Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.9 → 10.7

Time: 36.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.8132182994707265 \cdot 10^{172}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{elif}\;t \le -4.3441848100341449 \cdot 10^{-82}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;t \le -2.0482729984780924 \cdot 10^{-118}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{elif}\;t \le -3.4045282169254156 \cdot 10^{-188}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(-a \cdot \left(i \cdot b\right)\right) + z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;t \le 8.3618244494638386 \cdot 10^{21}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r82785 = x;
        double r82786 = y;
        double r82787 = z;
        double r82788 = r82786 * r82787;
        double r82789 = t;
        double r82790 = a;
        double r82791 = r82789 * r82790;
        double r82792 = r82788 - r82791;
        double r82793 = r82785 * r82792;
        double r82794 = b;
        double r82795 = c;
        double r82796 = r82795 * r82787;
        double r82797 = i;
        double r82798 = r82797 * r82790;
        double r82799 = r82796 - r82798;
        double r82800 = r82794 * r82799;
        double r82801 = r82793 - r82800;
        double r82802 = j;
        double r82803 = r82795 * r82789;
        double r82804 = r82797 * r82786;
        double r82805 = r82803 - r82804;
        double r82806 = r82802 * r82805;
        double r82807 = r82801 + r82806;
        return r82807;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r82808 = t;
        double r82809 = -1.8132182994707265e+172;
        bool r82810 = r82808 <= r82809;
        double r82811 = x;
        double r82812 = y;
        double r82813 = z;
        double r82814 = r82812 * r82813;
        double r82815 = a;
        double r82816 = r82808 * r82815;
        double r82817 = r82814 - r82816;
        double r82818 = r82811 * r82817;
        double r82819 = b;
        double r82820 = r82813 * r82819;
        double r82821 = c;
        double r82822 = r82820 * r82821;
        double r82823 = i;
        double r82824 = -r82815;
        double r82825 = r82824 * r82819;
        double r82826 = r82823 * r82825;
        double r82827 = r82822 + r82826;
        double r82828 = r82818 - r82827;
        double r82829 = j;
        double r82830 = r82829 * r82821;
        double r82831 = r82808 * r82830;
        double r82832 = r82829 * r82812;
        double r82833 = r82823 * r82832;
        double r82834 = r82831 - r82833;
        double r82835 = r82828 + r82834;
        double r82836 = -4.344184810034145e-82;
        bool r82837 = r82808 <= r82836;
        double r82838 = r82813 * r82812;
        double r82839 = r82811 * r82838;
        double r82840 = r82811 * r82808;
        double r82841 = r82815 * r82840;
        double r82842 = r82839 - r82841;
        double r82843 = r82842 - r82827;
        double r82844 = r82821 * r82808;
        double r82845 = r82823 * r82812;
        double r82846 = r82844 - r82845;
        double r82847 = r82829 * r82846;
        double r82848 = r82843 + r82847;
        double r82849 = -2.0482729984780924e-118;
        bool r82850 = r82808 <= r82849;
        double r82851 = -3.4045282169254156e-188;
        bool r82852 = r82808 <= r82851;
        double r82853 = r82823 * r82819;
        double r82854 = r82815 * r82853;
        double r82855 = -r82854;
        double r82856 = r82819 * r82821;
        double r82857 = r82813 * r82856;
        double r82858 = r82855 + r82857;
        double r82859 = r82818 - r82858;
        double r82860 = r82859 + r82847;
        double r82861 = 8.361824449463839e+21;
        bool r82862 = r82808 <= r82861;
        double r82863 = r82862 ? r82848 : r82835;
        double r82864 = r82852 ? r82860 : r82863;
        double r82865 = r82850 ? r82835 : r82864;
        double r82866 = r82837 ? r82848 : r82865;
        double r82867 = r82810 ? r82835 : r82866;
        return r82867;
}

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 t < -1.8132182994707265e+172 or -4.344184810034145e-82 < t < -2.0482729984780924e-118 or 8.361824449463839e+21 < t

   1. Initial program 18.0

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

   3. Applied sub-neg 18.0

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

   4. Applied distribute-lft-in 18.0

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

   5. Simplified 18.2

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

   6. Simplified 18.2

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

   8. Applied distribute-rgt-neg-in 18.2

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

   9. Applied associate-*l* 18.2

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

   11. Applied associate-*r* 19.3

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

   12. Taylor expanded around inf 14.0

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

   if -1.8132182994707265e+172 < t < -4.344184810034145e-82 or -3.4045282169254156e-188 < t < 8.361824449463839e+21

   1. Initial program 9.7

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

   3. Applied sub-neg 9.7

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

   4. Applied distribute-lft-in 9.7

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

   5. Simplified 10.2

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

   6. Simplified 10.2

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

   8. Applied distribute-rgt-neg-in 10.2

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

   9. Applied associate-*l* 10.0

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

   11. Applied associate-*r* 9.5

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

   12. Taylor expanded around inf 9.6

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

   if -2.0482729984780924e-118 < t < -3.4045282169254156e-188

   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

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

   4. Applied distribute-lft-in 8.6

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

   5. Simplified 8.7

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

   6. Simplified 8.7

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

   8. Applied distribute-rgt-neg-in 8.7

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

   9. Applied associate-*l* 8.8

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

   11. Applied pow1 8.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + i \cdot \left(\left(-a\right) \cdot \color{blue}{{b}^{1}}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   12. Applied pow1 8.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + i \cdot \left(\color{blue}{{\left(-a\right)}^{1}} \cdot {b}^{1}\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   13. Applied pow-prod-down 8.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + i \cdot \color{blue}{{\left(\left(-a\right) \cdot b\right)}^{1}}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   14. Applied pow1 8.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{{i}^{1}} \cdot {\left(\left(-a\right) \cdot b\right)}^{1}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   15. Applied pow-prod-down 8.8

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{{\left(i \cdot \left(\left(-a\right) \cdot b\right)\right)}^{1}}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   16. Simplified 7.9

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + {\color{blue}{\left(-a \cdot \left(i \cdot b\right)\right)}}^{1}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.8132182994707265 \cdot 10^{172}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{elif}\;t \le -4.3441848100341449 \cdot 10^{-82}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;t \le -2.0482729984780924 \cdot 10^{-118}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{elif}\;t \le -3.4045282169254156 \cdot 10^{-188}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(-a \cdot \left(i \cdot b\right)\right) + z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;t \le 8.3618244494638386 \cdot 10^{21}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + i \cdot \left(\left(-a\right) \cdot b\right)\right)\right) + \left(t \cdot \left(j \cdot c\right) - i \cdot \left(j \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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)))))