Linear.Matrix:det44 from linear-1.19.1.3

Average Error: 25.7 → 26.6

Time: 4.8m

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}\;j \le -1.0670051752273667 \cdot 10^{-95}:\\ \;\;\;\;\left(y4 \cdot y1 - y5 \cdot y0\right) \cdot \left(k \cdot y2 - y3 \cdot j\right) + \left(\left(\left(t \cdot j - y \cdot k\right) \cdot \left(b \cdot y4 - y5 \cdot i\right) + \left(\left(y1 \cdot \left(\left(z \cdot y3 - x \cdot y2\right) \cdot a\right) - \left(y0 \cdot c\right) \cdot \left(z \cdot y3\right)\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right)\right)\right) - \left(t \cdot y2 - y3 \cdot y\right) \cdot \left(c \cdot y4 - y5 \cdot a\right)\right)\\ \mathbf{elif}\;j \le 1.009426874676023 \cdot 10^{-48}:\\ \;\;\;\;\left(\left(\left(y3 \cdot j - k \cdot y2\right) \cdot y0\right) \cdot y5 - y1 \cdot \left(y4 \cdot \left(y3 \cdot j\right)\right)\right) + \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 - z \cdot k\right) \cdot \left(b \cdot y0 - 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(b \cdot y4 - y5 \cdot i\right)\right) - \left(t \cdot y2 - y3 \cdot y\right) \cdot \left(c \cdot y4 - y5 \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot j - y \cdot k\right) \cdot \left(b \cdot y4 - y5 \cdot i\right) + \left(\left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right) + \left(\sqrt[3]{\left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)} \cdot \sqrt[3]{\left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)}\right) \cdot \sqrt[3]{\left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)}\right)\right) - \left(t \cdot y2 - y3 \cdot y\right) \cdot \left(c \cdot y4 - y5 \cdot a\right)\right) + \left(y4 \cdot y1 - y5 \cdot y0\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 r27294234 = x;
        double r27294235 = y;
        double r27294236 = r27294234 * r27294235;
        double r27294237 = z;
        double r27294238 = t;
        double r27294239 = r27294237 * r27294238;
        double r27294240 = r27294236 - r27294239;
        double r27294241 = a;
        double r27294242 = b;
        double r27294243 = r27294241 * r27294242;
        double r27294244 = c;
        double r27294245 = i;
        double r27294246 = r27294244 * r27294245;
        double r27294247 = r27294243 - r27294246;
        double r27294248 = r27294240 * r27294247;
        double r27294249 = j;
        double r27294250 = r27294234 * r27294249;
        double r27294251 = k;
        double r27294252 = r27294237 * r27294251;
        double r27294253 = r27294250 - r27294252;
        double r27294254 = y0;
        double r27294255 = r27294254 * r27294242;
        double r27294256 = y1;
        double r27294257 = r27294256 * r27294245;
        double r27294258 = r27294255 - r27294257;
        double r27294259 = r27294253 * r27294258;
        double r27294260 = r27294248 - r27294259;
        double r27294261 = y2;
        double r27294262 = r27294234 * r27294261;
        double r27294263 = y3;
        double r27294264 = r27294237 * r27294263;
        double r27294265 = r27294262 - r27294264;
        double r27294266 = r27294254 * r27294244;
        double r27294267 = r27294256 * r27294241;
        double r27294268 = r27294266 - r27294267;
        double r27294269 = r27294265 * r27294268;
        double r27294270 = r27294260 + r27294269;
        double r27294271 = r27294238 * r27294249;
        double r27294272 = r27294235 * r27294251;
        double r27294273 = r27294271 - r27294272;
        double r27294274 = y4;
        double r27294275 = r27294274 * r27294242;
        double r27294276 = y5;
        double r27294277 = r27294276 * r27294245;
        double r27294278 = r27294275 - r27294277;
        double r27294279 = r27294273 * r27294278;
        double r27294280 = r27294270 + r27294279;
        double r27294281 = r27294238 * r27294261;
        double r27294282 = r27294235 * r27294263;
        double r27294283 = r27294281 - r27294282;
        double r27294284 = r27294274 * r27294244;
        double r27294285 = r27294276 * r27294241;
        double r27294286 = r27294284 - r27294285;
        double r27294287 = r27294283 * r27294286;
        double r27294288 = r27294280 - r27294287;
        double r27294289 = r27294251 * r27294261;
        double r27294290 = r27294249 * r27294263;
        double r27294291 = r27294289 - r27294290;
        double r27294292 = r27294274 * r27294256;
        double r27294293 = r27294276 * r27294254;
        double r27294294 = r27294292 - r27294293;
        double r27294295 = r27294291 * r27294294;
        double r27294296 = r27294288 + r27294295;
        return r27294296;
}

↓

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 r27294297 = j;
        double r27294298 = -1.0670051752273667e-95;
        bool r27294299 = r27294297 <= r27294298;
        double r27294300 = y4;
        double r27294301 = y1;
        double r27294302 = r27294300 * r27294301;
        double r27294303 = y5;
        double r27294304 = y0;
        double r27294305 = r27294303 * r27294304;
        double r27294306 = r27294302 - r27294305;
        double r27294307 = k;
        double r27294308 = y2;
        double r27294309 = r27294307 * r27294308;
        double r27294310 = y3;
        double r27294311 = r27294310 * r27294297;
        double r27294312 = r27294309 - r27294311;
        double r27294313 = r27294306 * r27294312;
        double r27294314 = t;
        double r27294315 = r27294314 * r27294297;
        double r27294316 = y;
        double r27294317 = r27294316 * r27294307;
        double r27294318 = r27294315 - r27294317;
        double r27294319 = b;
        double r27294320 = r27294319 * r27294300;
        double r27294321 = i;
        double r27294322 = r27294303 * r27294321;
        double r27294323 = r27294320 - r27294322;
        double r27294324 = r27294318 * r27294323;
        double r27294325 = z;
        double r27294326 = r27294325 * r27294310;
        double r27294327 = x;
        double r27294328 = r27294327 * r27294308;
        double r27294329 = r27294326 - r27294328;
        double r27294330 = a;
        double r27294331 = r27294329 * r27294330;
        double r27294332 = r27294301 * r27294331;
        double r27294333 = c;
        double r27294334 = r27294304 * r27294333;
        double r27294335 = r27294334 * r27294326;
        double r27294336 = r27294332 - r27294335;
        double r27294337 = r27294327 * r27294316;
        double r27294338 = r27294314 * r27294325;
        double r27294339 = r27294337 - r27294338;
        double r27294340 = r27294330 * r27294319;
        double r27294341 = r27294321 * r27294333;
        double r27294342 = r27294340 - r27294341;
        double r27294343 = r27294339 * r27294342;
        double r27294344 = r27294327 * r27294297;
        double r27294345 = r27294325 * r27294307;
        double r27294346 = r27294344 - r27294345;
        double r27294347 = r27294319 * r27294304;
        double r27294348 = r27294301 * r27294321;
        double r27294349 = r27294347 - r27294348;
        double r27294350 = r27294346 * r27294349;
        double r27294351 = r27294343 - r27294350;
        double r27294352 = r27294336 + r27294351;
        double r27294353 = r27294324 + r27294352;
        double r27294354 = r27294314 * r27294308;
        double r27294355 = r27294310 * r27294316;
        double r27294356 = r27294354 - r27294355;
        double r27294357 = r27294333 * r27294300;
        double r27294358 = r27294303 * r27294330;
        double r27294359 = r27294357 - r27294358;
        double r27294360 = r27294356 * r27294359;
        double r27294361 = r27294353 - r27294360;
        double r27294362 = r27294313 + r27294361;
        double r27294363 = 1.009426874676023e-48;
        bool r27294364 = r27294297 <= r27294363;
        double r27294365 = r27294311 - r27294309;
        double r27294366 = r27294365 * r27294304;
        double r27294367 = r27294366 * r27294303;
        double r27294368 = r27294300 * r27294311;
        double r27294369 = r27294301 * r27294368;
        double r27294370 = r27294367 - r27294369;
        double r27294371 = r27294328 - r27294326;
        double r27294372 = r27294301 * r27294330;
        double r27294373 = r27294334 - r27294372;
        double r27294374 = r27294371 * r27294373;
        double r27294375 = r27294351 + r27294374;
        double r27294376 = r27294375 + r27294324;
        double r27294377 = r27294376 - r27294360;
        double r27294378 = r27294370 + r27294377;
        double r27294379 = cbrt(r27294374);
        double r27294380 = r27294379 * r27294379;
        double r27294381 = r27294380 * r27294379;
        double r27294382 = r27294351 + r27294381;
        double r27294383 = r27294324 + r27294382;
        double r27294384 = r27294383 - r27294360;
        double r27294385 = r27294384 + r27294313;
        double r27294386 = r27294364 ? r27294378 : r27294385;
        double r27294387 = r27294299 ? r27294362 : r27294386;
        return r27294387;
}

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 j < -1.0670051752273667e-95

   1. Initial program 26.1

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

   2. Taylor expanded around inf 28.4

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

   3. Simplified 27.7

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

   if -1.0670051752273667e-95 < j < 1.009426874676023e-48

   1. Initial program 24.9

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

   2. Taylor expanded around -inf 26.5

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

   3. Simplified 25.9

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

   if 1.009426874676023e-48 < j

   1. Initial program 26.8

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

   3. Applied add-cube-cbrt 26.8

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

4. Final simplification 26.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1.0670051752273667 \cdot 10^{-95}:\\ \;\;\;\;\left(y4 \cdot y1 - y5 \cdot y0\right) \cdot \left(k \cdot y2 - y3 \cdot j\right) + \left(\left(\left(t \cdot j - y \cdot k\right) \cdot \left(b \cdot y4 - y5 \cdot i\right) + \left(\left(y1 \cdot \left(\left(z \cdot y3 - x \cdot y2\right) \cdot a\right) - \left(y0 \cdot c\right) \cdot \left(z \cdot y3\right)\right) + \left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right)\right)\right) - \left(t \cdot y2 - y3 \cdot y\right) \cdot \left(c \cdot y4 - y5 \cdot a\right)\right)\\ \mathbf{elif}\;j \le 1.009426874676023 \cdot 10^{-48}:\\ \;\;\;\;\left(\left(\left(y3 \cdot j - k \cdot y2\right) \cdot y0\right) \cdot y5 - y1 \cdot \left(y4 \cdot \left(y3 \cdot j\right)\right)\right) + \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 - z \cdot k\right) \cdot \left(b \cdot y0 - 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(b \cdot y4 - y5 \cdot i\right)\right) - \left(t \cdot y2 - y3 \cdot y\right) \cdot \left(c \cdot y4 - y5 \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot j - y \cdot k\right) \cdot \left(b \cdot y4 - y5 \cdot i\right) + \left(\left(\left(x \cdot y - t \cdot z\right) \cdot \left(a \cdot b - i \cdot c\right) - \left(x \cdot j - z \cdot k\right) \cdot \left(b \cdot y0 - y1 \cdot i\right)\right) + \left(\sqrt[3]{\left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)} \cdot \sqrt[3]{\left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)}\right) \cdot \sqrt[3]{\left(x \cdot y2 - z \cdot y3\right) \cdot \left(y0 \cdot c - y1 \cdot a\right)}\right)\right) - \left(t \cdot y2 - y3 \cdot y\right) \cdot \left(c \cdot y4 - y5 \cdot a\right)\right) + \left(y4 \cdot y1 - y5 \cdot y0\right) \cdot \left(k \cdot y2 - y3 \cdot j\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))))