Linear.Matrix:det44 from linear-1.19.1.3

Average Error: 25.7 → 26.7

Time: 1.9m

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 r4517416 = x;
        double r4517417 = y;
        double r4517418 = r4517416 * r4517417;
        double r4517419 = z;
        double r4517420 = t;
        double r4517421 = r4517419 * r4517420;
        double r4517422 = r4517418 - r4517421;
        double r4517423 = a;
        double r4517424 = b;
        double r4517425 = r4517423 * r4517424;
        double r4517426 = c;
        double r4517427 = i;
        double r4517428 = r4517426 * r4517427;
        double r4517429 = r4517425 - r4517428;
        double r4517430 = r4517422 * r4517429;
        double r4517431 = j;
        double r4517432 = r4517416 * r4517431;
        double r4517433 = k;
        double r4517434 = r4517419 * r4517433;
        double r4517435 = r4517432 - r4517434;
        double r4517436 = y0;
        double r4517437 = r4517436 * r4517424;
        double r4517438 = y1;
        double r4517439 = r4517438 * r4517427;
        double r4517440 = r4517437 - r4517439;
        double r4517441 = r4517435 * r4517440;
        double r4517442 = r4517430 - r4517441;
        double r4517443 = y2;
        double r4517444 = r4517416 * r4517443;
        double r4517445 = y3;
        double r4517446 = r4517419 * r4517445;
        double r4517447 = r4517444 - r4517446;
        double r4517448 = r4517436 * r4517426;
        double r4517449 = r4517438 * r4517423;
        double r4517450 = r4517448 - r4517449;
        double r4517451 = r4517447 * r4517450;
        double r4517452 = r4517442 + r4517451;
        double r4517453 = r4517420 * r4517431;
        double r4517454 = r4517417 * r4517433;
        double r4517455 = r4517453 - r4517454;
        double r4517456 = y4;
        double r4517457 = r4517456 * r4517424;
        double r4517458 = y5;
        double r4517459 = r4517458 * r4517427;
        double r4517460 = r4517457 - r4517459;
        double r4517461 = r4517455 * r4517460;
        double r4517462 = r4517452 + r4517461;
        double r4517463 = r4517420 * r4517443;
        double r4517464 = r4517417 * r4517445;
        double r4517465 = r4517463 - r4517464;
        double r4517466 = r4517456 * r4517426;
        double r4517467 = r4517458 * r4517423;
        double r4517468 = r4517466 - r4517467;
        double r4517469 = r4517465 * r4517468;
        double r4517470 = r4517462 - r4517469;
        double r4517471 = r4517433 * r4517443;
        double r4517472 = r4517431 * r4517445;
        double r4517473 = r4517471 - r4517472;
        double r4517474 = r4517456 * r4517438;
        double r4517475 = r4517458 * r4517436;
        double r4517476 = r4517474 - r4517475;
        double r4517477 = r4517473 * r4517476;
        double r4517478 = r4517470 + r4517477;
        return r4517478;
}

↓

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 r4517479 = a;
        double r4517480 = -0.00798469123475429;
        bool r4517481 = r4517479 <= r4517480;
        double r4517482 = c;
        double r4517483 = y0;
        double r4517484 = r4517482 * r4517483;
        double r4517485 = y1;
        double r4517486 = r4517479 * r4517485;
        double r4517487 = r4517484 - r4517486;
        double r4517488 = x;
        double r4517489 = y2;
        double r4517490 = r4517488 * r4517489;
        double r4517491 = y3;
        double r4517492 = z;
        double r4517493 = r4517491 * r4517492;
        double r4517494 = r4517490 - r4517493;
        double r4517495 = r4517487 * r4517494;
        double r4517496 = y;
        double r4517497 = r4517488 * r4517496;
        double r4517498 = t;
        double r4517499 = r4517498 * r4517492;
        double r4517500 = r4517497 - r4517499;
        double r4517501 = b;
        double r4517502 = r4517479 * r4517501;
        double r4517503 = i;
        double r4517504 = r4517503 * r4517482;
        double r4517505 = r4517502 - r4517504;
        double r4517506 = r4517500 * r4517505;
        double r4517507 = k;
        double r4517508 = r4517507 * r4517485;
        double r4517509 = r4517492 * r4517508;
        double r4517510 = r4517503 * r4517509;
        double r4517511 = r4517488 * r4517485;
        double r4517512 = j;
        double r4517513 = r4517511 * r4517512;
        double r4517514 = r4517513 * r4517503;
        double r4517515 = r4517501 * r4517483;
        double r4517516 = r4517515 * r4517492;
        double r4517517 = r4517516 * r4517507;
        double r4517518 = r4517514 + r4517517;
        double r4517519 = r4517510 - r4517518;
        double r4517520 = r4517506 - r4517519;
        double r4517521 = r4517495 + r4517520;
        double r4517522 = y4;
        double r4517523 = r4517522 * r4517501;
        double r4517524 = y5;
        double r4517525 = r4517503 * r4517524;
        double r4517526 = r4517523 - r4517525;
        double r4517527 = r4517512 * r4517498;
        double r4517528 = r4517496 * r4517507;
        double r4517529 = r4517527 - r4517528;
        double r4517530 = r4517526 * r4517529;
        double r4517531 = r4517521 + r4517530;
        double r4517532 = r4517522 * r4517482;
        double r4517533 = r4517524 * r4517479;
        double r4517534 = r4517532 - r4517533;
        double r4517535 = r4517498 * r4517489;
        double r4517536 = r4517491 * r4517496;
        double r4517537 = r4517535 - r4517536;
        double r4517538 = r4517534 * r4517537;
        double r4517539 = r4517531 - r4517538;
        double r4517540 = r4517489 * r4517507;
        double r4517541 = r4517491 * r4517512;
        double r4517542 = r4517540 - r4517541;
        double r4517543 = r4517485 * r4517522;
        double r4517544 = r4517524 * r4517483;
        double r4517545 = r4517543 - r4517544;
        double r4517546 = r4517542 * r4517545;
        double r4517547 = r4517539 + r4517546;
        double r4517548 = -2.0073425297372163e-147;
        bool r4517549 = r4517479 <= r4517548;
        double r4517550 = r4517488 * r4517512;
        double r4517551 = r4517507 * r4517492;
        double r4517552 = r4517550 - r4517551;
        double r4517553 = r4517485 * r4517503;
        double r4517554 = r4517515 - r4517553;
        double r4517555 = -r4517554;
        double r4517556 = r4517552 * r4517555;
        double r4517557 = r4517556 + r4517495;
        double r4517558 = r4517530 + r4517557;
        double r4517559 = r4517558 - r4517538;
        double r4517560 = r4517559 + r4517546;
        double r4517561 = r4517552 * r4517554;
        double r4517562 = r4517506 - r4517561;
        double r4517563 = r4517562 + r4517495;
        double r4517564 = r4517563 + r4517530;
        double r4517565 = cbrt(r4517538);
        double r4517566 = cbrt(r4517534);
        double r4517567 = r4517566 * r4517566;
        double r4517568 = r4517567 * r4517537;
        double r4517569 = r4517568 * r4517566;
        double r4517570 = cbrt(r4517569);
        double r4517571 = r4517565 * r4517570;
        double r4517572 = r4517565 * r4517571;
        double r4517573 = r4517564 - r4517572;
        double r4517574 = r4517573 + r4517546;
        double r4517575 = r4517549 ? r4517560 : r4517574;
        double r4517576 = r4517481 ? r4517547 : r4517575;
        return r4517576;
}

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)))))