Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.0 → 9.9

Time: 13.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.39963790607126303166727796248420519442 \cdot 10^{194}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \le 1.791240738306205660841110248460150700112 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\left(z \cdot b\right) \cdot c + \left(-a \cdot \left(i \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r206756 = x;
        double r206757 = y;
        double r206758 = z;
        double r206759 = r206757 * r206758;
        double r206760 = t;
        double r206761 = a;
        double r206762 = r206760 * r206761;
        double r206763 = r206759 - r206762;
        double r206764 = r206756 * r206763;
        double r206765 = b;
        double r206766 = c;
        double r206767 = r206766 * r206758;
        double r206768 = i;
        double r206769 = r206768 * r206761;
        double r206770 = r206767 - r206769;
        double r206771 = r206765 * r206770;
        double r206772 = r206764 - r206771;
        double r206773 = j;
        double r206774 = r206766 * r206760;
        double r206775 = r206768 * r206757;
        double r206776 = r206774 - r206775;
        double r206777 = r206773 * r206776;
        double r206778 = r206772 + r206777;
        return r206778;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r206779 = b;
        double r206780 = -4.399637906071263e+194;
        bool r206781 = r206779 <= r206780;
        double r206782 = c;
        double r206783 = t;
        double r206784 = r206782 * r206783;
        double r206785 = i;
        double r206786 = y;
        double r206787 = r206785 * r206786;
        double r206788 = r206784 - r206787;
        double r206789 = j;
        double r206790 = x;
        double r206791 = z;
        double r206792 = a;
        double r206793 = r206792 * r206783;
        double r206794 = -r206793;
        double r206795 = fma(r206786, r206791, r206794);
        double r206796 = r206790 * r206795;
        double r206797 = -r206792;
        double r206798 = fma(r206797, r206783, r206793);
        double r206799 = r206790 * r206798;
        double r206800 = r206796 + r206799;
        double r206801 = cbrt(r206779);
        double r206802 = r206801 * r206801;
        double r206803 = r206782 * r206791;
        double r206804 = r206785 * r206792;
        double r206805 = r206803 - r206804;
        double r206806 = r206801 * r206805;
        double r206807 = r206802 * r206806;
        double r206808 = r206800 - r206807;
        double r206809 = fma(r206788, r206789, r206808);
        double r206810 = 1.7912407383062057e-06;
        bool r206811 = r206779 <= r206810;
        double r206812 = r206791 * r206779;
        double r206813 = r206812 * r206782;
        double r206814 = r206785 * r206779;
        double r206815 = r206792 * r206814;
        double r206816 = -r206815;
        double r206817 = r206813 + r206816;
        double r206818 = r206800 - r206817;
        double r206819 = fma(r206788, r206789, r206818);
        double r206820 = sqrt(r206779);
        double r206821 = r206820 * r206805;
        double r206822 = r206820 * r206821;
        double r206823 = r206800 - r206822;
        double r206824 = fma(r206788, r206789, r206823);
        double r206825 = r206811 ? r206819 : r206824;
        double r206826 = r206781 ? r206809 : r206825;
        return r206826;
}

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 3 regimes

2. if b < -4.399637906071263e+194

   1. Initial program 5.7

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

   2. Simplified 5.7

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

   4. Applied prod-diff 5.7

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

   5. Applied distribute-lft-in 5.7

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

   7. Applied add-cube-cbrt 6.3

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

   8. Applied associate-*l* 6.3

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

   if -4.399637906071263e+194 < b < 1.7912407383062057e-06

   1. Initial program 13.7

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

   2. Simplified 13.7

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

   4. Applied prod-diff 13.7

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

   5. Applied distribute-lft-in 13.7

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

   7. Applied add-cube-cbrt 13.9

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

   8. Applied associate-*l* 13.9

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

   10. Applied sub-neg 13.9

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

   11. Applied distribute-lft-in 13.9

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

   12. Applied distribute-lft-in 13.9

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

   13. Simplified 12.6

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

   14. Simplified 10.8

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

   16. Applied associate-*r* 10.8

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

   if 1.7912407383062057e-06 < b

   1. Initial program 7.3

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

   2. Simplified 7.3

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

   4. Applied prod-diff 7.3

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

   5. Applied distribute-lft-in 7.3

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

   7. Applied add-sqr-sqrt 7.4

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

   8. Applied associate-*l* 7.4

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.39963790607126303166727796248420519442 \cdot 10^{194}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \le 1.791240738306205660841110248460150700112 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\left(z \cdot b\right) \cdot c + \left(-a \cdot \left(i \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - i \cdot a\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))