Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 29.2 → 29.2

Time: 26.8s

Precision: 64

Math

\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 1.924953770095044578091510096483797474331 \cdot 10^{240}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625} \cdot \sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625}\right) \cdot \left(\sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625} \cdot y\right) + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r72842 = x;
        double r72843 = y;
        double r72844 = r72842 * r72843;
        double r72845 = z;
        double r72846 = r72844 + r72845;
        double r72847 = r72846 * r72843;
        double r72848 = 27464.7644705;
        double r72849 = r72847 + r72848;
        double r72850 = r72849 * r72843;
        double r72851 = 230661.510616;
        double r72852 = r72850 + r72851;
        double r72853 = r72852 * r72843;
        double r72854 = t;
        double r72855 = r72853 + r72854;
        double r72856 = a;
        double r72857 = r72843 + r72856;
        double r72858 = r72857 * r72843;
        double r72859 = b;
        double r72860 = r72858 + r72859;
        double r72861 = r72860 * r72843;
        double r72862 = c;
        double r72863 = r72861 + r72862;
        double r72864 = r72863 * r72843;
        double r72865 = i;
        double r72866 = r72864 + r72865;
        double r72867 = r72855 / r72866;
        return r72867;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r72868 = x;
        double r72869 = y;
        double r72870 = r72868 * r72869;
        double r72871 = z;
        double r72872 = r72870 + r72871;
        double r72873 = r72872 * r72869;
        double r72874 = 27464.7644705;
        double r72875 = r72873 + r72874;
        double r72876 = r72875 * r72869;
        double r72877 = 230661.510616;
        double r72878 = r72876 + r72877;
        double r72879 = r72878 * r72869;
        double r72880 = t;
        double r72881 = r72879 + r72880;
        double r72882 = a;
        double r72883 = r72869 + r72882;
        double r72884 = r72883 * r72869;
        double r72885 = b;
        double r72886 = r72884 + r72885;
        double r72887 = r72886 * r72869;
        double r72888 = c;
        double r72889 = r72887 + r72888;
        double r72890 = r72889 * r72869;
        double r72891 = i;
        double r72892 = r72890 + r72891;
        double r72893 = r72881 / r72892;
        double r72894 = 1.9249537700950446e+240;
        bool r72895 = r72893 <= r72894;
        double r72896 = cbrt(r72875);
        double r72897 = r72896 * r72896;
        double r72898 = r72896 * r72869;
        double r72899 = r72897 * r72898;
        double r72900 = r72899 + r72877;
        double r72901 = r72900 * r72869;
        double r72902 = r72901 + r72880;
        double r72903 = r72902 / r72892;
        double r72904 = 0.0;
        double r72905 = r72895 ? r72903 : r72904;
        return r72905;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)) < 1.9249537700950446e+240

   1. Initial program 5.7

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 5.8

      \[\leadsto \frac{\left(\color{blue}{\left(\left(\sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625} \cdot \sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625}\right) \cdot \sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625}\right)} \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

   4. Applied associate-*l* 5.8

      \[\leadsto \frac{\left(\color{blue}{\left(\sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625} \cdot \sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625}\right) \cdot \left(\sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625} \cdot y\right)} + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

   if 1.9249537700950446e+240 < (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))

   1. Initial program 62.1

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

   2. Taylor expanded around 0 61.8

      \[\leadsto \color{blue}{0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 29.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \le 1.924953770095044578091510096483797474331 \cdot 10^{240}:\\ \;\;\;\;\frac{\left(\left(\sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625} \cdot \sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625}\right) \cdot \left(\sqrt[3]{\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625} \cdot y\right) + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))