Toniolo and Linder, Equation (13)

Average Error: 32.8 → 25.5

Time: 46.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -9.944037691838278 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \sqrt[3]{U - U*} \cdot \left(n \cdot \left(\left(\sqrt[3]{U - U*} \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le -1.1144092498290265 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \frac{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \left(U - U*\right) \cdot \left(\sqrt[3]{n} \cdot \left(\left(\sqrt[3]{n} \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{n} \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r811028 = 2.0;
        double r811029 = n;
        double r811030 = r811028 * r811029;
        double r811031 = U;
        double r811032 = r811030 * r811031;
        double r811033 = t;
        double r811034 = l;
        double r811035 = r811034 * r811034;
        double r811036 = Om;
        double r811037 = r811035 / r811036;
        double r811038 = r811028 * r811037;
        double r811039 = r811033 - r811038;
        double r811040 = r811034 / r811036;
        double r811041 = pow(r811040, r811028);
        double r811042 = r811029 * r811041;
        double r811043 = U_;
        double r811044 = r811031 - r811043;
        double r811045 = r811042 * r811044;
        double r811046 = r811039 - r811045;
        double r811047 = r811032 * r811046;
        double r811048 = sqrt(r811047);
        return r811048;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r811049 = n;
        double r811050 = -9.944037691838278e-75;
        bool r811051 = r811049 <= r811050;
        double r811052 = 2.0;
        double r811053 = r811052 * r811049;
        double r811054 = U;
        double r811055 = r811053 * r811054;
        double r811056 = t;
        double r811057 = l;
        double r811058 = Om;
        double r811059 = r811057 / r811058;
        double r811060 = r811057 * r811059;
        double r811061 = r811052 * r811060;
        double r811062 = r811056 - r811061;
        double r811063 = U_;
        double r811064 = r811054 - r811063;
        double r811065 = cbrt(r811064);
        double r811066 = r811065 * r811059;
        double r811067 = r811066 * r811066;
        double r811068 = r811049 * r811067;
        double r811069 = r811065 * r811068;
        double r811070 = r811062 - r811069;
        double r811071 = r811055 * r811070;
        double r811072 = sqrt(r811071);
        double r811073 = -1.1144092498290265e-301;
        bool r811074 = r811049 <= r811073;
        double r811075 = r811059 * r811052;
        double r811076 = r811060 * r811049;
        double r811077 = r811076 * r811064;
        double r811078 = r811077 / r811058;
        double r811079 = fma(r811057, r811075, r811078);
        double r811080 = r811056 - r811079;
        double r811081 = r811054 * r811080;
        double r811082 = r811081 * r811053;
        double r811083 = sqrt(r811082);
        double r811084 = cbrt(r811049);
        double r811085 = r811084 * r811059;
        double r811086 = r811085 * r811085;
        double r811087 = r811084 * r811086;
        double r811088 = r811064 * r811087;
        double r811089 = fma(r811057, r811075, r811088);
        double r811090 = r811056 - r811089;
        double r811091 = r811054 * r811090;
        double r811092 = sqrt(r811091);
        double r811093 = sqrt(r811053);
        double r811094 = r811092 * r811093;
        double r811095 = r811074 ? r811083 : r811094;
        double r811096 = r811051 ? r811072 : r811095;
        return r811096;
}

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 n < -9.944037691838278e-75

   1. Initial program 30.7

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

   3. Applied *-un-lft-identity 30.7

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

   4. Applied times-frac 28.2

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

   5. Simplified 28.2

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

   7. Applied add-cube-cbrt 28.2

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

   8. Applied associate-*r* 28.2

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

   9. Simplified 27.1

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

   if -9.944037691838278e-75 < n < -1.1144092498290265e-301

   1. Initial program 34.9

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

   3. Applied *-un-lft-identity 34.9

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

   4. Applied times-frac 32.4

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

   5. Simplified 32.4

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

   7. Applied associate-*l* 31.2

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

   8. Simplified 31.2

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

   10. Applied associate-*r/ 31.2

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

   11. Applied associate-*l/ 30.2

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

   12. Applied associate-*l/ 30.1

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

   if -1.1144092498290265e-301 < n

   1. Initial program 32.9

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

   3. Applied *-un-lft-identity 32.9

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

   4. Applied times-frac 30.3

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

   5. Simplified 30.3

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

   7. Applied associate-*l* 29.9

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

   8. Simplified 29.9

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

   10. Applied add-cube-cbrt 30.0

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

   11. Applied associate-*r* 30.0

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

   12. Simplified 29.2

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

   14. Applied sqrt-prod 22.6

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

4. Final simplification 25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -9.944037691838278 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \sqrt[3]{U - U*} \cdot \left(n \cdot \left(\left(\sqrt[3]{U - U*} \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le -1.1144092498290265 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \frac{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \left(U - U*\right) \cdot \left(\sqrt[3]{n} \cdot \left(\left(\sqrt[3]{n} \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{n} \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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*))))))