Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.9

Time: 37.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le 2.9259008121443 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)} \cdot \left(\sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)} \cdot \sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot n\right) \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{Om}\right)\right)\right) \cdot 2\right)} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1488121 = 2.0;
        double r1488122 = n;
        double r1488123 = r1488121 * r1488122;
        double r1488124 = U;
        double r1488125 = r1488123 * r1488124;
        double r1488126 = t;
        double r1488127 = l;
        double r1488128 = r1488127 * r1488127;
        double r1488129 = Om;
        double r1488130 = r1488128 / r1488129;
        double r1488131 = r1488121 * r1488130;
        double r1488132 = r1488126 - r1488131;
        double r1488133 = r1488127 / r1488129;
        double r1488134 = pow(r1488133, r1488121);
        double r1488135 = r1488122 * r1488134;
        double r1488136 = U_;
        double r1488137 = r1488124 - r1488136;
        double r1488138 = r1488135 * r1488137;
        double r1488139 = r1488132 - r1488138;
        double r1488140 = r1488125 * r1488139;
        double r1488141 = sqrt(r1488140);
        return r1488141;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1488142 = U;
        double r1488143 = 2.9259008121443e-311;
        bool r1488144 = r1488142 <= r1488143;
        double r1488145 = t;
        double r1488146 = l;
        double r1488147 = 2.0;
        double r1488148 = r1488146 * r1488147;
        double r1488149 = U_;
        double r1488150 = r1488149 - r1488142;
        double r1488151 = cbrt(r1488146);
        double r1488152 = r1488151 * r1488151;
        double r1488153 = Om;
        double r1488154 = r1488151 / r1488153;
        double r1488155 = n;
        double r1488156 = r1488154 * r1488155;
        double r1488157 = r1488152 * r1488156;
        double r1488158 = r1488150 * r1488157;
        double r1488159 = cbrt(r1488158);
        double r1488160 = r1488159 * r1488159;
        double r1488161 = r1488159 * r1488160;
        double r1488162 = r1488148 - r1488161;
        double r1488163 = r1488146 / r1488153;
        double r1488164 = r1488162 * r1488163;
        double r1488165 = r1488145 - r1488164;
        double r1488166 = r1488165 * r1488147;
        double r1488167 = r1488166 * r1488155;
        double r1488168 = r1488142 * r1488167;
        double r1488169 = sqrt(r1488168);
        double r1488170 = r1488151 * r1488155;
        double r1488171 = r1488152 / r1488153;
        double r1488172 = r1488170 * r1488171;
        double r1488173 = r1488150 * r1488172;
        double r1488174 = r1488148 - r1488173;
        double r1488175 = r1488163 * r1488174;
        double r1488176 = r1488145 - r1488175;
        double r1488177 = r1488176 * r1488147;
        double r1488178 = r1488155 * r1488177;
        double r1488179 = sqrt(r1488178);
        double r1488180 = sqrt(r1488142);
        double r1488181 = r1488179 * r1488180;
        double r1488182 = r1488144 ? r1488169 : r1488181;
        return r1488182;
}

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 U < 2.9259008121443e-311

   1. Initial program 33.5

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

   2. Simplified 30.0

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

   4. Applied div-inv 30.0

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

   5. Applied add-cube-cbrt 30.0

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{Om \cdot \frac{1}{n}}\right)\right)\right)}\]

   6. Applied times-frac 29.5

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{Om} \cdot \frac{\sqrt[3]{\ell}}{\frac{1}{n}}\right)}\right)\right)\right)}\]

   7. Simplified 29.5

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

   9. Applied associate-*l* 29.6

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

   11. Applied div-inv 29.6

       \[\leadsto \sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \frac{1}{Om}\right)} \cdot \left(n \cdot \sqrt[3]{\ell}\right)\right)\right)\right)\right)\right)}\]

   12. Applied associate-*l* 29.7

       \[\leadsto \sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{1}{Om} \cdot \left(n \cdot \sqrt[3]{\ell}\right)\right)\right)}\right)\right)\right)\right)}\]

   13. Simplified 29.6

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

   15. Applied add-cube-cbrt 29.6

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

   if 2.9259008121443e-311 < U

   1. Initial program 33.0

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

   2. Simplified 30.1

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

   4. Applied div-inv 30.1

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

   5. Applied add-cube-cbrt 30.1

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{Om \cdot \frac{1}{n}}\right)\right)\right)}\]

   6. Applied times-frac 29.7

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{Om} \cdot \frac{\sqrt[3]{\ell}}{\frac{1}{n}}\right)}\right)\right)\right)}\]

   7. Simplified 29.7

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

   9. Applied associate-*l* 29.1

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

   11. Applied sqrt-prod 22.2

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

4. Final simplification 25.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 2.9259008121443 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)} \cdot \left(\sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)} \cdot \sqrt[3]{\left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left(\frac{\sqrt[3]{\ell}}{Om} \cdot n\right)\right)}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(\left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\sqrt[3]{\ell} \cdot n\right) \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{Om}\right)\right)\right) \cdot 2\right)} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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*))))))