Toniolo and Linder, Equation (13)

Average Error: 34.7 → 30.2

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.214046473047511182262273069897395979677 \cdot 10^{201} \lor t \le -6.212072691974284363372137380706930578219 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t \le 5.824775899044275048393983934519217716302 \cdot 10^{-226} \lor t \le 436193475525558403072:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.976343330438959153714939343152765464593 \cdot 10^{262}:\\ \;\;\;\;\sqrt{{\left(\left(\left(2 \cdot n\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot \frac{2}{2}}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot \frac{2}{2}}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot U\right)}^{1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r233994 = 2.0;
        double r233995 = n;
        double r233996 = r233994 * r233995;
        double r233997 = U;
        double r233998 = r233996 * r233997;
        double r233999 = t;
        double r234000 = l;
        double r234001 = r234000 * r234000;
        double r234002 = Om;
        double r234003 = r234001 / r234002;
        double r234004 = r233994 * r234003;
        double r234005 = r233999 - r234004;
        double r234006 = r234000 / r234002;
        double r234007 = pow(r234006, r233994);
        double r234008 = r233995 * r234007;
        double r234009 = U_;
        double r234010 = r233997 - r234009;
        double r234011 = r234008 * r234010;
        double r234012 = r234005 - r234011;
        double r234013 = r233998 * r234012;
        double r234014 = sqrt(r234013);
        return r234014;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r234015 = t;
        double r234016 = -2.2140464730475112e+201;
        bool r234017 = r234015 <= r234016;
        double r234018 = -6.212072691974284e-200;
        bool r234019 = r234015 <= r234018;
        bool r234020 = r234017 || r234019;
        double r234021 = 2.0;
        double r234022 = n;
        double r234023 = r234021 * r234022;
        double r234024 = U;
        double r234025 = r234023 * r234024;
        double r234026 = l;
        double r234027 = Om;
        double r234028 = r234027 / r234026;
        double r234029 = r234026 / r234028;
        double r234030 = r234021 * r234029;
        double r234031 = r234015 - r234030;
        double r234032 = r234026 / r234027;
        double r234033 = 2.0;
        double r234034 = r234021 / r234033;
        double r234035 = pow(r234032, r234034);
        double r234036 = r234022 * r234035;
        double r234037 = U_;
        double r234038 = r234024 - r234037;
        double r234039 = r234035 * r234038;
        double r234040 = r234036 * r234039;
        double r234041 = r234031 - r234040;
        double r234042 = r234025 * r234041;
        double r234043 = sqrt(r234042);
        double r234044 = 5.824775899044275e-226;
        bool r234045 = r234015 <= r234044;
        double r234046 = 4.361934755255584e+20;
        bool r234047 = r234015 <= r234046;
        bool r234048 = r234045 || r234047;
        double r234049 = 1.976343330438959e+262;
        bool r234050 = r234015 <= r234049;
        double r234051 = r234033 * r234034;
        double r234052 = r234051 / r234033;
        double r234053 = pow(r234032, r234052);
        double r234054 = r234022 * r234053;
        double r234055 = r234054 * r234053;
        double r234056 = -r234055;
        double r234057 = r234056 * r234038;
        double r234058 = r234030 - r234057;
        double r234059 = r234015 - r234058;
        double r234060 = r234023 * r234059;
        double r234061 = r234060 * r234024;
        double r234062 = 1.0;
        double r234063 = pow(r234061, r234062);
        double r234064 = sqrt(r234063);
        double r234065 = sqrt(r234025);
        double r234066 = r234036 * r234035;
        double r234067 = r234066 * r234038;
        double r234068 = r234031 - r234067;
        double r234069 = sqrt(r234068);
        double r234070 = r234065 * r234069;
        double r234071 = r234050 ? r234064 : r234070;
        double r234072 = r234048 ? r234043 : r234071;
        double r234073 = r234020 ? r234043 : r234072;
        return r234073;
}

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 t < -2.2140464730475112e+201 or -6.212072691974284e-200 < t < 5.824775899044275e-226 or 4.361934755255584e+20 < t < 1.976343330438959e+262

   1. Initial program 36.1

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

   3. Applied associate-/l* 33.4

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

   5. Applied sqr-pow 33.4

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

   6. Applied associate-*r* 32.8

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

   8. Applied pow1 32.8

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

   9. Applied pow1 32.8

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

   10. Applied pow1 32.8

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

   11. Applied pow1 32.8

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

   12. Applied pow-prod-down 32.8

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

   13. Applied pow-prod-down 32.8

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

   14. Applied pow-prod-down 32.8

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

   15. Simplified 33.1

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

   17. Applied sqr-pow 33.1

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

   18. Applied associate-*r* 32.0

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

   if -2.2140464730475112e+201 < t < -6.212072691974284e-200 or 5.824775899044275e-226 < t < 4.361934755255584e+20

   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* 30.5

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

   5. Applied sqr-pow 30.5

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

   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* 29.3

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

   if 1.976343330438959e+262 < t

   1. Initial program 42.1

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

   3. Applied associate-/l* 40.3

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

   5. Applied sqr-pow 40.3

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

   6. Applied associate-*r* 40.1

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

   8. Applied sqrt-prod 22.0

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

4. Final simplification 30.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.214046473047511182262273069897395979677 \cdot 10^{201} \lor t \le -6.212072691974284363372137380706930578219 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t \le 5.824775899044275048393983934519217716302 \cdot 10^{-226} \lor t \le 436193475525558403072:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t \le 1.976343330438959153714939343152765464593 \cdot 10^{262}:\\ \;\;\;\;\sqrt{{\left(\left(\left(2 \cdot n\right) \cdot \left(t - \left(2 \cdot \frac{\ell}{\frac{Om}{\ell}} - \left(-\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot \frac{2}{2}}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2 \cdot \frac{2}{2}}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot U\right)}^{1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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*))))))