Linear.Matrix:det44 from linear-1.19.1.3

Average Error: 27.0 → 27.3

Time: 39.9s

Precision: 64

Math

\[\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;y0 \le -5.491209746310479243911953043107916795698 \cdot 10^{-87}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(y0 \cdot \left(y3 \cdot \left(j \cdot y5\right)\right) - \left(y0 \cdot \left(y2 \cdot \left(k \cdot y5\right)\right) + y1 \cdot \left(y3 \cdot \left(j \cdot y4\right)\right)\right)\right)\\ \mathbf{elif}\;y0 \le -5.930087093633848733827530546660597620228 \cdot 10^{-132}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(a \cdot \left(y3 \cdot \left(y \cdot y5\right)\right) - \left(y \cdot \left(y3 \cdot \left(y4 \cdot c\right)\right) + y5 \cdot \left(a \cdot \left(y2 \cdot t\right)\right)\right)\right)\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\\ \mathbf{elif}\;y0 \le 7213927107.58885097503662109375:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(\sqrt[3]{k \cdot y2 - j \cdot y3} \cdot \sqrt[3]{k \cdot y2 - j \cdot y3}\right) \cdot \left(\sqrt[3]{k \cdot y2 - j \cdot y3} \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(y0 \cdot \left(y3 \cdot \left(j \cdot y5\right)\right) - \left(y0 \cdot \left(y2 \cdot \left(k \cdot y5\right)\right) + y1 \cdot \left(y3 \cdot \left(j \cdot y4\right)\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k, double y0, double y1, double y2, double y3, double y4, double y5) {
        double r149816 = x;
        double r149817 = y;
        double r149818 = r149816 * r149817;
        double r149819 = z;
        double r149820 = t;
        double r149821 = r149819 * r149820;
        double r149822 = r149818 - r149821;
        double r149823 = a;
        double r149824 = b;
        double r149825 = r149823 * r149824;
        double r149826 = c;
        double r149827 = i;
        double r149828 = r149826 * r149827;
        double r149829 = r149825 - r149828;
        double r149830 = r149822 * r149829;
        double r149831 = j;
        double r149832 = r149816 * r149831;
        double r149833 = k;
        double r149834 = r149819 * r149833;
        double r149835 = r149832 - r149834;
        double r149836 = y0;
        double r149837 = r149836 * r149824;
        double r149838 = y1;
        double r149839 = r149838 * r149827;
        double r149840 = r149837 - r149839;
        double r149841 = r149835 * r149840;
        double r149842 = r149830 - r149841;
        double r149843 = y2;
        double r149844 = r149816 * r149843;
        double r149845 = y3;
        double r149846 = r149819 * r149845;
        double r149847 = r149844 - r149846;
        double r149848 = r149836 * r149826;
        double r149849 = r149838 * r149823;
        double r149850 = r149848 - r149849;
        double r149851 = r149847 * r149850;
        double r149852 = r149842 + r149851;
        double r149853 = r149820 * r149831;
        double r149854 = r149817 * r149833;
        double r149855 = r149853 - r149854;
        double r149856 = y4;
        double r149857 = r149856 * r149824;
        double r149858 = y5;
        double r149859 = r149858 * r149827;
        double r149860 = r149857 - r149859;
        double r149861 = r149855 * r149860;
        double r149862 = r149852 + r149861;
        double r149863 = r149820 * r149843;
        double r149864 = r149817 * r149845;
        double r149865 = r149863 - r149864;
        double r149866 = r149856 * r149826;
        double r149867 = r149858 * r149823;
        double r149868 = r149866 - r149867;
        double r149869 = r149865 * r149868;
        double r149870 = r149862 - r149869;
        double r149871 = r149833 * r149843;
        double r149872 = r149831 * r149845;
        double r149873 = r149871 - r149872;
        double r149874 = r149856 * r149838;
        double r149875 = r149858 * r149836;
        double r149876 = r149874 - r149875;
        double r149877 = r149873 * r149876;
        double r149878 = r149870 + r149877;
        return r149878;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k, double y0, double y1, double y2, double y3, double y4, double y5) {
        double r149879 = y0;
        double r149880 = -5.491209746310479e-87;
        bool r149881 = r149879 <= r149880;
        double r149882 = x;
        double r149883 = y;
        double r149884 = r149882 * r149883;
        double r149885 = z;
        double r149886 = t;
        double r149887 = r149885 * r149886;
        double r149888 = r149884 - r149887;
        double r149889 = a;
        double r149890 = b;
        double r149891 = r149889 * r149890;
        double r149892 = c;
        double r149893 = i;
        double r149894 = r149892 * r149893;
        double r149895 = r149891 - r149894;
        double r149896 = r149888 * r149895;
        double r149897 = j;
        double r149898 = r149882 * r149897;
        double r149899 = k;
        double r149900 = r149885 * r149899;
        double r149901 = r149898 - r149900;
        double r149902 = r149879 * r149890;
        double r149903 = y1;
        double r149904 = r149903 * r149893;
        double r149905 = r149902 - r149904;
        double r149906 = r149901 * r149905;
        double r149907 = r149896 - r149906;
        double r149908 = y2;
        double r149909 = r149882 * r149908;
        double r149910 = y3;
        double r149911 = r149885 * r149910;
        double r149912 = r149909 - r149911;
        double r149913 = r149879 * r149892;
        double r149914 = r149903 * r149889;
        double r149915 = r149913 - r149914;
        double r149916 = r149912 * r149915;
        double r149917 = r149907 + r149916;
        double r149918 = r149886 * r149897;
        double r149919 = r149883 * r149899;
        double r149920 = r149918 - r149919;
        double r149921 = y4;
        double r149922 = r149921 * r149890;
        double r149923 = y5;
        double r149924 = r149923 * r149893;
        double r149925 = r149922 - r149924;
        double r149926 = r149920 * r149925;
        double r149927 = r149917 + r149926;
        double r149928 = r149886 * r149908;
        double r149929 = r149883 * r149910;
        double r149930 = r149928 - r149929;
        double r149931 = r149921 * r149892;
        double r149932 = r149923 * r149889;
        double r149933 = r149931 - r149932;
        double r149934 = r149930 * r149933;
        double r149935 = r149927 - r149934;
        double r149936 = r149897 * r149923;
        double r149937 = r149910 * r149936;
        double r149938 = r149879 * r149937;
        double r149939 = r149899 * r149923;
        double r149940 = r149908 * r149939;
        double r149941 = r149879 * r149940;
        double r149942 = r149897 * r149921;
        double r149943 = r149910 * r149942;
        double r149944 = r149903 * r149943;
        double r149945 = r149941 + r149944;
        double r149946 = r149938 - r149945;
        double r149947 = r149935 + r149946;
        double r149948 = -5.930087093633849e-132;
        bool r149949 = r149879 <= r149948;
        double r149950 = r149883 * r149923;
        double r149951 = r149910 * r149950;
        double r149952 = r149889 * r149951;
        double r149953 = r149910 * r149931;
        double r149954 = r149883 * r149953;
        double r149955 = r149908 * r149886;
        double r149956 = r149889 * r149955;
        double r149957 = r149923 * r149956;
        double r149958 = r149954 + r149957;
        double r149959 = r149952 - r149958;
        double r149960 = r149927 - r149959;
        double r149961 = r149899 * r149908;
        double r149962 = r149897 * r149910;
        double r149963 = r149961 - r149962;
        double r149964 = r149921 * r149903;
        double r149965 = r149923 * r149879;
        double r149966 = r149964 - r149965;
        double r149967 = r149963 * r149966;
        double r149968 = r149960 + r149967;
        double r149969 = 7213927107.588851;
        bool r149970 = r149879 <= r149969;
        double r149971 = cbrt(r149963);
        double r149972 = r149971 * r149971;
        double r149973 = r149971 * r149966;
        double r149974 = r149972 * r149973;
        double r149975 = r149935 + r149974;
        double r149976 = r149970 ? r149975 : r149947;
        double r149977 = r149949 ? r149968 : r149976;
        double r149978 = r149881 ? r149947 : r149977;
        return r149978;
}

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

Bits error versus j

Bits error versus k

Bits error versus y0

Bits error versus y1

Bits error versus y2

Bits error versus y3

Bits error versus y4

Bits error versus y5

Derivation

1. Split input into 3 regimes

2. if y0 < -5.491209746310479e-87 or 7213927107.588851 < y0

   1. Initial program 27.5

      \[\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\]

   2. Taylor expanded around inf 27.5

      \[\leadsto \left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \color{blue}{\left(y0 \cdot \left(y3 \cdot \left(j \cdot y5\right)\right) - \left(y0 \cdot \left(y2 \cdot \left(k \cdot y5\right)\right) + y1 \cdot \left(y3 \cdot \left(j \cdot y4\right)\right)\right)\right)}\]

   if -5.491209746310479e-87 < y0 < -5.930087093633849e-132

   1. Initial program 27.2

      \[\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\]

   2. Taylor expanded around inf 30.1

      \[\leadsto \left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \color{blue}{\left(a \cdot \left(y3 \cdot \left(y \cdot y5\right)\right) - \left(y \cdot \left(y3 \cdot \left(y4 \cdot c\right)\right) + y5 \cdot \left(a \cdot \left(y2 \cdot t\right)\right)\right)\right)}\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\]

   if -5.930087093633849e-132 < y0 < 7213927107.588851

   1. Initial program 26.7

      \[\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 26.8

      \[\leadsto \left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \color{blue}{\left(\left(\sqrt[3]{k \cdot y2 - j \cdot y3} \cdot \sqrt[3]{k \cdot y2 - j \cdot y3}\right) \cdot \sqrt[3]{k \cdot y2 - j \cdot y3}\right)} \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\]

   4. Applied associate-*l* 26.8

      \[\leadsto \left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \color{blue}{\left(\sqrt[3]{k \cdot y2 - j \cdot y3} \cdot \sqrt[3]{k \cdot y2 - j \cdot y3}\right) \cdot \left(\sqrt[3]{k \cdot y2 - j \cdot y3} \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y0 \le -5.491209746310479243911953043107916795698 \cdot 10^{-87}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(y0 \cdot \left(y3 \cdot \left(j \cdot y5\right)\right) - \left(y0 \cdot \left(y2 \cdot \left(k \cdot y5\right)\right) + y1 \cdot \left(y3 \cdot \left(j \cdot y4\right)\right)\right)\right)\\ \mathbf{elif}\;y0 \le -5.930087093633848733827530546660597620228 \cdot 10^{-132}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(a \cdot \left(y3 \cdot \left(y \cdot y5\right)\right) - \left(y \cdot \left(y3 \cdot \left(y4 \cdot c\right)\right) + y5 \cdot \left(a \cdot \left(y2 \cdot t\right)\right)\right)\right)\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\\ \mathbf{elif}\;y0 \le 7213927107.58885097503662109375:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(\sqrt[3]{k \cdot y2 - j \cdot y3} \cdot \sqrt[3]{k \cdot y2 - j \cdot y3}\right) \cdot \left(\sqrt[3]{k \cdot y2 - j \cdot y3} \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(y0 \cdot \left(y3 \cdot \left(j \cdot y5\right)\right) - \left(y0 \cdot \left(y2 \cdot \left(k \cdot y5\right)\right) + y1 \cdot \left(y3 \cdot \left(j \cdot y4\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (x y z t a b c i j k y0 y1 y2 y3 y4 y5)
  :name "Linear.Matrix:det44 from linear-1.19.1.3"
  :precision binary64
  (+ (- (+ (+ (- (* (- (* x y) (* z t)) (- (* a b) (* c i))) (* (- (* x j) (* z k)) (- (* y0 b) (* y1 i)))) (* (- (* x y2) (* z y3)) (- (* y0 c) (* y1 a)))) (* (- (* t j) (* y k)) (- (* y4 b) (* y5 i)))) (* (- (* t y2) (* y y3)) (- (* y4 c) (* y5 a)))) (* (- (* k y2) (* j y3)) (- (* y4 y1) (* y5 y0)))))