Linear.Matrix:det44 from linear-1.19.1.3

Average Error: 27.1 → 28.5

Time: 47.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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}\;z \le -3.2649420662695442 \cdot 10^{-224}:\\ \;\;\;\;\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(a \cdot \left(y3 \cdot \left(y1 \cdot z\right)\right) - \left(y0 \cdot \left(z \cdot \left(y3 \cdot c\right)\right) + a \cdot \left(x \cdot \left(y2 \cdot y1\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)\\ \mathbf{elif}\;z \le 2.85563530538145874 \cdot 10^{-168}:\\ \;\;\;\;\left(\left(\left(\left(\left(t \cdot \left(i \cdot \left(z \cdot c\right)\right) - \left(i \cdot \left(c \cdot \left(y \cdot x\right)\right) + a \cdot \left(t \cdot \left(z \cdot b\right)\right)\right)\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) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\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(\left(t \cdot y2 - y \cdot y3\right) \cdot \left(\sqrt[3]{y4 \cdot c - y5 \cdot a} \cdot \sqrt[3]{y4 \cdot c - y5 \cdot a}\right)\right) \cdot \sqrt[3]{y4 \cdot c - y5 \cdot a}\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\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 r168125 = x;
        double r168126 = y;
        double r168127 = r168125 * r168126;
        double r168128 = z;
        double r168129 = t;
        double r168130 = r168128 * r168129;
        double r168131 = r168127 - r168130;
        double r168132 = a;
        double r168133 = b;
        double r168134 = r168132 * r168133;
        double r168135 = c;
        double r168136 = i;
        double r168137 = r168135 * r168136;
        double r168138 = r168134 - r168137;
        double r168139 = r168131 * r168138;
        double r168140 = j;
        double r168141 = r168125 * r168140;
        double r168142 = k;
        double r168143 = r168128 * r168142;
        double r168144 = r168141 - r168143;
        double r168145 = y0;
        double r168146 = r168145 * r168133;
        double r168147 = y1;
        double r168148 = r168147 * r168136;
        double r168149 = r168146 - r168148;
        double r168150 = r168144 * r168149;
        double r168151 = r168139 - r168150;
        double r168152 = y2;
        double r168153 = r168125 * r168152;
        double r168154 = y3;
        double r168155 = r168128 * r168154;
        double r168156 = r168153 - r168155;
        double r168157 = r168145 * r168135;
        double r168158 = r168147 * r168132;
        double r168159 = r168157 - r168158;
        double r168160 = r168156 * r168159;
        double r168161 = r168151 + r168160;
        double r168162 = r168129 * r168140;
        double r168163 = r168126 * r168142;
        double r168164 = r168162 - r168163;
        double r168165 = y4;
        double r168166 = r168165 * r168133;
        double r168167 = y5;
        double r168168 = r168167 * r168136;
        double r168169 = r168166 - r168168;
        double r168170 = r168164 * r168169;
        double r168171 = r168161 + r168170;
        double r168172 = r168129 * r168152;
        double r168173 = r168126 * r168154;
        double r168174 = r168172 - r168173;
        double r168175 = r168165 * r168135;
        double r168176 = r168167 * r168132;
        double r168177 = r168175 - r168176;
        double r168178 = r168174 * r168177;
        double r168179 = r168171 - r168178;
        double r168180 = r168142 * r168152;
        double r168181 = r168140 * r168154;
        double r168182 = r168180 - r168181;
        double r168183 = r168165 * r168147;
        double r168184 = r168167 * r168145;
        double r168185 = r168183 - r168184;
        double r168186 = r168182 * r168185;
        double r168187 = r168179 + r168186;
        return r168187;
}

↓

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 r168188 = z;
        double r168189 = -3.264942066269544e-224;
        bool r168190 = r168188 <= r168189;
        double r168191 = x;
        double r168192 = y;
        double r168193 = r168191 * r168192;
        double r168194 = t;
        double r168195 = r168188 * r168194;
        double r168196 = r168193 - r168195;
        double r168197 = a;
        double r168198 = b;
        double r168199 = r168197 * r168198;
        double r168200 = c;
        double r168201 = i;
        double r168202 = r168200 * r168201;
        double r168203 = r168199 - r168202;
        double r168204 = r168196 * r168203;
        double r168205 = j;
        double r168206 = r168191 * r168205;
        double r168207 = k;
        double r168208 = r168188 * r168207;
        double r168209 = r168206 - r168208;
        double r168210 = y0;
        double r168211 = r168210 * r168198;
        double r168212 = y1;
        double r168213 = r168212 * r168201;
        double r168214 = r168211 - r168213;
        double r168215 = r168209 * r168214;
        double r168216 = r168204 - r168215;
        double r168217 = y3;
        double r168218 = r168212 * r168188;
        double r168219 = r168217 * r168218;
        double r168220 = r168197 * r168219;
        double r168221 = r168217 * r168200;
        double r168222 = r168188 * r168221;
        double r168223 = r168210 * r168222;
        double r168224 = y2;
        double r168225 = r168224 * r168212;
        double r168226 = r168191 * r168225;
        double r168227 = r168197 * r168226;
        double r168228 = r168223 + r168227;
        double r168229 = r168220 - r168228;
        double r168230 = r168216 + r168229;
        double r168231 = r168194 * r168205;
        double r168232 = r168192 * r168207;
        double r168233 = r168231 - r168232;
        double r168234 = y4;
        double r168235 = r168234 * r168198;
        double r168236 = y5;
        double r168237 = r168236 * r168201;
        double r168238 = r168235 - r168237;
        double r168239 = r168233 * r168238;
        double r168240 = r168230 + r168239;
        double r168241 = r168194 * r168224;
        double r168242 = r168192 * r168217;
        double r168243 = r168241 - r168242;
        double r168244 = r168234 * r168200;
        double r168245 = r168236 * r168197;
        double r168246 = r168244 - r168245;
        double r168247 = r168243 * r168246;
        double r168248 = r168240 - r168247;
        double r168249 = r168207 * r168224;
        double r168250 = r168205 * r168217;
        double r168251 = r168249 - r168250;
        double r168252 = r168234 * r168212;
        double r168253 = r168236 * r168210;
        double r168254 = r168252 - r168253;
        double r168255 = r168251 * r168254;
        double r168256 = r168248 + r168255;
        double r168257 = 2.8556353053814587e-168;
        bool r168258 = r168188 <= r168257;
        double r168259 = r168188 * r168200;
        double r168260 = r168201 * r168259;
        double r168261 = r168194 * r168260;
        double r168262 = r168192 * r168191;
        double r168263 = r168200 * r168262;
        double r168264 = r168201 * r168263;
        double r168265 = r168188 * r168198;
        double r168266 = r168194 * r168265;
        double r168267 = r168197 * r168266;
        double r168268 = r168264 + r168267;
        double r168269 = r168261 - r168268;
        double r168270 = r168269 - r168215;
        double r168271 = r168191 * r168224;
        double r168272 = r168188 * r168217;
        double r168273 = r168271 - r168272;
        double r168274 = r168210 * r168200;
        double r168275 = r168212 * r168197;
        double r168276 = r168274 - r168275;
        double r168277 = r168273 * r168276;
        double r168278 = r168270 + r168277;
        double r168279 = cbrt(r168238);
        double r168280 = r168279 * r168279;
        double r168281 = r168233 * r168280;
        double r168282 = r168281 * r168279;
        double r168283 = r168278 + r168282;
        double r168284 = r168283 - r168247;
        double r168285 = r168284 + r168255;
        double r168286 = r168216 + r168277;
        double r168287 = r168286 + r168239;
        double r168288 = cbrt(r168246);
        double r168289 = r168288 * r168288;
        double r168290 = r168243 * r168289;
        double r168291 = r168290 * r168288;
        double r168292 = r168287 - r168291;
        double r168293 = r168292 + r168255;
        double r168294 = r168258 ? r168285 : r168293;
        double r168295 = r168190 ? r168256 : r168294;
        return r168295;
}

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 z < -3.264942066269544e-224

   1. Initial program 27.3

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

   2. Taylor expanded around inf 30.3

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

   if -3.264942066269544e-224 < z < 2.8556353053814587e-168

   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. Using strategy rm

   3. Applied add-cube-cbrt 27.4

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

   4. Applied associate-*r* 27.4

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

   5. Taylor expanded around inf 27.6

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

   if 2.8556353053814587e-168 < z

   1. Initial program 26.9

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

   3. Applied add-cube-cbrt 26.9

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

   4. Applied associate-*r* 26.9

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

4. Final simplification 28.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.2649420662695442 \cdot 10^{-224}:\\ \;\;\;\;\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(a \cdot \left(y3 \cdot \left(y1 \cdot z\right)\right) - \left(y0 \cdot \left(z \cdot \left(y3 \cdot c\right)\right) + a \cdot \left(x \cdot \left(y2 \cdot y1\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)\\ \mathbf{elif}\;z \le 2.85563530538145874 \cdot 10^{-168}:\\ \;\;\;\;\left(\left(\left(\left(\left(t \cdot \left(i \cdot \left(z \cdot c\right)\right) - \left(i \cdot \left(c \cdot \left(y \cdot x\right)\right) + a \cdot \left(t \cdot \left(z \cdot b\right)\right)\right)\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) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\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(\left(t \cdot y2 - y \cdot y3\right) \cdot \left(\sqrt[3]{y4 \cdot c - y5 \cdot a} \cdot \sqrt[3]{y4 \cdot c - y5 \cdot a}\right)\right) \cdot \sqrt[3]{y4 \cdot c - y5 \cdot a}\right) + \left(k \cdot y2 - j \cdot y3\right) \cdot \left(y4 \cdot y1 - y5 \cdot y0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))