Linear.Matrix:det44 from linear-1.19.1.3

Average Error: 26.1 → 26.0

Time: 3.4m

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}\;k \le -4.883143427445851 \cdot 10^{+79}:\\ \;\;\;\;\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right) + \left(\left(\left(k \cdot \left(y \cdot \left(y5 \cdot i - b \cdot y4\right)\right) - \left(y5 \cdot j\right) \cdot \left(i \cdot t\right)\right) + \left(\left(y0 \cdot c - y1 \cdot a\right) \cdot \left(x \cdot y2 - z \cdot y3\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - k \cdot z\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right)\right)\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y \cdot y3\right)\right)\\ \mathbf{elif}\;k \le 1.4318064591012303 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(\left(\left(y0 \cdot c - y1 \cdot a\right) \cdot \left(x \cdot y2 - z \cdot y3\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - k \cdot z\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right)\right) + \left(\left(t \cdot j - y \cdot k\right) \cdot \left(\sqrt[3]{b \cdot y4 - y5 \cdot i} \cdot \sqrt[3]{b \cdot y4 - y5 \cdot i}\right)\right) \cdot \sqrt[3]{b \cdot y4 - y5 \cdot i}\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y \cdot y3\right)\right) + \sqrt[3]{\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)} \cdot \left(\sqrt[3]{\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)} \cdot \sqrt[3]{\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right) + \left(\left(\left(k \cdot \left(y \cdot \left(y5 \cdot i - b \cdot y4\right)\right) - \left(y5 \cdot j\right) \cdot \left(i \cdot t\right)\right) + \left(\left(y0 \cdot c - y1 \cdot a\right) \cdot \left(x \cdot y2 - z \cdot y3\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - k \cdot z\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right)\right)\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y \cdot y3\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 r23330945 = x;
        double r23330946 = y;
        double r23330947 = r23330945 * r23330946;
        double r23330948 = z;
        double r23330949 = t;
        double r23330950 = r23330948 * r23330949;
        double r23330951 = r23330947 - r23330950;
        double r23330952 = a;
        double r23330953 = b;
        double r23330954 = r23330952 * r23330953;
        double r23330955 = c;
        double r23330956 = i;
        double r23330957 = r23330955 * r23330956;
        double r23330958 = r23330954 - r23330957;
        double r23330959 = r23330951 * r23330958;
        double r23330960 = j;
        double r23330961 = r23330945 * r23330960;
        double r23330962 = k;
        double r23330963 = r23330948 * r23330962;
        double r23330964 = r23330961 - r23330963;
        double r23330965 = y0;
        double r23330966 = r23330965 * r23330953;
        double r23330967 = y1;
        double r23330968 = r23330967 * r23330956;
        double r23330969 = r23330966 - r23330968;
        double r23330970 = r23330964 * r23330969;
        double r23330971 = r23330959 - r23330970;
        double r23330972 = y2;
        double r23330973 = r23330945 * r23330972;
        double r23330974 = y3;
        double r23330975 = r23330948 * r23330974;
        double r23330976 = r23330973 - r23330975;
        double r23330977 = r23330965 * r23330955;
        double r23330978 = r23330967 * r23330952;
        double r23330979 = r23330977 - r23330978;
        double r23330980 = r23330976 * r23330979;
        double r23330981 = r23330971 + r23330980;
        double r23330982 = r23330949 * r23330960;
        double r23330983 = r23330946 * r23330962;
        double r23330984 = r23330982 - r23330983;
        double r23330985 = y4;
        double r23330986 = r23330985 * r23330953;
        double r23330987 = y5;
        double r23330988 = r23330987 * r23330956;
        double r23330989 = r23330986 - r23330988;
        double r23330990 = r23330984 * r23330989;
        double r23330991 = r23330981 + r23330990;
        double r23330992 = r23330949 * r23330972;
        double r23330993 = r23330946 * r23330974;
        double r23330994 = r23330992 - r23330993;
        double r23330995 = r23330985 * r23330955;
        double r23330996 = r23330987 * r23330952;
        double r23330997 = r23330995 - r23330996;
        double r23330998 = r23330994 * r23330997;
        double r23330999 = r23330991 - r23330998;
        double r23331000 = r23330962 * r23330972;
        double r23331001 = r23330960 * r23330974;
        double r23331002 = r23331000 - r23331001;
        double r23331003 = r23330985 * r23330967;
        double r23331004 = r23330987 * r23330965;
        double r23331005 = r23331003 - r23331004;
        double r23331006 = r23331002 * r23331005;
        double r23331007 = r23330999 + r23331006;
        return r23331007;
}

↓

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 r23331008 = k;
        double r23331009 = -4.883143427445851e+79;
        bool r23331010 = r23331008 <= r23331009;
        double r23331011 = y2;
        double r23331012 = r23331011 * r23331008;
        double r23331013 = y3;
        double r23331014 = j;
        double r23331015 = r23331013 * r23331014;
        double r23331016 = r23331012 - r23331015;
        double r23331017 = y4;
        double r23331018 = y1;
        double r23331019 = r23331017 * r23331018;
        double r23331020 = y0;
        double r23331021 = y5;
        double r23331022 = r23331020 * r23331021;
        double r23331023 = r23331019 - r23331022;
        double r23331024 = r23331016 * r23331023;
        double r23331025 = y;
        double r23331026 = i;
        double r23331027 = r23331021 * r23331026;
        double r23331028 = b;
        double r23331029 = r23331028 * r23331017;
        double r23331030 = r23331027 - r23331029;
        double r23331031 = r23331025 * r23331030;
        double r23331032 = r23331008 * r23331031;
        double r23331033 = r23331021 * r23331014;
        double r23331034 = t;
        double r23331035 = r23331026 * r23331034;
        double r23331036 = r23331033 * r23331035;
        double r23331037 = r23331032 - r23331036;
        double r23331038 = c;
        double r23331039 = r23331020 * r23331038;
        double r23331040 = a;
        double r23331041 = r23331018 * r23331040;
        double r23331042 = r23331039 - r23331041;
        double r23331043 = x;
        double r23331044 = r23331043 * r23331011;
        double r23331045 = z;
        double r23331046 = r23331045 * r23331013;
        double r23331047 = r23331044 - r23331046;
        double r23331048 = r23331042 * r23331047;
        double r23331049 = r23331043 * r23331025;
        double r23331050 = r23331034 * r23331045;
        double r23331051 = r23331049 - r23331050;
        double r23331052 = r23331040 * r23331028;
        double r23331053 = r23331026 * r23331038;
        double r23331054 = r23331052 - r23331053;
        double r23331055 = r23331051 * r23331054;
        double r23331056 = r23331043 * r23331014;
        double r23331057 = r23331008 * r23331045;
        double r23331058 = r23331056 - r23331057;
        double r23331059 = r23331028 * r23331020;
        double r23331060 = r23331018 * r23331026;
        double r23331061 = r23331059 - r23331060;
        double r23331062 = r23331058 * r23331061;
        double r23331063 = r23331055 - r23331062;
        double r23331064 = r23331048 + r23331063;
        double r23331065 = r23331037 + r23331064;
        double r23331066 = r23331017 * r23331038;
        double r23331067 = r23331021 * r23331040;
        double r23331068 = r23331066 - r23331067;
        double r23331069 = r23331034 * r23331011;
        double r23331070 = r23331025 * r23331013;
        double r23331071 = r23331069 - r23331070;
        double r23331072 = r23331068 * r23331071;
        double r23331073 = r23331065 - r23331072;
        double r23331074 = r23331024 + r23331073;
        double r23331075 = 1.4318064591012303e+56;
        bool r23331076 = r23331008 <= r23331075;
        double r23331077 = r23331034 * r23331014;
        double r23331078 = r23331025 * r23331008;
        double r23331079 = r23331077 - r23331078;
        double r23331080 = r23331029 - r23331027;
        double r23331081 = cbrt(r23331080);
        double r23331082 = r23331081 * r23331081;
        double r23331083 = r23331079 * r23331082;
        double r23331084 = r23331083 * r23331081;
        double r23331085 = r23331064 + r23331084;
        double r23331086 = r23331085 - r23331072;
        double r23331087 = cbrt(r23331024);
        double r23331088 = r23331087 * r23331087;
        double r23331089 = r23331087 * r23331088;
        double r23331090 = r23331086 + r23331089;
        double r23331091 = r23331076 ? r23331090 : r23331074;
        double r23331092 = r23331010 ? r23331074 : r23331091;
        return r23331092;
}

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 2 regimes

2. if k < -4.883143427445851e+79 or 1.4318064591012303e+56 < k

   1. Initial program 28.7

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

   2. Taylor expanded around inf 31.2

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

   3. Simplified 27.6

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

   if -4.883143427445851e+79 < k < 1.4318064591012303e+56

   1. Initial program 25.3

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

   3. Applied add-cube-cbrt 25.4

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

   4. Applied associate-*r* 25.4

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

   6. Applied add-cube-cbrt 25.4

      \[\leadsto \left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(y0 \cdot b - y1 \cdot i\right)\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(\left(t \cdot j - y \cdot k\right) \cdot \left(\sqrt[3]{y4 \cdot b - y5 \cdot i} \cdot \sqrt[3]{y4 \cdot b - y5 \cdot i}\right)\right) \cdot \sqrt[3]{y4 \cdot b - y5 \cdot i}\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \color{blue}{\left(\sqrt[3]{\left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)} \cdot \sqrt[3]{\left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)}\right) \cdot \sqrt[3]{\left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.883143427445851 \cdot 10^{+79}:\\ \;\;\;\;\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right) + \left(\left(\left(k \cdot \left(y \cdot \left(y5 \cdot i - b \cdot y4\right)\right) - \left(y5 \cdot j\right) \cdot \left(i \cdot t\right)\right) + \left(\left(y0 \cdot c - y1 \cdot a\right) \cdot \left(x \cdot y2 - z \cdot y3\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - k \cdot z\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right)\right)\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y \cdot y3\right)\right)\\ \mathbf{elif}\;k \le 1.4318064591012303 \cdot 10^{+56}:\\ \;\;\;\;\left(\left(\left(\left(y0 \cdot c - y1 \cdot a\right) \cdot \left(x \cdot y2 - z \cdot y3\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - k \cdot z\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right)\right) + \left(\left(t \cdot j - y \cdot k\right) \cdot \left(\sqrt[3]{b \cdot y4 - y5 \cdot i} \cdot \sqrt[3]{b \cdot y4 - y5 \cdot i}\right)\right) \cdot \sqrt[3]{b \cdot y4 - y5 \cdot i}\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y \cdot y3\right)\right) + \sqrt[3]{\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)} \cdot \left(\sqrt[3]{\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)} \cdot \sqrt[3]{\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right) + \left(\left(\left(k \cdot \left(y \cdot \left(y5 \cdot i - b \cdot y4\right)\right) - \left(y5 \cdot j\right) \cdot \left(i \cdot t\right)\right) + \left(\left(y0 \cdot c - y1 \cdot a\right) \cdot \left(x \cdot y2 - z \cdot y3\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - k \cdot z\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right)\right)\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y \cdot y3\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(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"
  (+ (- (+ (+ (- (* (- (* 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)))))