Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.8 → 11.5

Time: 33.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.8711162379413496 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(\left(z \cdot y\right) \cdot x + \left(-\left(x \cdot t\right) \cdot a\right)\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 1.4698479981983464 \cdot 10^{+66}:\\ \;\;\;\;\left(\left(z \cdot \left(y \cdot x\right) + \left(-\left(x \cdot t\right) \cdot a\right)\right) - \left(\left(\left(-a\right) \cdot i\right) \cdot b + b \cdot \left(c \cdot z\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 9.229327516273808 \cdot 10^{+187}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(z \cdot y - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-\left(x \cdot t\right) \cdot a\right) + y \cdot \left(z \cdot x\right)\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right) + \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r3346733 = x;
        double r3346734 = y;
        double r3346735 = z;
        double r3346736 = r3346734 * r3346735;
        double r3346737 = t;
        double r3346738 = a;
        double r3346739 = r3346737 * r3346738;
        double r3346740 = r3346736 - r3346739;
        double r3346741 = r3346733 * r3346740;
        double r3346742 = b;
        double r3346743 = c;
        double r3346744 = r3346743 * r3346735;
        double r3346745 = i;
        double r3346746 = r3346745 * r3346738;
        double r3346747 = r3346744 - r3346746;
        double r3346748 = r3346742 * r3346747;
        double r3346749 = r3346741 - r3346748;
        double r3346750 = j;
        double r3346751 = r3346743 * r3346737;
        double r3346752 = r3346745 * r3346734;
        double r3346753 = r3346751 - r3346752;
        double r3346754 = r3346750 * r3346753;
        double r3346755 = r3346749 + r3346754;
        return r3346755;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r3346756 = x;
        double r3346757 = -1.8711162379413496e+44;
        bool r3346758 = r3346756 <= r3346757;
        double r3346759 = z;
        double r3346760 = y;
        double r3346761 = r3346759 * r3346760;
        double r3346762 = r3346761 * r3346756;
        double r3346763 = t;
        double r3346764 = r3346756 * r3346763;
        double r3346765 = a;
        double r3346766 = r3346764 * r3346765;
        double r3346767 = -r3346766;
        double r3346768 = r3346762 + r3346767;
        double r3346769 = c;
        double r3346770 = r3346769 * r3346759;
        double r3346771 = i;
        double r3346772 = r3346771 * r3346765;
        double r3346773 = r3346770 - r3346772;
        double r3346774 = b;
        double r3346775 = r3346773 * r3346774;
        double r3346776 = r3346768 - r3346775;
        double r3346777 = j;
        double r3346778 = r3346769 * r3346763;
        double r3346779 = r3346771 * r3346760;
        double r3346780 = r3346778 - r3346779;
        double r3346781 = r3346777 * r3346780;
        double r3346782 = r3346776 + r3346781;
        double r3346783 = 1.4698479981983464e+66;
        bool r3346784 = r3346756 <= r3346783;
        double r3346785 = r3346760 * r3346756;
        double r3346786 = r3346759 * r3346785;
        double r3346787 = r3346786 + r3346767;
        double r3346788 = -r3346765;
        double r3346789 = r3346788 * r3346771;
        double r3346790 = r3346789 * r3346774;
        double r3346791 = r3346774 * r3346770;
        double r3346792 = r3346790 + r3346791;
        double r3346793 = r3346787 - r3346792;
        double r3346794 = r3346793 + r3346781;
        double r3346795 = 9.229327516273808e+187;
        bool r3346796 = r3346756 <= r3346795;
        double r3346797 = cbrt(r3346756);
        double r3346798 = r3346797 * r3346797;
        double r3346799 = r3346763 * r3346765;
        double r3346800 = r3346761 - r3346799;
        double r3346801 = r3346797 * r3346800;
        double r3346802 = r3346798 * r3346801;
        double r3346803 = r3346781 + r3346802;
        double r3346804 = r3346759 * r3346756;
        double r3346805 = r3346760 * r3346804;
        double r3346806 = r3346767 + r3346805;
        double r3346807 = r3346806 - r3346775;
        double r3346808 = cbrt(r3346781);
        double r3346809 = r3346808 * r3346808;
        double r3346810 = r3346808 * r3346809;
        double r3346811 = r3346807 + r3346810;
        double r3346812 = r3346796 ? r3346803 : r3346811;
        double r3346813 = r3346784 ? r3346794 : r3346812;
        double r3346814 = r3346758 ? r3346782 : r3346813;
        return r3346814;
}

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 4 regimes

2. if x < -1.8711162379413496e+44

   1. Initial program 7.1

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

   3. Applied add-cube-cbrt 7.6

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

   4. Applied associate-*l* 7.6

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

   6. Applied sub-neg 7.6

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

   7. Applied distribute-rgt-in 7.6

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

   8. Applied distribute-rgt-in 7.6

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

   9. Simplified 12.6

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

   10. Simplified 16.9

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

   11. Taylor expanded around -inf 11.9

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

   if -1.8711162379413496e+44 < x < 1.4698479981983464e+66

   1. Initial program 13.8

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

   3. Applied add-cube-cbrt 14.0

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

   4. Applied associate-*l* 14.0

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

   6. Applied sub-neg 14.0

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

   7. Applied distribute-rgt-in 14.0

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

   8. Applied distribute-rgt-in 14.0

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

   9. Simplified 12.0

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

   10. Simplified 9.7

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

   12. Applied sub-neg 9.7

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

   13. Applied distribute-lft-in 9.7

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

   if 1.4698479981983464e+66 < x < 9.229327516273808e+187

   1. Initial program 7.5

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

   3. Applied add-cube-cbrt 8.0

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

   4. Applied associate-*l* 8.0

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

   5. Taylor expanded around 0 19.7

      \[\leadsto \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \color{blue}{0}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   if 9.229327516273808e+187 < x

   1. Initial program 6.6

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

   3. Applied add-cube-cbrt 7.2

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

   4. Applied associate-*l* 7.2

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

   6. Applied sub-neg 7.2

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

   7. Applied distribute-rgt-in 7.2

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

   8. Applied distribute-rgt-in 7.2

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

   9. Simplified 15.0

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

   10. Simplified 21.6

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

   12. Applied associate-*r* 21.3

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

   14. Applied add-cube-cbrt 21.4

       \[\leadsto \left(\left(\left(z \cdot x\right) \cdot y + \left(\left(-x\right) \cdot t\right) \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.8711162379413496 \cdot 10^{+44}:\\ \;\;\;\;\left(\left(\left(z \cdot y\right) \cdot x + \left(-\left(x \cdot t\right) \cdot a\right)\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 1.4698479981983464 \cdot 10^{+66}:\\ \;\;\;\;\left(\left(z \cdot \left(y \cdot x\right) + \left(-\left(x \cdot t\right) \cdot a\right)\right) - \left(\left(\left(-a\right) \cdot i\right) \cdot b + b \cdot \left(c \cdot z\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 9.229327516273808 \cdot 10^{+187}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(z \cdot y - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-\left(x \cdot t\right) \cdot a\right) + y \cdot \left(z \cdot x\right)\right) - \left(c \cdot z - i \cdot a\right) \cdot b\right) + \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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)))))