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

Average Error: 12.1 → 11.0

Time: 27.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.43573431022587006539678668819771121502 \cdot 10^{-55}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + y \cdot \left(-i \cdot j\right)\right)\\ \mathbf{elif}\;b \le -3.247942119715740045247268518672381664306 \cdot 10^{-183}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le -1.795486041934680815088450859297908251741 \cdot 10^{-305}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) + \left(a \cdot \left(j \cdot c\right) + y \cdot \left(-i \cdot j\right)\right)\\ \mathbf{elif}\;b \le 2.370330019284661706130848263045197366618 \cdot 10^{-276}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot 0\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le 40241249613545970408093533024125341138940:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + \left(a \cdot \left(j \cdot c\right) + y \cdot \left(-i \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r531703 = x;
        double r531704 = y;
        double r531705 = z;
        double r531706 = r531704 * r531705;
        double r531707 = t;
        double r531708 = a;
        double r531709 = r531707 * r531708;
        double r531710 = r531706 - r531709;
        double r531711 = r531703 * r531710;
        double r531712 = b;
        double r531713 = c;
        double r531714 = r531713 * r531705;
        double r531715 = i;
        double r531716 = r531707 * r531715;
        double r531717 = r531714 - r531716;
        double r531718 = r531712 * r531717;
        double r531719 = r531711 - r531718;
        double r531720 = j;
        double r531721 = r531713 * r531708;
        double r531722 = r531704 * r531715;
        double r531723 = r531721 - r531722;
        double r531724 = r531720 * r531723;
        double r531725 = r531719 + r531724;
        return r531725;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r531726 = b;
        double r531727 = -2.43573431022587e-55;
        bool r531728 = r531726 <= r531727;
        double r531729 = x;
        double r531730 = z;
        double r531731 = r531729 * r531730;
        double r531732 = y;
        double r531733 = r531731 * r531732;
        double r531734 = a;
        double r531735 = t;
        double r531736 = r531729 * r531735;
        double r531737 = r531734 * r531736;
        double r531738 = -r531737;
        double r531739 = r531733 + r531738;
        double r531740 = c;
        double r531741 = r531740 * r531730;
        double r531742 = i;
        double r531743 = r531735 * r531742;
        double r531744 = r531741 - r531743;
        double r531745 = r531726 * r531744;
        double r531746 = r531739 - r531745;
        double r531747 = j;
        double r531748 = r531747 * r531740;
        double r531749 = r531734 * r531748;
        double r531750 = r531742 * r531747;
        double r531751 = -r531750;
        double r531752 = r531732 * r531751;
        double r531753 = r531749 + r531752;
        double r531754 = r531746 + r531753;
        double r531755 = -3.24794211971574e-183;
        bool r531756 = r531726 <= r531755;
        double r531757 = r531730 * r531732;
        double r531758 = r531729 * r531757;
        double r531759 = r531729 * r531734;
        double r531760 = r531735 * r531759;
        double r531761 = -r531760;
        double r531762 = r531758 + r531761;
        double r531763 = r531762 - r531745;
        double r531764 = r531740 * r531734;
        double r531765 = r531732 * r531742;
        double r531766 = r531764 - r531765;
        double r531767 = r531747 * r531766;
        double r531768 = r531763 + r531767;
        double r531769 = -1.7954860419346808e-305;
        bool r531770 = r531726 <= r531769;
        double r531771 = r531758 + r531738;
        double r531772 = r531771 + r531753;
        double r531773 = 2.3703300192846617e-276;
        bool r531774 = r531726 <= r531773;
        double r531775 = r531732 * r531730;
        double r531776 = r531735 * r531734;
        double r531777 = r531775 - r531776;
        double r531778 = r531729 * r531777;
        double r531779 = 0.0;
        double r531780 = r531726 * r531779;
        double r531781 = r531778 - r531780;
        double r531782 = r531781 + r531767;
        double r531783 = 4.024124961354597e+40;
        bool r531784 = r531726 <= r531783;
        double r531785 = r531726 * r531740;
        double r531786 = r531730 * r531785;
        double r531787 = -r531743;
        double r531788 = r531787 * r531726;
        double r531789 = r531786 + r531788;
        double r531790 = r531771 - r531789;
        double r531791 = r531790 + r531753;
        double r531792 = -r531765;
        double r531793 = r531792 * r531747;
        double r531794 = r531749 + r531793;
        double r531795 = r531746 + r531794;
        double r531796 = r531784 ? r531791 : r531795;
        double r531797 = r531774 ? r531782 : r531796;
        double r531798 = r531770 ? r531772 : r531797;
        double r531799 = r531756 ? r531768 : r531798;
        double r531800 = r531728 ? r531754 : r531799;
        return r531800;
}

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-rgt-neg-in 8.4

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

   14. Applied associate-*l* 8.2

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

   15. Simplified 8.2

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

   17. Applied associate-*r* 7.6

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

   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. Taylor expanded around inf 14.6

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

   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-rgt-neg-in 15.9

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

   14. Applied associate-*l* 16.5

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

   15. Simplified 16.5

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

   16. Taylor expanded around 0 17.3

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

   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

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

   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-rgt-neg-in 13.7

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

   14. Applied associate-*l* 13.0

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

   15. Simplified 13.0

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

   17. Applied sub-neg 13.0

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

   18. Applied distribute-lft-in 13.0

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

   19. Simplified 11.0

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

   20. Simplified 11.0

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.43573431022587006539678668819771121502 \cdot 10^{-55}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + y \cdot \left(-i \cdot j\right)\right)\\ \mathbf{elif}\;b \le -3.247942119715740045247268518672381664306 \cdot 10^{-183}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le -1.795486041934680815088450859297908251741 \cdot 10^{-305}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) + \left(a \cdot \left(j \cdot c\right) + y \cdot \left(-i \cdot j\right)\right)\\ \mathbf{elif}\;b \le 2.370330019284661706130848263045197366618 \cdot 10^{-276}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot 0\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le 40241249613545970408093533024125341138940:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot i\right) \cdot b\right)\right) + \left(a \cdot \left(j \cdot c\right) + y \cdot \left(-i \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\\ \end{array}\]

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)))))