Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.2 → 10.6

Time: 29.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4053319781872422142178495508913324032:\\ \;\;\;\;\left(c \cdot \left(t \cdot j\right) + \left(-\sqrt[3]{y}\right) \cdot \left(\sqrt[3]{\left(i \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot j} \cdot \left(\sqrt[3]{\left(i \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot j} \cdot \sqrt[3]{\left(i \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot j}\right)\right)\right) + \left(\mathsf{fma}\left(a \cdot x, -t, x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;b \le -8.403626242362648477213397681914174771732 \cdot 10^{-198}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(b \cdot \left(c \cdot z\right) + \left(-a\right) \cdot \left(i \cdot b\right)\right)\right) + \left(c \cdot t - y \cdot i\right) \cdot j\\ \mathbf{elif}\;b \le -4.687249819554670663506382100649210607577 \cdot 10^{-238}:\\ \;\;\;\;\left(y \cdot \left(i \cdot \left(-j\right)\right) + c \cdot \left(t \cdot j\right)\right) + \left(y \cdot z - a \cdot t\right) \cdot x\\ \mathbf{elif}\;b \le 2.880247879572147946040031720732651251947 \cdot 10^{-95}:\\ \;\;\;\;\left(c \cdot \left(t \cdot j\right) + \left(\left(i \cdot j\right) \cdot \left(\sqrt[3]{y} \cdot \left(-\sqrt[3]{y}\right)\right)\right) \cdot \sqrt[3]{y}\right) + \left(\mathsf{fma}\left(a \cdot x, -t, x \cdot \left(y \cdot z\right)\right) - \left(c \cdot \left(b \cdot z\right) + \left(-i\right) \cdot \left(a \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(y \cdot \left(x \cdot z\right) + \left(a \cdot \left(-t\right)\right) \cdot x\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r86094 = x;
        double r86095 = y;
        double r86096 = z;
        double r86097 = r86095 * r86096;
        double r86098 = t;
        double r86099 = a;
        double r86100 = r86098 * r86099;
        double r86101 = r86097 - r86100;
        double r86102 = r86094 * r86101;
        double r86103 = b;
        double r86104 = c;
        double r86105 = r86104 * r86096;
        double r86106 = i;
        double r86107 = r86106 * r86099;
        double r86108 = r86105 - r86107;
        double r86109 = r86103 * r86108;
        double r86110 = r86102 - r86109;
        double r86111 = j;
        double r86112 = r86104 * r86098;
        double r86113 = r86106 * r86095;
        double r86114 = r86112 - r86113;
        double r86115 = r86111 * r86114;
        double r86116 = r86110 + r86115;
        return r86116;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r86117 = b;
        double r86118 = -4.053319781872422e+36;
        bool r86119 = r86117 <= r86118;
        double r86120 = c;
        double r86121 = t;
        double r86122 = j;
        double r86123 = r86121 * r86122;
        double r86124 = r86120 * r86123;
        double r86125 = y;
        double r86126 = cbrt(r86125);
        double r86127 = -r86126;
        double r86128 = i;
        double r86129 = r86126 * r86126;
        double r86130 = r86128 * r86129;
        double r86131 = r86130 * r86122;
        double r86132 = cbrt(r86131);
        double r86133 = r86132 * r86132;
        double r86134 = r86132 * r86133;
        double r86135 = r86127 * r86134;
        double r86136 = r86124 + r86135;
        double r86137 = a;
        double r86138 = x;
        double r86139 = r86137 * r86138;
        double r86140 = -r86121;
        double r86141 = z;
        double r86142 = r86125 * r86141;
        double r86143 = r86138 * r86142;
        double r86144 = fma(r86139, r86140, r86143);
        double r86145 = r86120 * r86141;
        double r86146 = r86128 * r86137;
        double r86147 = r86145 - r86146;
        double r86148 = r86117 * r86147;
        double r86149 = r86144 - r86148;
        double r86150 = r86136 + r86149;
        double r86151 = -8.403626242362648e-198;
        bool r86152 = r86117 <= r86151;
        double r86153 = r86137 * r86121;
        double r86154 = r86142 - r86153;
        double r86155 = r86154 * r86138;
        double r86156 = r86117 * r86145;
        double r86157 = -r86137;
        double r86158 = r86128 * r86117;
        double r86159 = r86157 * r86158;
        double r86160 = r86156 + r86159;
        double r86161 = r86155 - r86160;
        double r86162 = r86120 * r86121;
        double r86163 = r86125 * r86128;
        double r86164 = r86162 - r86163;
        double r86165 = r86164 * r86122;
        double r86166 = r86161 + r86165;
        double r86167 = -4.687249819554671e-238;
        bool r86168 = r86117 <= r86167;
        double r86169 = -r86122;
        double r86170 = r86128 * r86169;
        double r86171 = r86125 * r86170;
        double r86172 = r86171 + r86124;
        double r86173 = r86172 + r86155;
        double r86174 = 2.880247879572148e-95;
        bool r86175 = r86117 <= r86174;
        double r86176 = r86128 * r86122;
        double r86177 = r86126 * r86127;
        double r86178 = r86176 * r86177;
        double r86179 = r86178 * r86126;
        double r86180 = r86124 + r86179;
        double r86181 = r86117 * r86141;
        double r86182 = r86120 * r86181;
        double r86183 = -r86128;
        double r86184 = r86137 * r86117;
        double r86185 = r86183 * r86184;
        double r86186 = r86182 + r86185;
        double r86187 = r86144 - r86186;
        double r86188 = r86180 + r86187;
        double r86189 = r86138 * r86141;
        double r86190 = r86125 * r86189;
        double r86191 = r86137 * r86140;
        double r86192 = r86191 * r86138;
        double r86193 = r86190 + r86192;
        double r86194 = r86193 - r86148;
        double r86195 = r86165 + r86194;
        double r86196 = r86175 ? r86188 : r86195;
        double r86197 = r86168 ? r86173 : r86196;
        double r86198 = r86152 ? r86166 : r86197;
        double r86199 = r86119 ? r86150 : r86198;
        return r86199;
}

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 < -4.053319781872422e+36

   1. Initial program 7.2

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

   3. Applied sub-neg 7.2

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

   4. Applied distribute-lft-in 7.2

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

   5. Simplified 8.2

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

   6. Simplified 8.6

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

   7. Taylor expanded around inf 9.5

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

   8. Simplified 8.7

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

   10. Applied add-cube-cbrt 8.8

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

   11. Applied distribute-lft-neg-in 8.8

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

   12. Applied associate-*r* 8.8

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

   13. Simplified 8.8

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

   15. Applied add-cube-cbrt 8.8

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

   16. Simplified 9.4

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

   17. Simplified 8.0

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

   if -4.053319781872422e+36 < b < -8.403626242362648e-198

   1. Initial program 12.7

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

   3. Applied sub-neg 12.7

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

   4. Applied distribute-lft-in 12.7

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

   5. Simplified 11.1

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

   if -8.403626242362648e-198 < b < -4.687249819554671e-238

   1. Initial program 19.4

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

   3. Applied sub-neg 19.4

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

   4. Applied distribute-lft-in 19.4

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

   5. Simplified 20.2

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

   6. Simplified 19.9

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

   7. Taylor expanded around 0 21.4

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

   if -4.687249819554671e-238 < b < 2.880247879572148e-95

   1. Initial program 17.0

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

   3. Applied sub-neg 17.0

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

   4. Applied distribute-lft-in 17.0

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

   5. Simplified 17.4

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

   6. Simplified 17.3

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

   7. Taylor expanded around inf 17.8

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

   8. Simplified 17.7

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

   10. Applied add-cube-cbrt 17.9

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

   11. Applied distribute-lft-neg-in 17.9

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

   12. Applied associate-*r* 17.9

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

   13. Simplified 17.9

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

   15. Applied sub-neg 17.9

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

   16. Applied distribute-lft-in 17.9

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

   17. Simplified 14.6

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

   18. Simplified 11.7

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

   if 2.880247879572148e-95 < b

   1. Initial program 8.6

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

   3. Applied sub-neg 8.6

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

   4. Applied distribute-lft-in 8.6

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

   5. Simplified 8.9

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

   6. Simplified 8.9

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4053319781872422142178495508913324032:\\ \;\;\;\;\left(c \cdot \left(t \cdot j\right) + \left(-\sqrt[3]{y}\right) \cdot \left(\sqrt[3]{\left(i \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot j} \cdot \left(\sqrt[3]{\left(i \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot j} \cdot \sqrt[3]{\left(i \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot j}\right)\right)\right) + \left(\mathsf{fma}\left(a \cdot x, -t, x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \mathbf{elif}\;b \le -8.403626242362648477213397681914174771732 \cdot 10^{-198}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(b \cdot \left(c \cdot z\right) + \left(-a\right) \cdot \left(i \cdot b\right)\right)\right) + \left(c \cdot t - y \cdot i\right) \cdot j\\ \mathbf{elif}\;b \le -4.687249819554670663506382100649210607577 \cdot 10^{-238}:\\ \;\;\;\;\left(y \cdot \left(i \cdot \left(-j\right)\right) + c \cdot \left(t \cdot j\right)\right) + \left(y \cdot z - a \cdot t\right) \cdot x\\ \mathbf{elif}\;b \le 2.880247879572147946040031720732651251947 \cdot 10^{-95}:\\ \;\;\;\;\left(c \cdot \left(t \cdot j\right) + \left(\left(i \cdot j\right) \cdot \left(\sqrt[3]{y} \cdot \left(-\sqrt[3]{y}\right)\right)\right) \cdot \sqrt[3]{y}\right) + \left(\mathsf{fma}\left(a \cdot x, -t, x \cdot \left(y \cdot z\right)\right) - \left(c \cdot \left(b \cdot z\right) + \left(-i\right) \cdot \left(a \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot t - y \cdot i\right) \cdot j + \left(\left(y \cdot \left(x \cdot z\right) + \left(a \cdot \left(-t\right)\right) \cdot x\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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)))))