Toniolo and Linder, Equation (13)

Average Error: 34.9 → 30.0

Time: 39.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -5.85228649109029279 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;n \le 5.7542881375349792 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;n \le 1.1504183261425661 \cdot 10^{-17} \lor \neg \left(n \le 5.84977345983970428 \cdot 10^{38}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot n}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r171951 = 2.0;
        double r171952 = n;
        double r171953 = r171951 * r171952;
        double r171954 = U;
        double r171955 = r171953 * r171954;
        double r171956 = t;
        double r171957 = l;
        double r171958 = r171957 * r171957;
        double r171959 = Om;
        double r171960 = r171958 / r171959;
        double r171961 = r171951 * r171960;
        double r171962 = r171956 - r171961;
        double r171963 = r171957 / r171959;
        double r171964 = pow(r171963, r171951);
        double r171965 = r171952 * r171964;
        double r171966 = U_;
        double r171967 = r171954 - r171966;
        double r171968 = r171965 * r171967;
        double r171969 = r171962 - r171968;
        double r171970 = r171955 * r171969;
        double r171971 = sqrt(r171970);
        return r171971;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r171972 = n;
        double r171973 = -5.852286491090293e-126;
        bool r171974 = r171972 <= r171973;
        double r171975 = 2.0;
        double r171976 = r171975 * r171972;
        double r171977 = U;
        double r171978 = r171976 * r171977;
        double r171979 = t;
        double r171980 = l;
        double r171981 = Om;
        double r171982 = r171981 / r171980;
        double r171983 = r171980 / r171982;
        double r171984 = r171975 * r171983;
        double r171985 = r171979 - r171984;
        double r171986 = r171980 / r171981;
        double r171987 = 2.0;
        double r171988 = r171975 / r171987;
        double r171989 = pow(r171986, r171988);
        double r171990 = r171972 * r171989;
        double r171991 = U_;
        double r171992 = r171977 - r171991;
        double r171993 = r171989 * r171992;
        double r171994 = r171990 * r171993;
        double r171995 = r171985 - r171994;
        double r171996 = r171978 * r171995;
        double r171997 = sqrt(r171996);
        double r171998 = 5.754288137534979e-186;
        bool r171999 = r171972 <= r171998;
        double r172000 = pow(r171986, r171975);
        double r172001 = r171972 * r172000;
        double r172002 = r172001 * r171992;
        double r172003 = r171985 - r172002;
        double r172004 = r171976 * r172003;
        double r172005 = r171977 * r172004;
        double r172006 = sqrt(r172005);
        double r172007 = 1.1504183261425661e-17;
        bool r172008 = r171972 <= r172007;
        double r172009 = 5.849773459839704e+38;
        bool r172010 = r171972 <= r172009;
        double r172011 = !r172010;
        bool r172012 = r172008 || r172011;
        double r172013 = sqrt(r171978);
        double r172014 = r171987 * r171988;
        double r172015 = pow(r171986, r172014);
        double r172016 = r171992 * r172015;
        double r172017 = r172016 * r171972;
        double r172018 = r171985 - r172017;
        double r172019 = sqrt(r172018);
        double r172020 = r172013 * r172019;
        double r172021 = r172012 ? r171997 : r172020;
        double r172022 = r171999 ? r172006 : r172021;
        double r172023 = r171974 ? r171997 : r172022;
        return r172023;
}

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 < -5.852286491090293e-126 or 5.754288137534979e-186 < n < 1.1504183261425661e-17 or 5.849773459839704e+38 < n

   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. Using strategy rm

   3. Applied associate-/l* 30.4

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

   5. Applied sqr-pow 30.4

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

   6. Applied associate-*r* 29.6

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

   8. Applied associate-*l* 28.7

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

   if -5.852286491090293e-126 < n < 5.754288137534979e-186

   1. Initial program 39.0

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

   3. Applied associate-/l* 36.2

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

   5. Applied pow1 36.2

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

   6. Applied pow1 36.2

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

   7. Applied pow1 36.2

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

   8. Applied pow1 36.2

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

   9. Applied pow-prod-down 36.2

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

   10. Applied pow-prod-down 36.2

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

   11. Applied pow-prod-down 36.2

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

   12. Simplified 31.3

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

   if 1.1504183261425661e-17 < n < 5.849773459839704e+38

   1. Initial program 32.5

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

   3. Applied associate-/l* 28.0

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

   5. Applied sqr-pow 28.0

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

   6. Applied associate-*r* 27.9

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

   8. Applied associate-*l* 26.6

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

   10. Applied sqrt-prod 37.1

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

   11. Simplified 38.1

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

4. Final simplification 30.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5.85228649109029279 \cdot 10^{-126}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;n \le 5.7542881375349792 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;n \le 1.1504183261425661 \cdot 10^{-17} \lor \neg \left(n \le 5.84977345983970428 \cdot 10^{38}\right):\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))