Linear.Matrix:det44 from linear-1.19.1.3

Average Error: 25.7 → 26.7

Time: 2.1m

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}\;a \le -0.00798469123475429:\\ \;\;\;\;\left(\left(\left(\left(c \cdot y0 - a \cdot y1\right) \cdot \left(x \cdot y2 - y3 \cdot z\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(i \cdot \left(z \cdot \left(k \cdot y1\right)\right) - \left(\left(\left(x \cdot y1\right) \cdot j\right) \cdot i + \left(\left(b \cdot y0\right) \cdot z\right) \cdot k\right)\right)\right)\right) + \left(y4 \cdot b - i \cdot y5\right) \cdot \left(j \cdot t - y \cdot k\right)\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)\right) + \left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y1 \cdot y4 - y5 \cdot y0\right)\\ \mathbf{elif}\;a \le -2.0073425297372163 \cdot 10^{-147}:\\ \;\;\;\;\left(\left(\left(y4 \cdot b - i \cdot y5\right) \cdot \left(j \cdot t - y \cdot k\right) + \left(\left(x \cdot j - k \cdot z\right) \cdot \left(-\left(b \cdot y0 - y1 \cdot i\right)\right) + \left(c \cdot y0 - a \cdot y1\right) \cdot \left(x \cdot y2 - y3 \cdot z\right)\right)\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)\right) + \left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y1 \cdot y4 - y5 \cdot y0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\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) + \left(c \cdot y0 - a \cdot y1\right) \cdot \left(x \cdot y2 - y3 \cdot z\right)\right) + \left(y4 \cdot b - i \cdot y5\right) \cdot \left(j \cdot t - y \cdot k\right)\right) - \sqrt[3]{\left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)} \cdot \left(\sqrt[3]{\left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)} \cdot \sqrt[3]{\left(\left(\sqrt[3]{y4 \cdot c - y5 \cdot a} \cdot \sqrt[3]{y4 \cdot c - y5 \cdot a}\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)\right) \cdot \sqrt[3]{y4 \cdot c - y5 \cdot a}}\right)\right) + \left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y1 \cdot y4 - 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 r2344176 = x;
        double r2344177 = y;
        double r2344178 = r2344176 * r2344177;
        double r2344179 = z;
        double r2344180 = t;
        double r2344181 = r2344179 * r2344180;
        double r2344182 = r2344178 - r2344181;
        double r2344183 = a;
        double r2344184 = b;
        double r2344185 = r2344183 * r2344184;
        double r2344186 = c;
        double r2344187 = i;
        double r2344188 = r2344186 * r2344187;
        double r2344189 = r2344185 - r2344188;
        double r2344190 = r2344182 * r2344189;
        double r2344191 = j;
        double r2344192 = r2344176 * r2344191;
        double r2344193 = k;
        double r2344194 = r2344179 * r2344193;
        double r2344195 = r2344192 - r2344194;
        double r2344196 = y0;
        double r2344197 = r2344196 * r2344184;
        double r2344198 = y1;
        double r2344199 = r2344198 * r2344187;
        double r2344200 = r2344197 - r2344199;
        double r2344201 = r2344195 * r2344200;
        double r2344202 = r2344190 - r2344201;
        double r2344203 = y2;
        double r2344204 = r2344176 * r2344203;
        double r2344205 = y3;
        double r2344206 = r2344179 * r2344205;
        double r2344207 = r2344204 - r2344206;
        double r2344208 = r2344196 * r2344186;
        double r2344209 = r2344198 * r2344183;
        double r2344210 = r2344208 - r2344209;
        double r2344211 = r2344207 * r2344210;
        double r2344212 = r2344202 + r2344211;
        double r2344213 = r2344180 * r2344191;
        double r2344214 = r2344177 * r2344193;
        double r2344215 = r2344213 - r2344214;
        double r2344216 = y4;
        double r2344217 = r2344216 * r2344184;
        double r2344218 = y5;
        double r2344219 = r2344218 * r2344187;
        double r2344220 = r2344217 - r2344219;
        double r2344221 = r2344215 * r2344220;
        double r2344222 = r2344212 + r2344221;
        double r2344223 = r2344180 * r2344203;
        double r2344224 = r2344177 * r2344205;
        double r2344225 = r2344223 - r2344224;
        double r2344226 = r2344216 * r2344186;
        double r2344227 = r2344218 * r2344183;
        double r2344228 = r2344226 - r2344227;
        double r2344229 = r2344225 * r2344228;
        double r2344230 = r2344222 - r2344229;
        double r2344231 = r2344193 * r2344203;
        double r2344232 = r2344191 * r2344205;
        double r2344233 = r2344231 - r2344232;
        double r2344234 = r2344216 * r2344198;
        double r2344235 = r2344218 * r2344196;
        double r2344236 = r2344234 - r2344235;
        double r2344237 = r2344233 * r2344236;
        double r2344238 = r2344230 + r2344237;
        return r2344238;
}

↓

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 r2344239 = a;
        double r2344240 = -0.00798469123475429;
        bool r2344241 = r2344239 <= r2344240;
        double r2344242 = c;
        double r2344243 = y0;
        double r2344244 = r2344242 * r2344243;
        double r2344245 = y1;
        double r2344246 = r2344239 * r2344245;
        double r2344247 = r2344244 - r2344246;
        double r2344248 = x;
        double r2344249 = y2;
        double r2344250 = r2344248 * r2344249;
        double r2344251 = y3;
        double r2344252 = z;
        double r2344253 = r2344251 * r2344252;
        double r2344254 = r2344250 - r2344253;
        double r2344255 = r2344247 * r2344254;
        double r2344256 = y;
        double r2344257 = r2344248 * r2344256;
        double r2344258 = t;
        double r2344259 = r2344258 * r2344252;
        double r2344260 = r2344257 - r2344259;
        double r2344261 = b;
        double r2344262 = r2344239 * r2344261;
        double r2344263 = i;
        double r2344264 = r2344263 * r2344242;
        double r2344265 = r2344262 - r2344264;
        double r2344266 = r2344260 * r2344265;
        double r2344267 = k;
        double r2344268 = r2344267 * r2344245;
        double r2344269 = r2344252 * r2344268;
        double r2344270 = r2344263 * r2344269;
        double r2344271 = r2344248 * r2344245;
        double r2344272 = j;
        double r2344273 = r2344271 * r2344272;
        double r2344274 = r2344273 * r2344263;
        double r2344275 = r2344261 * r2344243;
        double r2344276 = r2344275 * r2344252;
        double r2344277 = r2344276 * r2344267;
        double r2344278 = r2344274 + r2344277;
        double r2344279 = r2344270 - r2344278;
        double r2344280 = r2344266 - r2344279;
        double r2344281 = r2344255 + r2344280;
        double r2344282 = y4;
        double r2344283 = r2344282 * r2344261;
        double r2344284 = y5;
        double r2344285 = r2344263 * r2344284;
        double r2344286 = r2344283 - r2344285;
        double r2344287 = r2344272 * r2344258;
        double r2344288 = r2344256 * r2344267;
        double r2344289 = r2344287 - r2344288;
        double r2344290 = r2344286 * r2344289;
        double r2344291 = r2344281 + r2344290;
        double r2344292 = r2344282 * r2344242;
        double r2344293 = r2344284 * r2344239;
        double r2344294 = r2344292 - r2344293;
        double r2344295 = r2344258 * r2344249;
        double r2344296 = r2344251 * r2344256;
        double r2344297 = r2344295 - r2344296;
        double r2344298 = r2344294 * r2344297;
        double r2344299 = r2344291 - r2344298;
        double r2344300 = r2344249 * r2344267;
        double r2344301 = r2344251 * r2344272;
        double r2344302 = r2344300 - r2344301;
        double r2344303 = r2344245 * r2344282;
        double r2344304 = r2344284 * r2344243;
        double r2344305 = r2344303 - r2344304;
        double r2344306 = r2344302 * r2344305;
        double r2344307 = r2344299 + r2344306;
        double r2344308 = -2.0073425297372163e-147;
        bool r2344309 = r2344239 <= r2344308;
        double r2344310 = r2344248 * r2344272;
        double r2344311 = r2344267 * r2344252;
        double r2344312 = r2344310 - r2344311;
        double r2344313 = r2344245 * r2344263;
        double r2344314 = r2344275 - r2344313;
        double r2344315 = -r2344314;
        double r2344316 = r2344312 * r2344315;
        double r2344317 = r2344316 + r2344255;
        double r2344318 = r2344290 + r2344317;
        double r2344319 = r2344318 - r2344298;
        double r2344320 = r2344319 + r2344306;
        double r2344321 = r2344312 * r2344314;
        double r2344322 = r2344266 - r2344321;
        double r2344323 = r2344322 + r2344255;
        double r2344324 = r2344323 + r2344290;
        double r2344325 = cbrt(r2344298);
        double r2344326 = cbrt(r2344294);
        double r2344327 = r2344326 * r2344326;
        double r2344328 = r2344327 * r2344297;
        double r2344329 = r2344328 * r2344326;
        double r2344330 = cbrt(r2344329);
        double r2344331 = r2344325 * r2344330;
        double r2344332 = r2344325 * r2344331;
        double r2344333 = r2344324 - r2344332;
        double r2344334 = r2344333 + r2344306;
        double r2344335 = r2344309 ? r2344320 : r2344334;
        double r2344336 = r2344241 ? r2344307 : r2344335;
        return r2344336;
}

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 a < -0.00798469123475429

   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. Taylor expanded around -inf 27.3

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

   if -0.00798469123475429 < a < -2.0073425297372163e-147

   1. Initial program 23.7

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

   2. Taylor expanded around 0 29.3

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

   if -2.0073425297372163e-147 < a

   1. Initial program 25.9

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

   3. Applied add-cube-cbrt 26.0

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

   5. Applied add-cube-cbrt 26.0

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

   6. Applied associate-*r* 26.0

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

4. Final simplification 26.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -0.00798469123475429:\\ \;\;\;\;\left(\left(\left(\left(c \cdot y0 - a \cdot y1\right) \cdot \left(x \cdot y2 - y3 \cdot z\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(i \cdot \left(z \cdot \left(k \cdot y1\right)\right) - \left(\left(\left(x \cdot y1\right) \cdot j\right) \cdot i + \left(\left(b \cdot y0\right) \cdot z\right) \cdot k\right)\right)\right)\right) + \left(y4 \cdot b - i \cdot y5\right) \cdot \left(j \cdot t - y \cdot k\right)\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)\right) + \left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y1 \cdot y4 - y5 \cdot y0\right)\\ \mathbf{elif}\;a \le -2.0073425297372163 \cdot 10^{-147}:\\ \;\;\;\;\left(\left(\left(y4 \cdot b - i \cdot y5\right) \cdot \left(j \cdot t - y \cdot k\right) + \left(\left(x \cdot j - k \cdot z\right) \cdot \left(-\left(b \cdot y0 - y1 \cdot i\right)\right) + \left(c \cdot y0 - a \cdot y1\right) \cdot \left(x \cdot y2 - y3 \cdot z\right)\right)\right) - \left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)\right) + \left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y1 \cdot y4 - y5 \cdot y0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\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) + \left(c \cdot y0 - a \cdot y1\right) \cdot \left(x \cdot y2 - y3 \cdot z\right)\right) + \left(y4 \cdot b - i \cdot y5\right) \cdot \left(j \cdot t - y \cdot k\right)\right) - \sqrt[3]{\left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)} \cdot \left(\sqrt[3]{\left(y4 \cdot c - y5 \cdot a\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)} \cdot \sqrt[3]{\left(\left(\sqrt[3]{y4 \cdot c - y5 \cdot a} \cdot \sqrt[3]{y4 \cdot c - y5 \cdot a}\right) \cdot \left(t \cdot y2 - y3 \cdot y\right)\right) \cdot \sqrt[3]{y4 \cdot c - y5 \cdot a}}\right)\right) + \left(y2 \cdot k - y3 \cdot j\right) \cdot \left(y1 \cdot y4 - y5 \cdot y0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))))