Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.6 → 11.3

Time: 23.3s

Precision: 64

Math

\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;c \le -1.911661663231193133152425417101608962071 \cdot 10^{-99}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \left(t \cdot j\right)\right) \cdot \sqrt[3]{c} + \left(-i \cdot \left(y \cdot j\right)\right)\right)\\ \mathbf{elif}\;c \le 3.281737004707663056029424521453613958063 \cdot 10^{145}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;c \le 2.662534141559160429989964984960950739743 \cdot 10^{222}:\\ \;\;\;\;\left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(y \cdot j\right)\right)\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;c \le 4.731713770906781313248457864535070930729 \cdot 10^{243}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(y \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-\sqrt[3]{{\left(i \cdot \left(y \cdot j\right)\right)}^{3}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r458418 = x;
        double r458419 = y;
        double r458420 = z;
        double r458421 = r458419 * r458420;
        double r458422 = t;
        double r458423 = a;
        double r458424 = r458422 * r458423;
        double r458425 = r458421 - r458424;
        double r458426 = r458418 * r458425;
        double r458427 = b;
        double r458428 = c;
        double r458429 = r458428 * r458420;
        double r458430 = i;
        double r458431 = r458430 * r458423;
        double r458432 = r458429 - r458431;
        double r458433 = r458427 * r458432;
        double r458434 = r458426 - r458433;
        double r458435 = j;
        double r458436 = r458428 * r458422;
        double r458437 = r458430 * r458419;
        double r458438 = r458436 - r458437;
        double r458439 = r458435 * r458438;
        double r458440 = r458434 + r458439;
        return r458440;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r458441 = c;
        double r458442 = -1.911661663231193e-99;
        bool r458443 = r458441 <= r458442;
        double r458444 = x;
        double r458445 = y;
        double r458446 = z;
        double r458447 = r458445 * r458446;
        double r458448 = t;
        double r458449 = a;
        double r458450 = r458448 * r458449;
        double r458451 = r458447 - r458450;
        double r458452 = r458444 * r458451;
        double r458453 = b;
        double r458454 = r458441 * r458446;
        double r458455 = i;
        double r458456 = r458455 * r458449;
        double r458457 = r458454 - r458456;
        double r458458 = r458453 * r458457;
        double r458459 = r458452 - r458458;
        double r458460 = cbrt(r458441);
        double r458461 = r458460 * r458460;
        double r458462 = j;
        double r458463 = r458448 * r458462;
        double r458464 = r458461 * r458463;
        double r458465 = r458464 * r458460;
        double r458466 = r458445 * r458462;
        double r458467 = r458455 * r458466;
        double r458468 = -r458467;
        double r458469 = r458465 + r458468;
        double r458470 = r458459 + r458469;
        double r458471 = 3.281737004707663e+145;
        bool r458472 = r458441 <= r458471;
        double r458473 = cbrt(r458458);
        double r458474 = r458473 * r458473;
        double r458475 = r458474 * r458473;
        double r458476 = r458452 - r458475;
        double r458477 = r458441 * r458448;
        double r458478 = r458455 * r458445;
        double r458479 = r458477 - r458478;
        double r458480 = r458462 * r458479;
        double r458481 = r458476 + r458480;
        double r458482 = 2.6625341415591604e+222;
        bool r458483 = r458441 <= r458482;
        double r458484 = r458463 * r458441;
        double r458485 = r458484 + r458468;
        double r458486 = -r458458;
        double r458487 = r458485 + r458486;
        double r458488 = 4.731713770906781e+243;
        bool r458489 = r458441 <= r458488;
        double r458490 = r458453 * r458441;
        double r458491 = r458446 * r458490;
        double r458492 = -r458456;
        double r458493 = r458492 * r458453;
        double r458494 = r458491 + r458493;
        double r458495 = r458452 - r458494;
        double r458496 = r458462 * r458441;
        double r458497 = r458448 * r458496;
        double r458498 = r458497 + r458468;
        double r458499 = r458495 + r458498;
        double r458500 = 3.0;
        double r458501 = pow(r458467, r458500);
        double r458502 = cbrt(r458501);
        double r458503 = -r458502;
        double r458504 = r458484 + r458503;
        double r458505 = r458459 + r458504;
        double r458506 = r458489 ? r458499 : r458505;
        double r458507 = r458483 ? r458487 : r458506;
        double r458508 = r458472 ? r458481 : r458507;
        double r458509 = r458443 ? r458470 : r458508;
        return r458509;
}

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

Target

Original	11.6
Target	15.6
Herbie	11.3

\[\begin{array}{l} \mathbf{if}\;t \lt -8.12097891919591218149793027759825150959 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.712553818218485141757938537793350881052 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.633533346031583686060259351057142920433 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.053588855745548710002760210539645467715 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if c < -1.911661663231193e-99

   1. Initial program 13.5

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   2. Using strategy rm

   3. Applied sub-neg 13.5

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]

   4. Applied distribute-lft-in 13.5

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]

   5. Simplified 13.9

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{t \cdot \left(j \cdot c\right)} + j \cdot \left(-i \cdot y\right)\right)\]
   6. Using strategy rm

   7. Applied distribute-rgt-neg-out 13.9

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \color{blue}{\left(-j \cdot \left(i \cdot y\right)\right)}\right)\]

   8. Simplified 13.4

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\color{blue}{i \cdot \left(y \cdot j\right)}\right)\right)\]
   9. Using strategy rm

   10. Applied associate-*r* 11.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(t \cdot j\right) \cdot c} + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 12.0

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot \color{blue}{\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}\right)} + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   13. Applied associate-*r* 12.0

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(\left(t \cdot j\right) \cdot \left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right)\right) \cdot \sqrt[3]{c}} + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   14. Simplified 12.0

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \left(t \cdot j\right)\right)} \cdot \sqrt[3]{c} + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   if -1.911661663231193e-99 < c < 3.281737004707663e+145

   1. Initial program 9.3

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 9.6

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(\sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   if 3.281737004707663e+145 < c < 2.6625341415591604e+222

   1. Initial program 21.8

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   2. Using strategy rm

   3. Applied sub-neg 21.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]

   4. Applied distribute-lft-in 21.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]

   5. Simplified 24.7

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{t \cdot \left(j \cdot c\right)} + j \cdot \left(-i \cdot y\right)\right)\]
   6. Using strategy rm

   7. Applied distribute-rgt-neg-out 24.7

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \color{blue}{\left(-j \cdot \left(i \cdot y\right)\right)}\right)\]

   8. Simplified 22.8

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\color{blue}{i \cdot \left(y \cdot j\right)}\right)\right)\]
   9. Using strategy rm

   10. Applied associate-*r* 15.2

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(t \cdot j\right) \cdot c} + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   11. Taylor expanded around 0 23.6

       \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   if 2.6625341415591604e+222 < c < 4.731713770906781e+243

   1. Initial program 26.1

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   2. Using strategy rm

   3. Applied sub-neg 26.1

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]

   4. Applied distribute-lft-in 26.1

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]

   5. Simplified 22.1

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{t \cdot \left(j \cdot c\right)} + j \cdot \left(-i \cdot y\right)\right)\]
   6. Using strategy rm

   7. Applied distribute-rgt-neg-out 22.1

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \color{blue}{\left(-j \cdot \left(i \cdot y\right)\right)}\right)\]

   8. Simplified 20.5

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\color{blue}{i \cdot \left(y \cdot j\right)}\right)\right)\]
   9. Using strategy rm

   10. Applied sub-neg 20.5

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z + \left(-i \cdot a\right)\right)}\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   11. Applied distribute-lft-in 20.5

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{\left(b \cdot \left(c \cdot z\right) + b \cdot \left(-i \cdot a\right)\right)}\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   12. Simplified 18.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\color{blue}{z \cdot \left(b \cdot c\right)} + b \cdot \left(-i \cdot a\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   13. Simplified 18.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{\left(-i \cdot a\right) \cdot b}\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]

   if 4.731713770906781e+243 < c

   1. Initial program 21.6

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   2. Using strategy rm

   3. Applied sub-neg 21.6

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \color{blue}{\left(c \cdot t + \left(-i \cdot y\right)\right)}\]

   4. Applied distribute-lft-in 21.6

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(j \cdot \left(c \cdot t\right) + j \cdot \left(-i \cdot y\right)\right)}\]

   5. Simplified 24.4

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{t \cdot \left(j \cdot c\right)} + j \cdot \left(-i \cdot y\right)\right)\]
   6. Using strategy rm

   7. Applied distribute-rgt-neg-out 24.4

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \color{blue}{\left(-j \cdot \left(i \cdot y\right)\right)}\right)\]

   8. Simplified 22.3

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\color{blue}{i \cdot \left(y \cdot j\right)}\right)\right)\]
   9. Using strategy rm

   10. Applied associate-*r* 13.6

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\color{blue}{\left(t \cdot j\right) \cdot c} + \left(-i \cdot \left(y \cdot j\right)\right)\right)\]
   11. Using strategy rm

   12. Applied add-cbrt-cube 19.6

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(y \cdot \color{blue}{\sqrt[3]{\left(j \cdot j\right) \cdot j}}\right)\right)\right)\]

   13. Applied add-cbrt-cube 32.0

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(\color{blue}{\sqrt[3]{\left(y \cdot y\right) \cdot y}} \cdot \sqrt[3]{\left(j \cdot j\right) \cdot j}\right)\right)\right)\]

   14. Applied cbrt-unprod 32.5

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \color{blue}{\sqrt[3]{\left(\left(y \cdot y\right) \cdot y\right) \cdot \left(\left(j \cdot j\right) \cdot j\right)}}\right)\right)\]

   15. Applied add-cbrt-cube 40.1

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-\color{blue}{\sqrt[3]{\left(i \cdot i\right) \cdot i}} \cdot \sqrt[3]{\left(\left(y \cdot y\right) \cdot y\right) \cdot \left(\left(j \cdot j\right) \cdot j\right)}\right)\right)\]

   16. Applied cbrt-unprod 40.5

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-\color{blue}{\sqrt[3]{\left(\left(i \cdot i\right) \cdot i\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot \left(\left(j \cdot j\right) \cdot j\right)\right)}}\right)\right)\]

   17. Simplified 19.1

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-\sqrt[3]{\color{blue}{{\left(i \cdot \left(y \cdot j\right)\right)}^{3}}}\right)\right)\]
3. Recombined 5 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.911661663231193133152425417101608962071 \cdot 10^{-99}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \left(t \cdot j\right)\right) \cdot \sqrt[3]{c} + \left(-i \cdot \left(y \cdot j\right)\right)\right)\\ \mathbf{elif}\;c \le 3.281737004707663056029424521453613958063 \cdot 10^{145}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;c \le 2.662534141559160429989964984960950739743 \cdot 10^{222}:\\ \;\;\;\;\left(\left(t \cdot j\right) \cdot c + \left(-i \cdot \left(y \cdot j\right)\right)\right) + \left(-b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;c \le 4.731713770906781313248457864535070930729 \cdot 10^{243}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(y \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(t \cdot j\right) \cdot c + \left(-\sqrt[3]{{\left(i \cdot \left(y \cdot j\right)\right)}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.1209789191959122e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.7125538182184851e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.63353334603158369e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))