Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.5 → 10.8

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.0398945104776677 \cdot 10^{41}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + 1 \cdot \left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;a \le 1.4419120624263323 \cdot 10^{57}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + 1 \cdot \left(-1 \cdot {\left(t \cdot \left(x \cdot a\right)\right)}^{1}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + 1 \cdot \left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r89862 = x;
        double r89863 = y;
        double r89864 = z;
        double r89865 = r89863 * r89864;
        double r89866 = t;
        double r89867 = a;
        double r89868 = r89866 * r89867;
        double r89869 = r89865 - r89868;
        double r89870 = r89862 * r89869;
        double r89871 = b;
        double r89872 = c;
        double r89873 = r89872 * r89864;
        double r89874 = i;
        double r89875 = r89874 * r89867;
        double r89876 = r89873 - r89875;
        double r89877 = r89871 * r89876;
        double r89878 = r89870 - r89877;
        double r89879 = j;
        double r89880 = r89872 * r89866;
        double r89881 = r89874 * r89863;
        double r89882 = r89880 - r89881;
        double r89883 = r89879 * r89882;
        double r89884 = r89878 + r89883;
        return r89884;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r89885 = a;
        double r89886 = -1.0398945104776677e+41;
        bool r89887 = r89885 <= r89886;
        double r89888 = x;
        double r89889 = y;
        double r89890 = r89888 * r89889;
        double r89891 = z;
        double r89892 = r89890 * r89891;
        double r89893 = 1.0;
        double r89894 = -1.0;
        double r89895 = t;
        double r89896 = r89888 * r89895;
        double r89897 = r89885 * r89896;
        double r89898 = r89894 * r89897;
        double r89899 = r89893 * r89898;
        double r89900 = r89892 + r89899;
        double r89901 = b;
        double r89902 = c;
        double r89903 = r89902 * r89891;
        double r89904 = i;
        double r89905 = r89904 * r89885;
        double r89906 = r89903 - r89905;
        double r89907 = r89901 * r89906;
        double r89908 = r89900 - r89907;
        double r89909 = j;
        double r89910 = cbrt(r89909);
        double r89911 = r89910 * r89910;
        double r89912 = r89902 * r89895;
        double r89913 = r89904 * r89889;
        double r89914 = r89912 - r89913;
        double r89915 = r89910 * r89914;
        double r89916 = r89911 * r89915;
        double r89917 = r89908 + r89916;
        double r89918 = 1.4419120624263323e+57;
        bool r89919 = r89885 <= r89918;
        double r89920 = r89888 * r89885;
        double r89921 = r89895 * r89920;
        double r89922 = pow(r89921, r89893);
        double r89923 = r89894 * r89922;
        double r89924 = r89893 * r89923;
        double r89925 = r89892 + r89924;
        double r89926 = r89925 - r89907;
        double r89927 = r89909 * r89914;
        double r89928 = r89926 + r89927;
        double r89929 = cbrt(r89927);
        double r89930 = r89929 * r89929;
        double r89931 = r89930 * r89929;
        double r89932 = r89908 + r89931;
        double r89933 = r89919 ? r89928 : r89932;
        double r89934 = r89887 ? r89917 : r89933;
        return r89934;
}

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 a < -1.0398945104776677e+41

   1. Initial program 17.6

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

   3. Applied sub-neg 17.6

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

   4. Applied distribute-lft-in 17.6

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

   6. Applied *-un-lft-identity 17.6

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

   7. Applied associate-*l* 17.6

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

   8. Simplified 13.3

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

   10. Applied associate-*r* 13.1

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

   12. Applied add-cube-cbrt 13.3

       \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + 1 \cdot \left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \sqrt[3]{j}\right)} \cdot \left(c \cdot t - i \cdot y\right)\]

   13. Applied associate-*l* 13.3

       \[\leadsto \left(\left(\left(x \cdot y\right) \cdot z + 1 \cdot \left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)}\]

   if -1.0398945104776677e+41 < a < 1.4419120624263323e+57

   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. Using strategy rm

   6. Applied *-un-lft-identity 8.6

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

   7. Applied associate-*l* 8.6

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

   8. Simplified 11.1

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

   10. Applied associate-*r* 11.6

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

   12. Applied pow1 11.6

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

   13. Applied pow1 11.6

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

   14. Applied pow-prod-down 11.6

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

   15. Applied pow1 11.6

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

   16. Applied pow-prod-down 11.6

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

   17. Simplified 9.6

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

   if 1.4419120624263323e+57 < a

   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

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

   4. Applied distribute-lft-in 18.0

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

   6. Applied *-un-lft-identity 18.0

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

   7. Applied associate-*l* 18.0

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

   8. Simplified 14.0

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

   10. Applied associate-*r* 13.5

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

   12. Applied add-cube-cbrt 13.7

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

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.0398945104776677 \cdot 10^{41}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + 1 \cdot \left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;a \le 1.4419120624263323 \cdot 10^{57}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + 1 \cdot \left(-1 \cdot {\left(t \cdot \left(x \cdot a\right)\right)}^{1}\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot y\right) \cdot z + 1 \cdot \left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot t - i \cdot y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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)))))