Toniolo and Linder, Equation (13)

Average Error: 33.1 → 27.5

Time: 51.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -5.889660218949 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}\right) - \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}\right) \cdot \sqrt[3]{U - U*}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}, \left(\frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}} \cdot 0 - \frac{\frac{U - U*}{\frac{Om}{\ell}}}{\frac{\frac{Om}{\ell}}{n}}\right) + t\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2990972 = 2.0;
        double r2990973 = n;
        double r2990974 = r2990972 * r2990973;
        double r2990975 = U;
        double r2990976 = r2990974 * r2990975;
        double r2990977 = t;
        double r2990978 = l;
        double r2990979 = r2990978 * r2990978;
        double r2990980 = Om;
        double r2990981 = r2990979 / r2990980;
        double r2990982 = r2990972 * r2990981;
        double r2990983 = r2990977 - r2990982;
        double r2990984 = r2990978 / r2990980;
        double r2990985 = pow(r2990984, r2990972);
        double r2990986 = r2990973 * r2990985;
        double r2990987 = U_;
        double r2990988 = r2990975 - r2990987;
        double r2990989 = r2990986 * r2990988;
        double r2990990 = r2990983 - r2990989;
        double r2990991 = r2990976 * r2990990;
        double r2990992 = sqrt(r2990991);
        return r2990992;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2990993 = U;
        double r2990994 = -5.889660218949e-311;
        bool r2990995 = r2990993 <= r2990994;
        double r2990996 = 2.0;
        double r2990997 = n;
        double r2990998 = r2990996 * r2990997;
        double r2990999 = r2990998 * r2990993;
        double r2991000 = t;
        double r2991001 = l;
        double r2991002 = Om;
        double r2991003 = cbrt(r2991002);
        double r2991004 = r2991001 / r2991003;
        double r2991005 = r2991004 * r2991004;
        double r2991006 = r2991005 / r2991003;
        double r2991007 = r2990996 * r2991006;
        double r2991008 = r2991000 - r2991007;
        double r2991009 = U_;
        double r2991010 = r2990993 - r2991009;
        double r2991011 = cbrt(r2991010);
        double r2991012 = r2991011 * r2991011;
        double r2991013 = r2991002 / r2991001;
        double r2991014 = r2991013 * r2991013;
        double r2991015 = r2990997 / r2991014;
        double r2991016 = r2991012 * r2991015;
        double r2991017 = r2991016 * r2991011;
        double r2991018 = r2991008 - r2991017;
        double r2991019 = r2990999 * r2991018;
        double r2991020 = sqrt(r2991019);
        double r2991021 = sqrt(r2990993);
        double r2991022 = -2.0;
        double r2991023 = 0.0;
        double r2991024 = r2991006 * r2991023;
        double r2991025 = r2991010 / r2991013;
        double r2991026 = r2991013 / r2990997;
        double r2991027 = r2991025 / r2991026;
        double r2991028 = r2991024 - r2991027;
        double r2991029 = r2991028 + r2991000;
        double r2991030 = fma(r2991022, r2991006, r2991029);
        double r2991031 = r2990998 * r2991030;
        double r2991032 = sqrt(r2991031);
        double r2991033 = r2991021 * r2991032;
        double r2991034 = r2990995 ? r2991020 : r2991033;
        return r2991034;
}

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 < -5.889660218949e-311

   1. Initial program 33.4

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

   3. Applied add-cube-cbrt 33.5

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

   4. Applied associate-/r* 33.5

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

   5. Simplified 31.5

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

   6. Taylor expanded around 0 37.2

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

   7. Simplified 31.5

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

   9. Applied add-cube-cbrt 31.5

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

   10. Applied associate-*r* 31.5

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

   if -5.889660218949e-311 < U

   1. Initial program 32.7

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

   3. Applied add-cube-cbrt 32.8

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

   4. Applied associate-/r* 32.8

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

   5. Simplified 30.6

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

   6. Taylor expanded around 0 36.1

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

   7. Simplified 30.5

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

   9. Applied add-sqr-sqrt 46.2

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

   10. Applied prod-diff 46.2

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

   11. Applied associate--l+ 46.2

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

   12. Applied distribute-lft-in 46.2

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

   13. Simplified 31.3

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

   14. Simplified 30.4

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

   16. Applied distribute-lft-out 30.4

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

   17. Applied sqrt-prod 23.5

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

   18. Simplified 23.5

       \[\leadsto \sqrt{U} \cdot \color{blue}{\sqrt{\mathsf{fma}\left(-2, \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}, t + \left(\frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}} \cdot 0 - \frac{\frac{U - U*}{\frac{Om}{\ell}}}{\frac{\frac{Om}{\ell}}{n}}\right)\right) \cdot \left(n \cdot 2\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 27.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -5.889660218949 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}\right) - \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}\right) \cdot \sqrt[3]{U - U*}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}, \left(\frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}} \cdot 0 - \frac{\frac{U - U*}{\frac{Om}{\ell}}}{\frac{\frac{Om}{\ell}}{n}}\right) + t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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*))))))