Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 9.8

Time: 24.5s

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}\;b \le -5.40768492793686555941761981699772927749 \cdot 10^{-39}:\\ \;\;\;\;\left(j \cdot \left(t \cdot c - y \cdot i\right) + \left(y \cdot \left(x \cdot z\right) - t \cdot \left(x \cdot a\right)\right)\right) + \left(a \cdot i - c \cdot z\right) \cdot b\\ \mathbf{elif}\;b \le -1.310708033188256852286497632350011018004 \cdot 10^{-167}:\\ \;\;\;\;\left(\left(j \cdot \left(-y \cdot i\right) + \left(j \cdot c\right) \cdot t\right) + \left(y \cdot \left(x \cdot z\right) - \left(t \cdot x\right) \cdot a\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le -5.939141620508838471044724084871683968029 \cdot 10^{-276}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(t \cdot x\right) \cdot a\right) + j \cdot \left(t \cdot c - y \cdot i\right)\right) + \left(\sqrt[3]{\left(i \cdot b\right) \cdot a} \cdot \left(\sqrt[3]{\left(i \cdot b\right) \cdot a} \cdot \sqrt[3]{\left(i \cdot b\right) \cdot a}\right) - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le 2.248851513589900759233480267753595172809 \cdot 10^{-207}:\\ \;\;\;\;\left(\left(j \cdot \left(-y \cdot i\right) + \left(j \cdot c\right) \cdot t\right) + \left(y \cdot \left(x \cdot z\right) - \left(t \cdot x\right) \cdot a\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le 1.659927328976305946610149570336434936816 \cdot 10^{118}:\\ \;\;\;\;\left(j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z\right) - \sqrt[3]{a} \cdot \left(t \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot x\right)\right)\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot i - c \cdot z\right) \cdot b + \left(\left(\left(j \cdot c\right) \cdot t + y \cdot \left(\left(-i\right) \cdot j\right)\right) + \left(y \cdot z - t \cdot a\right) \cdot x\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 r88679 = x;
        double r88680 = y;
        double r88681 = z;
        double r88682 = r88680 * r88681;
        double r88683 = t;
        double r88684 = a;
        double r88685 = r88683 * r88684;
        double r88686 = r88682 - r88685;
        double r88687 = r88679 * r88686;
        double r88688 = b;
        double r88689 = c;
        double r88690 = r88689 * r88681;
        double r88691 = i;
        double r88692 = r88691 * r88684;
        double r88693 = r88690 - r88692;
        double r88694 = r88688 * r88693;
        double r88695 = r88687 - r88694;
        double r88696 = j;
        double r88697 = r88689 * r88683;
        double r88698 = r88691 * r88680;
        double r88699 = r88697 - r88698;
        double r88700 = r88696 * r88699;
        double r88701 = r88695 + r88700;
        return r88701;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r88702 = b;
        double r88703 = -5.4076849279368656e-39;
        bool r88704 = r88702 <= r88703;
        double r88705 = j;
        double r88706 = t;
        double r88707 = c;
        double r88708 = r88706 * r88707;
        double r88709 = y;
        double r88710 = i;
        double r88711 = r88709 * r88710;
        double r88712 = r88708 - r88711;
        double r88713 = r88705 * r88712;
        double r88714 = x;
        double r88715 = z;
        double r88716 = r88714 * r88715;
        double r88717 = r88709 * r88716;
        double r88718 = a;
        double r88719 = r88714 * r88718;
        double r88720 = r88706 * r88719;
        double r88721 = r88717 - r88720;
        double r88722 = r88713 + r88721;
        double r88723 = r88718 * r88710;
        double r88724 = r88707 * r88715;
        double r88725 = r88723 - r88724;
        double r88726 = r88725 * r88702;
        double r88727 = r88722 + r88726;
        double r88728 = -1.3107080331882569e-167;
        bool r88729 = r88702 <= r88728;
        double r88730 = -r88711;
        double r88731 = r88705 * r88730;
        double r88732 = r88705 * r88707;
        double r88733 = r88732 * r88706;
        double r88734 = r88731 + r88733;
        double r88735 = r88706 * r88714;
        double r88736 = r88735 * r88718;
        double r88737 = r88717 - r88736;
        double r88738 = r88734 + r88737;
        double r88739 = r88710 * r88702;
        double r88740 = r88739 * r88718;
        double r88741 = r88702 * r88715;
        double r88742 = r88741 * r88707;
        double r88743 = r88740 - r88742;
        double r88744 = r88738 + r88743;
        double r88745 = -5.9391416205088385e-276;
        bool r88746 = r88702 <= r88745;
        double r88747 = r88709 * r88715;
        double r88748 = r88714 * r88747;
        double r88749 = r88748 - r88736;
        double r88750 = r88749 + r88713;
        double r88751 = cbrt(r88740);
        double r88752 = r88751 * r88751;
        double r88753 = r88751 * r88752;
        double r88754 = r88753 - r88742;
        double r88755 = r88750 + r88754;
        double r88756 = 2.2488515135899008e-207;
        bool r88757 = r88702 <= r88756;
        double r88758 = 1.659927328976306e+118;
        bool r88759 = r88702 <= r88758;
        double r88760 = cbrt(r88718);
        double r88761 = r88760 * r88760;
        double r88762 = r88761 * r88714;
        double r88763 = r88706 * r88762;
        double r88764 = r88760 * r88763;
        double r88765 = r88748 - r88764;
        double r88766 = r88713 + r88765;
        double r88767 = r88766 + r88743;
        double r88768 = -r88710;
        double r88769 = r88768 * r88705;
        double r88770 = r88709 * r88769;
        double r88771 = r88733 + r88770;
        double r88772 = r88706 * r88718;
        double r88773 = r88747 - r88772;
        double r88774 = r88773 * r88714;
        double r88775 = r88771 + r88774;
        double r88776 = r88726 + r88775;
        double r88777 = r88759 ? r88767 : r88776;
        double r88778 = r88757 ? r88744 : r88777;
        double r88779 = r88746 ? r88755 : r88778;
        double r88780 = r88729 ? r88744 : r88779;
        double r88781 = r88704 ? r88727 : r88780;
        return r88781;
}

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

Derivation

1. Split input into 5 regimes

2. if b < -5.4076849279368656e-39

   1. Initial program 8.0

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

   2. Simplified 8.0

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

   3. Taylor expanded around inf 9.4

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

   4. Simplified 8.2

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

   if -5.4076849279368656e-39 < b < -1.3107080331882569e-167 or -5.9391416205088385e-276 < b < 2.2488515135899008e-207

   1. Initial program 15.8

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

   2. Simplified 15.8

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

   3. Taylor expanded around inf 10.5

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

   4. Simplified 10.4

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

   5. Taylor expanded around inf 10.2

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

   6. Simplified 10.2

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

   8. Applied sub-neg 10.2

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

   9. Applied distribute-lft-in 10.2

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

   10. Simplified 10.9

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

   11. Simplified 10.9

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

   13. Applied associate-*r* 10.5

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

   if -1.3107080331882569e-167 < b < -5.9391416205088385e-276

   1. Initial program 17.3

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

   2. Simplified 17.3

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

   3. Taylor expanded around inf 10.9

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

   4. Simplified 11.0

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

   5. Taylor expanded around inf 10.0

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

   6. Simplified 10.0

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

   8. Applied add-cube-cbrt 10.1

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

   9. Simplified 10.1

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

   10. Simplified 10.1

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

   if 2.2488515135899008e-207 < b < 1.659927328976306e+118

   1. Initial program 12.9

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

   2. Simplified 12.9

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

   3. Taylor expanded around inf 11.2

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

   4. Simplified 11.3

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

   5. Taylor expanded around inf 11.2

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

   6. Simplified 11.2

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

   8. Applied add-cube-cbrt 11.4

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

   9. Applied associate-*r* 11.4

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

   10. Simplified 11.3

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

   if 1.659927328976306e+118 < b

   1. Initial program 6.0

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

   2. Simplified 6.0

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

   4. Applied sub-neg 6.0

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

   5. Applied distribute-lft-in 6.0

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

   6. Simplified 6.5

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

   7. Simplified 6.6

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.40768492793686555941761981699772927749 \cdot 10^{-39}:\\ \;\;\;\;\left(j \cdot \left(t \cdot c - y \cdot i\right) + \left(y \cdot \left(x \cdot z\right) - t \cdot \left(x \cdot a\right)\right)\right) + \left(a \cdot i - c \cdot z\right) \cdot b\\ \mathbf{elif}\;b \le -1.310708033188256852286497632350011018004 \cdot 10^{-167}:\\ \;\;\;\;\left(\left(j \cdot \left(-y \cdot i\right) + \left(j \cdot c\right) \cdot t\right) + \left(y \cdot \left(x \cdot z\right) - \left(t \cdot x\right) \cdot a\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le -5.939141620508838471044724084871683968029 \cdot 10^{-276}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(t \cdot x\right) \cdot a\right) + j \cdot \left(t \cdot c - y \cdot i\right)\right) + \left(\sqrt[3]{\left(i \cdot b\right) \cdot a} \cdot \left(\sqrt[3]{\left(i \cdot b\right) \cdot a} \cdot \sqrt[3]{\left(i \cdot b\right) \cdot a}\right) - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le 2.248851513589900759233480267753595172809 \cdot 10^{-207}:\\ \;\;\;\;\left(\left(j \cdot \left(-y \cdot i\right) + \left(j \cdot c\right) \cdot t\right) + \left(y \cdot \left(x \cdot z\right) - \left(t \cdot x\right) \cdot a\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{elif}\;b \le 1.659927328976305946610149570336434936816 \cdot 10^{118}:\\ \;\;\;\;\left(j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z\right) - \sqrt[3]{a} \cdot \left(t \cdot \left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot x\right)\right)\right)\right) + \left(\left(i \cdot b\right) \cdot a - \left(b \cdot z\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot i - c \cdot z\right) \cdot b + \left(\left(\left(j \cdot c\right) \cdot t + y \cdot \left(\left(-i\right) \cdot j\right)\right) + \left(y \cdot z - t \cdot a\right) \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))