Toniolo and Linder, Equation (13)

Average Error: 34.3 → 27.5

Time: 59.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.712021468906686645219995359698905452875 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right), t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r4284104 = 2.0;
        double r4284105 = n;
        double r4284106 = r4284104 * r4284105;
        double r4284107 = U;
        double r4284108 = r4284106 * r4284107;
        double r4284109 = t;
        double r4284110 = l;
        double r4284111 = r4284110 * r4284110;
        double r4284112 = Om;
        double r4284113 = r4284111 / r4284112;
        double r4284114 = r4284104 * r4284113;
        double r4284115 = r4284109 - r4284114;
        double r4284116 = r4284110 / r4284112;
        double r4284117 = pow(r4284116, r4284104);
        double r4284118 = r4284105 * r4284117;
        double r4284119 = U_;
        double r4284120 = r4284107 - r4284119;
        double r4284121 = r4284118 * r4284120;
        double r4284122 = r4284115 - r4284121;
        double r4284123 = r4284108 * r4284122;
        double r4284124 = sqrt(r4284123);
        return r4284124;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r4284125 = n;
        double r4284126 = -1.7120214689066866e-307;
        bool r4284127 = r4284125 <= r4284126;
        double r4284128 = U;
        double r4284129 = U_;
        double r4284130 = r4284129 - r4284128;
        double r4284131 = l;
        double r4284132 = Om;
        double r4284133 = r4284131 / r4284132;
        double r4284134 = 2.0;
        double r4284135 = 2.0;
        double r4284136 = r4284134 / r4284135;
        double r4284137 = pow(r4284133, r4284136);
        double r4284138 = 1.0;
        double r4284139 = cbrt(r4284132);
        double r4284140 = r4284139 * r4284139;
        double r4284141 = r4284138 / r4284140;
        double r4284142 = pow(r4284141, r4284136);
        double r4284143 = r4284142 * r4284125;
        double r4284144 = r4284131 / r4284139;
        double r4284145 = pow(r4284144, r4284136);
        double r4284146 = r4284143 * r4284145;
        double r4284147 = r4284137 * r4284146;
        double r4284148 = t;
        double r4284149 = r4284131 * r4284133;
        double r4284150 = r4284134 * r4284149;
        double r4284151 = r4284148 - r4284150;
        double r4284152 = fma(r4284130, r4284147, r4284151);
        double r4284153 = r4284128 * r4284152;
        double r4284154 = r4284134 * r4284125;
        double r4284155 = r4284153 * r4284154;
        double r4284156 = sqrt(r4284155);
        double r4284157 = sqrt(r4284154);
        double r4284158 = r4284125 * r4284137;
        double r4284159 = r4284158 * r4284137;
        double r4284160 = fma(r4284130, r4284159, r4284151);
        double r4284161 = r4284160 * r4284128;
        double r4284162 = sqrt(r4284161);
        double r4284163 = r4284157 * r4284162;
        double r4284164 = r4284127 ? r4284156 : r4284163;
        return r4284164;
}

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 31.5

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

   4. Applied sqr-pow 31.5

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

   5. Applied associate-*r* 30.8

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

   7. Applied associate-*l* 31.7

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

   9. Applied add-cube-cbrt 31.7

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

   10. Applied *-un-lft-identity 31.7

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

   11. Applied times-frac 31.7

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

   12. Applied unpow-prod-down 31.7

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

   13. Applied associate-*r* 31.9

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

   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 31.1

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

   4. Applied sqr-pow 31.1

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

   5. Applied associate-*r* 30.3

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

   7. Applied associate-*l* 30.0

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

   9. Applied sqrt-prod 23.1

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

4. Final simplification 27.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.712021468906686645219995359698905452875 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right), t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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*))))))