Linear.Matrix:det44 from linear-1.19.1.3

Average Error: 25.4 → 26.3

Time: 1.7m

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 -2.5668315936367674 \cdot 10^{-277}:\\ \;\;\;\;\left(\left(\left(\left(c \cdot y0 - 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(b \cdot y0 - y1 \cdot i\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - i \cdot y5\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 - y3 \cdot j} \cdot \sqrt[3]{k \cdot y2 - y3 \cdot j}\right) \cdot \left(\sqrt[3]{k \cdot y2 - y3 \cdot j} \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)\right)\\ \mathbf{elif}\;y0 \le 6.418500696234417 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(\left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - i \cdot y5\right) + \left(\left(\left(\left(z \cdot y1\right) \cdot y3\right) \cdot a - \left(c \cdot \left(\left(y3 \cdot y0\right) \cdot z\right) + \left(x \cdot \left(y2 \cdot y1\right)\right) \cdot a\right)\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(b \cdot y0 - y1 \cdot i\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(y4 \cdot y1 - y0 \cdot y5\right) \cdot \left(k \cdot y2 - y3 \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(y5 \cdot k\right)\right) \cdot i - \left(t \cdot \left(\left(y5 \cdot j\right) \cdot i\right) + \left(\left(y4 \cdot b\right) \cdot y\right) \cdot k\right)\right) + \left(\left(c \cdot y0 - 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(b \cdot y0 - y1 \cdot i\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(y4 \cdot y1 - y0 \cdot y5\right) \cdot \left(k \cdot y2 - y3 \cdot j\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 r2218042 = x;
        double r2218043 = y;
        double r2218044 = r2218042 * r2218043;
        double r2218045 = z;
        double r2218046 = t;
        double r2218047 = r2218045 * r2218046;
        double r2218048 = r2218044 - r2218047;
        double r2218049 = a;
        double r2218050 = b;
        double r2218051 = r2218049 * r2218050;
        double r2218052 = c;
        double r2218053 = i;
        double r2218054 = r2218052 * r2218053;
        double r2218055 = r2218051 - r2218054;
        double r2218056 = r2218048 * r2218055;
        double r2218057 = j;
        double r2218058 = r2218042 * r2218057;
        double r2218059 = k;
        double r2218060 = r2218045 * r2218059;
        double r2218061 = r2218058 - r2218060;
        double r2218062 = y0;
        double r2218063 = r2218062 * r2218050;
        double r2218064 = y1;
        double r2218065 = r2218064 * r2218053;
        double r2218066 = r2218063 - r2218065;
        double r2218067 = r2218061 * r2218066;
        double r2218068 = r2218056 - r2218067;
        double r2218069 = y2;
        double r2218070 = r2218042 * r2218069;
        double r2218071 = y3;
        double r2218072 = r2218045 * r2218071;
        double r2218073 = r2218070 - r2218072;
        double r2218074 = r2218062 * r2218052;
        double r2218075 = r2218064 * r2218049;
        double r2218076 = r2218074 - r2218075;
        double r2218077 = r2218073 * r2218076;
        double r2218078 = r2218068 + r2218077;
        double r2218079 = r2218046 * r2218057;
        double r2218080 = r2218043 * r2218059;
        double r2218081 = r2218079 - r2218080;
        double r2218082 = y4;
        double r2218083 = r2218082 * r2218050;
        double r2218084 = y5;
        double r2218085 = r2218084 * r2218053;
        double r2218086 = r2218083 - r2218085;
        double r2218087 = r2218081 * r2218086;
        double r2218088 = r2218078 + r2218087;
        double r2218089 = r2218046 * r2218069;
        double r2218090 = r2218043 * r2218071;
        double r2218091 = r2218089 - r2218090;
        double r2218092 = r2218082 * r2218052;
        double r2218093 = r2218084 * r2218049;
        double r2218094 = r2218092 - r2218093;
        double r2218095 = r2218091 * r2218094;
        double r2218096 = r2218088 - r2218095;
        double r2218097 = r2218059 * r2218069;
        double r2218098 = r2218057 * r2218071;
        double r2218099 = r2218097 - r2218098;
        double r2218100 = r2218082 * r2218064;
        double r2218101 = r2218084 * r2218062;
        double r2218102 = r2218100 - r2218101;
        double r2218103 = r2218099 * r2218102;
        double r2218104 = r2218096 + r2218103;
        return r2218104;
}

↓

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 r2218105 = y0;
        double r2218106 = -2.5668315936367674e-277;
        bool r2218107 = r2218105 <= r2218106;
        double r2218108 = c;
        double r2218109 = r2218108 * r2218105;
        double r2218110 = y1;
        double r2218111 = a;
        double r2218112 = r2218110 * r2218111;
        double r2218113 = r2218109 - r2218112;
        double r2218114 = x;
        double r2218115 = y2;
        double r2218116 = r2218114 * r2218115;
        double r2218117 = z;
        double r2218118 = y3;
        double r2218119 = r2218117 * r2218118;
        double r2218120 = r2218116 - r2218119;
        double r2218121 = r2218113 * r2218120;
        double r2218122 = y;
        double r2218123 = r2218114 * r2218122;
        double r2218124 = t;
        double r2218125 = r2218124 * r2218117;
        double r2218126 = r2218123 - r2218125;
        double r2218127 = b;
        double r2218128 = r2218111 * r2218127;
        double r2218129 = i;
        double r2218130 = r2218129 * r2218108;
        double r2218131 = r2218128 - r2218130;
        double r2218132 = r2218126 * r2218131;
        double r2218133 = r2218127 * r2218105;
        double r2218134 = r2218110 * r2218129;
        double r2218135 = r2218133 - r2218134;
        double r2218136 = j;
        double r2218137 = r2218114 * r2218136;
        double r2218138 = k;
        double r2218139 = r2218117 * r2218138;
        double r2218140 = r2218137 - r2218139;
        double r2218141 = r2218135 * r2218140;
        double r2218142 = r2218132 - r2218141;
        double r2218143 = r2218121 + r2218142;
        double r2218144 = r2218124 * r2218136;
        double r2218145 = r2218122 * r2218138;
        double r2218146 = r2218144 - r2218145;
        double r2218147 = y4;
        double r2218148 = r2218147 * r2218127;
        double r2218149 = y5;
        double r2218150 = r2218129 * r2218149;
        double r2218151 = r2218148 - r2218150;
        double r2218152 = r2218146 * r2218151;
        double r2218153 = r2218143 + r2218152;
        double r2218154 = r2218124 * r2218115;
        double r2218155 = r2218122 * r2218118;
        double r2218156 = r2218154 - r2218155;
        double r2218157 = r2218147 * r2218108;
        double r2218158 = r2218149 * r2218111;
        double r2218159 = r2218157 - r2218158;
        double r2218160 = r2218156 * r2218159;
        double r2218161 = r2218153 - r2218160;
        double r2218162 = r2218138 * r2218115;
        double r2218163 = r2218118 * r2218136;
        double r2218164 = r2218162 - r2218163;
        double r2218165 = cbrt(r2218164);
        double r2218166 = r2218165 * r2218165;
        double r2218167 = r2218147 * r2218110;
        double r2218168 = r2218105 * r2218149;
        double r2218169 = r2218167 - r2218168;
        double r2218170 = r2218165 * r2218169;
        double r2218171 = r2218166 * r2218170;
        double r2218172 = r2218161 + r2218171;
        double r2218173 = 6.418500696234417e+26;
        bool r2218174 = r2218105 <= r2218173;
        double r2218175 = r2218117 * r2218110;
        double r2218176 = r2218175 * r2218118;
        double r2218177 = r2218176 * r2218111;
        double r2218178 = r2218118 * r2218105;
        double r2218179 = r2218178 * r2218117;
        double r2218180 = r2218108 * r2218179;
        double r2218181 = r2218115 * r2218110;
        double r2218182 = r2218114 * r2218181;
        double r2218183 = r2218182 * r2218111;
        double r2218184 = r2218180 + r2218183;
        double r2218185 = r2218177 - r2218184;
        double r2218186 = r2218185 + r2218142;
        double r2218187 = r2218152 + r2218186;
        double r2218188 = r2218187 - r2218160;
        double r2218189 = r2218169 * r2218164;
        double r2218190 = r2218188 + r2218189;
        double r2218191 = r2218149 * r2218138;
        double r2218192 = r2218122 * r2218191;
        double r2218193 = r2218192 * r2218129;
        double r2218194 = r2218149 * r2218136;
        double r2218195 = r2218194 * r2218129;
        double r2218196 = r2218124 * r2218195;
        double r2218197 = r2218148 * r2218122;
        double r2218198 = r2218197 * r2218138;
        double r2218199 = r2218196 + r2218198;
        double r2218200 = r2218193 - r2218199;
        double r2218201 = r2218200 + r2218143;
        double r2218202 = r2218201 - r2218160;
        double r2218203 = r2218202 + r2218189;
        double r2218204 = r2218174 ? r2218190 : r2218203;
        double r2218205 = r2218107 ? r2218172 : r2218204;
        return r2218205;
}

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 < -2.5668315936367674e-277

   1. Initial program 25.2

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

   3. Applied add-cube-cbrt 25.3

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

   4. Applied associate-*l* 25.3

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

   if -2.5668315936367674e-277 < y0 < 6.418500696234417e+26

   1. Initial program 24.8

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

   2. Taylor expanded around inf 26.6

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

   if 6.418500696234417e+26 < y0

   1. Initial program 28.0

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

   2. Taylor expanded around inf 29.1

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

4. Final simplification 26.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y0 \le -2.5668315936367674 \cdot 10^{-277}:\\ \;\;\;\;\left(\left(\left(\left(c \cdot y0 - 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(b \cdot y0 - y1 \cdot i\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right) + \left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - i \cdot y5\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 - y3 \cdot j} \cdot \sqrt[3]{k \cdot y2 - y3 \cdot j}\right) \cdot \left(\sqrt[3]{k \cdot y2 - y3 \cdot j} \cdot \left(y4 \cdot y1 - y0 \cdot y5\right)\right)\\ \mathbf{elif}\;y0 \le 6.418500696234417 \cdot 10^{+26}:\\ \;\;\;\;\left(\left(\left(t \cdot j - y \cdot k\right) \cdot \left(y4 \cdot b - i \cdot y5\right) + \left(\left(\left(\left(z \cdot y1\right) \cdot y3\right) \cdot a - \left(c \cdot \left(\left(y3 \cdot y0\right) \cdot z\right) + \left(x \cdot \left(y2 \cdot y1\right)\right) \cdot a\right)\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(b \cdot y0 - y1 \cdot i\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(y4 \cdot y1 - y0 \cdot y5\right) \cdot \left(k \cdot y2 - y3 \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(y5 \cdot k\right)\right) \cdot i - \left(t \cdot \left(\left(y5 \cdot j\right) \cdot i\right) + \left(\left(y4 \cdot b\right) \cdot y\right) \cdot k\right)\right) + \left(\left(c \cdot y0 - 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(b \cdot y0 - y1 \cdot i\right) \cdot \left(x \cdot j - z \cdot k\right)\right)\right)\right) - \left(t \cdot y2 - y \cdot y3\right) \cdot \left(y4 \cdot c - y5 \cdot a\right)\right) + \left(y4 \cdot y1 - y0 \cdot y5\right) \cdot \left(k \cdot y2 - y3 \cdot j\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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)))))