Linear.Matrix:det44 from linear-1.19.1.3

Average Error: 25.7 → 26.7

Time: 3.3m

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}\;y4 \le 4.5136545484096645 \cdot 10^{-287}:\\ \;\;\;\;\left(y2 \cdot k - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right) + \left(\left(\left(\left(\left(a \cdot \left(-y1\right)\right) \cdot \left(y2 \cdot x - y3 \cdot z\right) + \left(y0 \cdot c\right) \cdot \left(y2 \cdot x - y3 \cdot z\right)\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(y0 \cdot b - i \cdot y1\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right) + \left(b \cdot y4 - y5 \cdot i\right) \cdot \left(t \cdot j - y \cdot k\right)\right) - \left(c \cdot y4 - y5 \cdot a\right) \cdot \left(y2 \cdot t - y3 \cdot y\right)\right)\\ \mathbf{elif}\;y4 \le 2.717491665660854 \cdot 10^{-73}:\\ \;\;\;\;\left(\left(\left(b \cdot y4 - y5 \cdot i\right) \cdot \left(t \cdot j - y \cdot k\right) + \left(\left(y2 \cdot x - y3 \cdot z\right) \cdot \left(y0 \cdot c - y1 \cdot a\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(y0 \cdot b - i \cdot y1\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right)\right) - \left(c \cdot y4 - y5 \cdot a\right) \cdot \left(y2 \cdot t - y3 \cdot y\right)\right) + \left(\left(-k \cdot \left(\left(y0 \cdot y5\right) \cdot y2\right)\right) + y3 \cdot \left(\left(y0 \cdot y5 - y4 \cdot y1\right) \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(y2 \cdot x - y3 \cdot z\right) \cdot \left(y0 \cdot c - y1 \cdot a\right) + \left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right)\right) + \left(b \cdot y4 - y5 \cdot i\right) \cdot \left(t \cdot j - y \cdot k\right)\right) - \left(c \cdot y4 - y5 \cdot a\right) \cdot \left(y2 \cdot t - y3 \cdot y\right)\right) + \sqrt[3]{\left(y2 \cdot k - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)} \cdot \left(\sqrt[3]{\left(y2 \cdot k - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)} \cdot \sqrt[3]{\left(y2 \cdot k - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\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 r13704989 = x;
        double r13704990 = y;
        double r13704991 = r13704989 * r13704990;
        double r13704992 = z;
        double r13704993 = t;
        double r13704994 = r13704992 * r13704993;
        double r13704995 = r13704991 - r13704994;
        double r13704996 = a;
        double r13704997 = b;
        double r13704998 = r13704996 * r13704997;
        double r13704999 = c;
        double r13705000 = i;
        double r13705001 = r13704999 * r13705000;
        double r13705002 = r13704998 - r13705001;
        double r13705003 = r13704995 * r13705002;
        double r13705004 = j;
        double r13705005 = r13704989 * r13705004;
        double r13705006 = k;
        double r13705007 = r13704992 * r13705006;
        double r13705008 = r13705005 - r13705007;
        double r13705009 = y0;
        double r13705010 = r13705009 * r13704997;
        double r13705011 = y1;
        double r13705012 = r13705011 * r13705000;
        double r13705013 = r13705010 - r13705012;
        double r13705014 = r13705008 * r13705013;
        double r13705015 = r13705003 - r13705014;
        double r13705016 = y2;
        double r13705017 = r13704989 * r13705016;
        double r13705018 = y3;
        double r13705019 = r13704992 * r13705018;
        double r13705020 = r13705017 - r13705019;
        double r13705021 = r13705009 * r13704999;
        double r13705022 = r13705011 * r13704996;
        double r13705023 = r13705021 - r13705022;
        double r13705024 = r13705020 * r13705023;
        double r13705025 = r13705015 + r13705024;
        double r13705026 = r13704993 * r13705004;
        double r13705027 = r13704990 * r13705006;
        double r13705028 = r13705026 - r13705027;
        double r13705029 = y4;
        double r13705030 = r13705029 * r13704997;
        double r13705031 = y5;
        double r13705032 = r13705031 * r13705000;
        double r13705033 = r13705030 - r13705032;
        double r13705034 = r13705028 * r13705033;
        double r13705035 = r13705025 + r13705034;
        double r13705036 = r13704993 * r13705016;
        double r13705037 = r13704990 * r13705018;
        double r13705038 = r13705036 - r13705037;
        double r13705039 = r13705029 * r13704999;
        double r13705040 = r13705031 * r13704996;
        double r13705041 = r13705039 - r13705040;
        double r13705042 = r13705038 * r13705041;
        double r13705043 = r13705035 - r13705042;
        double r13705044 = r13705006 * r13705016;
        double r13705045 = r13705004 * r13705018;
        double r13705046 = r13705044 - r13705045;
        double r13705047 = r13705029 * r13705011;
        double r13705048 = r13705031 * r13705009;
        double r13705049 = r13705047 - r13705048;
        double r13705050 = r13705046 * r13705049;
        double r13705051 = r13705043 + r13705050;
        return r13705051;
}

↓

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 r13705052 = y4;
        double r13705053 = 4.5136545484096645e-287;
        bool r13705054 = r13705052 <= r13705053;
        double r13705055 = y2;
        double r13705056 = k;
        double r13705057 = r13705055 * r13705056;
        double r13705058 = j;
        double r13705059 = y3;
        double r13705060 = r13705058 * r13705059;
        double r13705061 = r13705057 - r13705060;
        double r13705062 = y1;
        double r13705063 = r13705052 * r13705062;
        double r13705064 = y0;
        double r13705065 = y5;
        double r13705066 = r13705064 * r13705065;
        double r13705067 = r13705063 - r13705066;
        double r13705068 = r13705061 * r13705067;
        double r13705069 = a;
        double r13705070 = -r13705062;
        double r13705071 = r13705069 * r13705070;
        double r13705072 = x;
        double r13705073 = r13705055 * r13705072;
        double r13705074 = z;
        double r13705075 = r13705059 * r13705074;
        double r13705076 = r13705073 - r13705075;
        double r13705077 = r13705071 * r13705076;
        double r13705078 = c;
        double r13705079 = r13705064 * r13705078;
        double r13705080 = r13705079 * r13705076;
        double r13705081 = r13705077 + r13705080;
        double r13705082 = y;
        double r13705083 = r13705072 * r13705082;
        double r13705084 = t;
        double r13705085 = r13705084 * r13705074;
        double r13705086 = r13705083 - r13705085;
        double r13705087 = b;
        double r13705088 = r13705069 * r13705087;
        double r13705089 = i;
        double r13705090 = r13705089 * r13705078;
        double r13705091 = r13705088 - r13705090;
        double r13705092 = r13705086 * r13705091;
        double r13705093 = r13705064 * r13705087;
        double r13705094 = r13705089 * r13705062;
        double r13705095 = r13705093 - r13705094;
        double r13705096 = r13705072 * r13705058;
        double r13705097 = r13705074 * r13705056;
        double r13705098 = r13705096 - r13705097;
        double r13705099 = r13705095 * r13705098;
        double r13705100 = r13705092 - r13705099;
        double r13705101 = r13705081 + r13705100;
        double r13705102 = r13705087 * r13705052;
        double r13705103 = r13705065 * r13705089;
        double r13705104 = r13705102 - r13705103;
        double r13705105 = r13705084 * r13705058;
        double r13705106 = r13705082 * r13705056;
        double r13705107 = r13705105 - r13705106;
        double r13705108 = r13705104 * r13705107;
        double r13705109 = r13705101 + r13705108;
        double r13705110 = r13705078 * r13705052;
        double r13705111 = r13705065 * r13705069;
        double r13705112 = r13705110 - r13705111;
        double r13705113 = r13705055 * r13705084;
        double r13705114 = r13705059 * r13705082;
        double r13705115 = r13705113 - r13705114;
        double r13705116 = r13705112 * r13705115;
        double r13705117 = r13705109 - r13705116;
        double r13705118 = r13705068 + r13705117;
        double r13705119 = 2.717491665660854e-73;
        bool r13705120 = r13705052 <= r13705119;
        double r13705121 = r13705062 * r13705069;
        double r13705122 = r13705079 - r13705121;
        double r13705123 = r13705076 * r13705122;
        double r13705124 = r13705123 + r13705100;
        double r13705125 = r13705108 + r13705124;
        double r13705126 = r13705125 - r13705116;
        double r13705127 = r13705066 * r13705055;
        double r13705128 = r13705056 * r13705127;
        double r13705129 = -r13705128;
        double r13705130 = r13705066 - r13705063;
        double r13705131 = r13705130 * r13705058;
        double r13705132 = r13705059 * r13705131;
        double r13705133 = r13705129 + r13705132;
        double r13705134 = r13705126 + r13705133;
        double r13705135 = r13705123 + r13705092;
        double r13705136 = r13705135 + r13705108;
        double r13705137 = r13705136 - r13705116;
        double r13705138 = cbrt(r13705068);
        double r13705139 = r13705138 * r13705138;
        double r13705140 = r13705138 * r13705139;
        double r13705141 = r13705137 + r13705140;
        double r13705142 = r13705120 ? r13705134 : r13705141;
        double r13705143 = r13705054 ? r13705118 : r13705142;
        return r13705143;
}

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 y4 < 4.5136545484096645e-287

   1. Initial program 25.4

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

   3. Applied sub-neg 25.4

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

   4. Applied distribute-lft-in 25.4

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

   if 4.5136545484096645e-287 < y4 < 2.717491665660854e-73

   1. Initial program 26.0

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

   2. Taylor expanded around inf 26.3

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

   3. Simplified 26.3

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

   if 2.717491665660854e-73 < y4

   1. Initial program 26.3

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

   3. Applied add-cube-cbrt 26.4

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

   4. Taylor expanded around 0 30.1

      \[\leadsto \left(\left(\left(\left(\left(x \cdot y - z \cdot t\right) \cdot \left(a \cdot b - c \cdot i\right) - \color{blue}{0}\right) + \left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - y5 \cdot i\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(\sqrt[3]{\left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)} \cdot \sqrt[3]{\left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)}\right) \cdot \sqrt[3]{\left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 26.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y4 \le 4.5136545484096645 \cdot 10^{-287}:\\ \;\;\;\;\left(y2 \cdot k - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right) + \left(\left(\left(\left(\left(a \cdot \left(-y1\right)\right) \cdot \left(y2 \cdot x - y3 \cdot z\right) + \left(y0 \cdot c\right) \cdot \left(y2 \cdot x - y3 \cdot z\right)\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(y0 \cdot b - i \cdot y1\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right) + \left(b \cdot y4 - y5 \cdot i\right) \cdot \left(t \cdot j - y \cdot k\right)\right) - \left(c \cdot y4 - y5 \cdot a\right) \cdot \left(y2 \cdot t - y3 \cdot y\right)\right)\\ \mathbf{elif}\;y4 \le 2.717491665660854 \cdot 10^{-73}:\\ \;\;\;\;\left(\left(\left(b \cdot y4 - y5 \cdot i\right) \cdot \left(t \cdot j - y \cdot k\right) + \left(\left(y2 \cdot x - y3 \cdot z\right) \cdot \left(y0 \cdot c - y1 \cdot a\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(y0 \cdot b - i \cdot y1\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right)\right) - \left(c \cdot y4 - y5 \cdot a\right) \cdot \left(y2 \cdot t - y3 \cdot y\right)\right) + \left(\left(-k \cdot \left(\left(y0 \cdot y5\right) \cdot y2\right)\right) + y3 \cdot \left(\left(y0 \cdot y5 - y4 \cdot y1\right) \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(y2 \cdot x - y3 \cdot z\right) \cdot \left(y0 \cdot c - y1 \cdot a\right) + \left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right)\right) + \left(b \cdot y4 - y5 \cdot i\right) \cdot \left(t \cdot j - y \cdot k\right)\right) - \left(c \cdot y4 - y5 \cdot a\right) \cdot \left(y2 \cdot t - y3 \cdot y\right)\right) + \sqrt[3]{\left(y2 \cdot k - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)} \cdot \left(\sqrt[3]{\left(y2 \cdot k - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)} \cdot \sqrt[3]{\left(y2 \cdot k - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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)))))