Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 12.1 → 11.0

Time: 27.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.43573431022587006539678668819771121502 \cdot 10^{-55}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y\right) \cdot \left(i \cdot j\right)\right)\\ \mathbf{elif}\;b \le -3.247942119715740045247268518672381664306 \cdot 10^{-183}:\\ \;\;\;\;\left(\left(\left(-t \cdot \left(x \cdot a\right)\right) + x \cdot \left(z \cdot y\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le -1.795486041934680815088450859297908251741 \cdot 10^{-305}:\\ \;\;\;\;\left(a \cdot \left(j \cdot c\right) + \left(-y\right) \cdot \left(i \cdot j\right)\right) + \left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{elif}\;b \le 2.370330019284661706130848263045197366618 \cdot 10^{-276}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;b \le 40241249613545970408093533024125341138940:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y\right) \cdot \left(i \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r595429 = x;
        double r595430 = y;
        double r595431 = z;
        double r595432 = r595430 * r595431;
        double r595433 = t;
        double r595434 = a;
        double r595435 = r595433 * r595434;
        double r595436 = r595432 - r595435;
        double r595437 = r595429 * r595436;
        double r595438 = b;
        double r595439 = c;
        double r595440 = r595439 * r595431;
        double r595441 = i;
        double r595442 = r595433 * r595441;
        double r595443 = r595440 - r595442;
        double r595444 = r595438 * r595443;
        double r595445 = r595437 - r595444;
        double r595446 = j;
        double r595447 = r595439 * r595434;
        double r595448 = r595430 * r595441;
        double r595449 = r595447 - r595448;
        double r595450 = r595446 * r595449;
        double r595451 = r595445 + r595450;
        return r595451;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r595452 = b;
        double r595453 = -2.43573431022587e-55;
        bool r595454 = r595452 <= r595453;
        double r595455 = x;
        double r595456 = z;
        double r595457 = r595455 * r595456;
        double r595458 = y;
        double r595459 = r595457 * r595458;
        double r595460 = a;
        double r595461 = t;
        double r595462 = r595455 * r595461;
        double r595463 = r595460 * r595462;
        double r595464 = -r595463;
        double r595465 = r595459 + r595464;
        double r595466 = c;
        double r595467 = r595466 * r595456;
        double r595468 = i;
        double r595469 = r595461 * r595468;
        double r595470 = r595467 - r595469;
        double r595471 = r595452 * r595470;
        double r595472 = r595465 - r595471;
        double r595473 = j;
        double r595474 = r595473 * r595466;
        double r595475 = r595460 * r595474;
        double r595476 = -r595458;
        double r595477 = r595468 * r595473;
        double r595478 = r595476 * r595477;
        double r595479 = r595475 + r595478;
        double r595480 = r595472 + r595479;
        double r595481 = -3.24794211971574e-183;
        bool r595482 = r595452 <= r595481;
        double r595483 = r595455 * r595460;
        double r595484 = r595461 * r595483;
        double r595485 = -r595484;
        double r595486 = r595456 * r595458;
        double r595487 = r595455 * r595486;
        double r595488 = r595485 + r595487;
        double r595489 = r595488 - r595471;
        double r595490 = r595466 * r595460;
        double r595491 = r595458 * r595468;
        double r595492 = r595490 - r595491;
        double r595493 = r595473 * r595492;
        double r595494 = r595489 + r595493;
        double r595495 = -1.7954860419346808e-305;
        bool r595496 = r595452 <= r595495;
        double r595497 = r595487 + r595464;
        double r595498 = r595479 + r595497;
        double r595499 = 2.3703300192846617e-276;
        bool r595500 = r595452 <= r595499;
        double r595501 = r595458 * r595456;
        double r595502 = r595461 * r595460;
        double r595503 = r595501 - r595502;
        double r595504 = r595455 * r595503;
        double r595505 = r595493 + r595504;
        double r595506 = 4.024124961354597e+40;
        bool r595507 = r595452 <= r595506;
        double r595508 = r595452 * r595466;
        double r595509 = r595456 * r595508;
        double r595510 = -r595469;
        double r595511 = r595510 * r595452;
        double r595512 = r595509 + r595511;
        double r595513 = r595497 - r595512;
        double r595514 = r595513 + r595479;
        double r595515 = -r595491;
        double r595516 = r595515 * r595473;
        double r595517 = r595475 + r595516;
        double r595518 = r595472 + r595517;
        double r595519 = r595507 ? r595514 : r595518;
        double r595520 = r595500 ? r595505 : r595519;
        double r595521 = r595496 ? r595498 : r595520;
        double r595522 = r595482 ? r595494 : r595521;
        double r595523 = r595454 ? r595480 : r595522;
        return r595523;
}

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	12.1
Target	19.7
Herbie	11.0

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 6 regimes

2. if b < -2.43573431022587e-55

   1. Initial program 8.6

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

   3. Applied sub-neg 8.6

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

   4. Applied distribute-lft-in 8.6

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

   5. Simplified 8.6

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

   6. Simplified 8.9

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

   8. Applied sub-neg 8.9

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

   9. Applied distribute-lft-in 8.9

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

   10. Simplified 8.4

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

   11. Simplified 8.4

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

   13. Applied distribute-lft-neg-in 8.4

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

   14. Applied associate-*l* 8.2

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

   16. Applied associate-*r* 7.6

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

   if -2.43573431022587e-55 < b < -3.24794211971574e-183

   1. Initial program 14.6

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

   3. Applied sub-neg 14.6

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

   4. Applied distribute-lft-in 14.6

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

   5. Simplified 14.6

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

   6. Simplified 15.3

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

   8. Applied *-un-lft-identity 15.3

      \[\leadsto \left(\left(x \cdot \left(z \cdot y\right) + \left(-\color{blue}{\left(1 \cdot a\right)} \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   9. Applied associate-*l* 15.3

      \[\leadsto \left(\left(x \cdot \left(z \cdot y\right) + \left(-\color{blue}{1 \cdot \left(a \cdot \left(x \cdot t\right)\right)}\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   10. Simplified 14.6

       \[\leadsto \left(\left(x \cdot \left(z \cdot y\right) + \left(-1 \cdot \color{blue}{\left(t \cdot \left(x \cdot a\right)\right)}\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   if -3.24794211971574e-183 < b < -1.7954860419346808e-305

   1. Initial program 16.6

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

   3. Applied sub-neg 16.6

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

   4. Applied distribute-lft-in 16.6

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

   5. Simplified 16.6

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

   6. Simplified 16.7

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

   8. Applied sub-neg 16.7

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

   9. Applied distribute-lft-in 16.7

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

   10. Simplified 15.9

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

   11. Simplified 15.9

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

   13. Applied distribute-lft-neg-in 15.9

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

   14. Applied associate-*l* 16.5

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

   15. Taylor expanded around 0 17.3

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

   if -1.7954860419346808e-305 < b < 2.3703300192846617e-276

   1. Initial program 20.5

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

   2. Taylor expanded around 0 16.1

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

   if 2.3703300192846617e-276 < b < 4.024124961354597e+40

   1. Initial program 13.7

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

   3. Applied sub-neg 13.7

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

   4. Applied distribute-lft-in 13.7

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

   5. Simplified 13.7

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

   6. Simplified 13.8

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

   8. Applied sub-neg 13.8

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

   9. Applied distribute-lft-in 13.8

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

   10. Simplified 13.7

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

   11. Simplified 13.7

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

   13. Applied distribute-lft-neg-in 13.7

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

   14. Applied associate-*l* 13.0

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

   16. Applied sub-neg 13.0

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

   17. Applied distribute-lft-in 13.0

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

   18. Simplified 11.0

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

   19. Simplified 11.0

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

   if 4.024124961354597e+40 < b

   1. Initial program 7.1

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

   3. Applied sub-neg 7.1

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

   4. Applied distribute-lft-in 7.1

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

   5. Simplified 7.1

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

   6. Simplified 7.0

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

   8. Applied sub-neg 7.0

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

   9. Applied distribute-lft-in 7.0

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

   10. Simplified 6.9

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

   11. Simplified 6.9

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

   13. Applied associate-*r* 6.9

       \[\leadsto \left(\left(\color{blue}{\left(x \cdot z\right) \cdot y} + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\]
3. Recombined 6 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.43573431022587006539678668819771121502 \cdot 10^{-55}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y\right) \cdot \left(i \cdot j\right)\right)\\ \mathbf{elif}\;b \le -3.247942119715740045247268518672381664306 \cdot 10^{-183}:\\ \;\;\;\;\left(\left(\left(-t \cdot \left(x \cdot a\right)\right) + x \cdot \left(z \cdot y\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le -1.795486041934680815088450859297908251741 \cdot 10^{-305}:\\ \;\;\;\;\left(a \cdot \left(j \cdot c\right) + \left(-y\right) \cdot \left(i \cdot j\right)\right) + \left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{elif}\;b \le 2.370330019284661706130848263045197366618 \cdot 10^{-276}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + x \cdot \left(y \cdot z - t \cdot a\right)\\ \mathbf{elif}\;b \le 40241249613545970408093533024125341138940:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y\right) \cdot \left(i \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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