-lft-identity 47.6

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

   12. Applied associate-*r* 47.6

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

   13. Simplified 47.5

       \[\leadsto \sqrt{{\left(\left(\color{blue}{\left(t - \frac{2 \cdot \ell + \left(U - U*\right) \cdot \frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)} \cdot U\right) \cdot \left(n \cdot 2\right)\right)}^{1}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 26.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 1.6344908840558971 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + 2 \cdot \ell}{\frac{Om}{\ell}}\right)} \cdot \sqrt{2 \cdot n}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 5.316465484859485 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right) + 2 \cdot \ell}{\frac{Om}{\ell}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(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*))))))