Toniolo and Linder, Equation (13)

Average Error: 33.6 → 28.6

Time: 51.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.092557959512387 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(0 \cdot \frac{\ell}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}, n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.1462142433560081 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\frac{\frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}, n \cdot \left(\frac{U* - U}{\sqrt[3]{Om}} \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1528788 = 2.0;
        double r1528789 = n;
        double r1528790 = r1528788 * r1528789;
        double r1528791 = U;
        double r1528792 = r1528790 * r1528791;
        double r1528793 = t;
        double r1528794 = l;
        double r1528795 = r1528794 * r1528794;
        double r1528796 = Om;
        double r1528797 = r1528795 / r1528796;
        double r1528798 = r1528788 * r1528797;
        double r1528799 = r1528793 - r1528798;
        double r1528800 = r1528794 / r1528796;
        double r1528801 = pow(r1528800, r1528788);
        double r1528802 = r1528789 * r1528801;
        double r1528803 = U_;
        double r1528804 = r1528791 - r1528803;
        double r1528805 = r1528802 * r1528804;
        double r1528806 = r1528799 - r1528805;
        double r1528807 = r1528792 * r1528806;
        double r1528808 = sqrt(r1528807);
        return r1528808;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1528809 = 2.0;
        double r1528810 = n;
        double r1528811 = r1528809 * r1528810;
        double r1528812 = U;
        double r1528813 = r1528811 * r1528812;
        double r1528814 = t;
        double r1528815 = l;
        double r1528816 = r1528815 * r1528815;
        double r1528817 = Om;
        double r1528818 = r1528816 / r1528817;
        double r1528819 = r1528818 * r1528809;
        double r1528820 = r1528814 - r1528819;
        double r1528821 = r1528815 / r1528817;
        double r1528822 = pow(r1528821, r1528809);
        double r1528823 = r1528810 * r1528822;
        double r1528824 = U_;
        double r1528825 = r1528812 - r1528824;
        double r1528826 = r1528823 * r1528825;
        double r1528827 = r1528820 - r1528826;
        double r1528828 = r1528813 * r1528827;
        double r1528829 = 1.092557959512387e-276;
        bool r1528830 = r1528828 <= r1528829;
        double r1528831 = 0.0;
        double r1528832 = cbrt(r1528817);
        double r1528833 = r1528832 * r1528832;
        double r1528834 = r1528815 / r1528833;
        double r1528835 = r1528831 * r1528834;
        double r1528836 = r1528821 * r1528815;
        double r1528837 = -2.0;
        double r1528838 = fma(r1528836, r1528837, r1528814);
        double r1528839 = fma(r1528835, r1528810, r1528838);
        double r1528840 = r1528812 * r1528839;
        double r1528841 = r1528840 * r1528811;
        double r1528842 = sqrt(r1528841);
        double r1528843 = 1.1462142433560081e+262;
        bool r1528844 = r1528828 <= r1528843;
        double r1528845 = sqrt(r1528828);
        double r1528846 = r1528815 / r1528832;
        double r1528847 = r1528846 / r1528832;
        double r1528848 = r1528824 - r1528812;
        double r1528849 = r1528848 / r1528832;
        double r1528850 = r1528849 * r1528821;
        double r1528851 = r1528810 * r1528850;
        double r1528852 = fma(r1528837, r1528836, r1528814);
        double r1528853 = fma(r1528847, r1528851, r1528852);
        double r1528854 = r1528812 * r1528853;
        double r1528855 = sqrt(r1528854);
        double r1528856 = sqrt(r1528811);
        double r1528857 = r1528855 * r1528856;
        double r1528858 = r1528844 ? r1528845 : r1528857;
        double r1528859 = r1528830 ? r1528842 : r1528858;
        return r1528859;
}

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 (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 1.092557959512387e-276

   1. Initial program 52.6

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

   2. Simplified 40.3

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

   4. Applied *-un-lft-identity 40.3

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

   5. Applied add-cube-cbrt 40.3

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

   6. Applied times-frac 40.3

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

   7. Applied times-frac 40.2

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

   8. Applied associate-*l* 38.4

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

   10. Applied *-un-lft-identity 38.4

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

   11. Applied times-frac 37.1

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

   12. Simplified 37.1

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

   13. Taylor expanded around 0 38.6

       \[\leadsto \sqrt{\left(U \cdot \mathsf{fma}\left(\frac{\frac{\ell}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}}{1} \cdot \color{blue}{0}, n, \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)\right) \cdot \left(2 \cdot n\right)}\]

   if 1.092557959512387e-276 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 1.1462142433560081e+262

   1. Initial program 1.3

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

   if 1.1462142433560081e+262 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

   1. Initial program 57.2

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

   2. Simplified 56.7

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

   4. Applied *-un-lft-identity 56.7

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

   5. Applied add-cube-cbrt 56.7

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

   6. Applied times-frac 56.6

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

   7. Applied times-frac 56.5

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

   8. Applied associate-*l* 55.9

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

   10. Applied sqrt-prod 54.8

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

   11. Simplified 51.5

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

4. Final simplification 28.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.092557959512387 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left(0 \cdot \frac{\ell}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}, n, \mathsf{fma}\left(\frac{\ell}{Om} \cdot \ell, -2, t\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.1462142433560081 \cdot 10^{+262}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \mathsf{fma}\left(\frac{\frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}, n \cdot \left(\frac{U* - U}{\sqrt[3]{Om}} \cdot \frac{\ell}{Om}\right), \mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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*))))))