Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 11.9 → 10.5

Time: 8.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.0454813922804694 \cdot 10^{83}:\\ \;\;\;\;\left(0 - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le 4.9067894261714203 \cdot 10^{-20}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{\left(b \cdot c\right) \cdot z} \cdot \sqrt[3]{\left(b \cdot c\right) \cdot z}\right) \cdot \sqrt[3]{\left(b \cdot c\right) \cdot z} + \left(b \cdot t\right) \cdot \left(-i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r837835 = x;
        double r837836 = y;
        double r837837 = z;
        double r837838 = r837836 * r837837;
        double r837839 = t;
        double r837840 = a;
        double r837841 = r837839 * r837840;
        double r837842 = r837838 - r837841;
        double r837843 = r837835 * r837842;
        double r837844 = b;
        double r837845 = c;
        double r837846 = r837845 * r837837;
        double r837847 = i;
        double r837848 = r837839 * r837847;
        double r837849 = r837846 - r837848;
        double r837850 = r837844 * r837849;
        double r837851 = r837843 - r837850;
        double r837852 = j;
        double r837853 = r837845 * r837840;
        double r837854 = r837836 * r837847;
        double r837855 = r837853 - r837854;
        double r837856 = r837852 * r837855;
        double r837857 = r837851 + r837856;
        return r837857;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r837858 = b;
        double r837859 = -1.0454813922804694e+83;
        bool r837860 = r837858 <= r837859;
        double r837861 = 0.0;
        double r837862 = c;
        double r837863 = z;
        double r837864 = r837862 * r837863;
        double r837865 = r837858 * r837864;
        double r837866 = t;
        double r837867 = i;
        double r837868 = r837866 * r837867;
        double r837869 = -r837868;
        double r837870 = r837858 * r837869;
        double r837871 = r837865 + r837870;
        double r837872 = r837861 - r837871;
        double r837873 = j;
        double r837874 = a;
        double r837875 = r837862 * r837874;
        double r837876 = y;
        double r837877 = r837876 * r837867;
        double r837878 = r837875 - r837877;
        double r837879 = r837873 * r837878;
        double r837880 = r837872 + r837879;
        double r837881 = 4.90678942617142e-20;
        bool r837882 = r837858 <= r837881;
        double r837883 = x;
        double r837884 = r837876 * r837863;
        double r837885 = r837866 * r837874;
        double r837886 = r837884 - r837885;
        double r837887 = r837883 * r837886;
        double r837888 = r837858 * r837862;
        double r837889 = r837888 * r837863;
        double r837890 = cbrt(r837889);
        double r837891 = r837890 * r837890;
        double r837892 = r837891 * r837890;
        double r837893 = r837858 * r837866;
        double r837894 = -r837867;
        double r837895 = r837893 * r837894;
        double r837896 = r837892 + r837895;
        double r837897 = r837887 - r837896;
        double r837898 = r837897 + r837879;
        double r837899 = cbrt(r837886);
        double r837900 = r837899 * r837899;
        double r837901 = r837883 * r837900;
        double r837902 = r837901 * r837899;
        double r837903 = r837902 - r837871;
        double r837904 = r837903 + r837879;
        double r837905 = r837882 ? r837898 : r837904;
        double r837906 = r837860 ? r837880 : r837905;
        return r837906;
}

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

Target

Original	11.9
Target	19.8
Herbie	10.5

\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -1.0454813922804694e+83

   1. Initial program 6.9

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

   3. Applied sub-neg 6.9

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

   4. Applied distribute-lft-in 6.9

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

   5. Taylor expanded around 0 17.9

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

   if -1.0454813922804694e+83 < b < 4.90678942617142e-20

   1. Initial program 14.1

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

   3. Applied sub-neg 14.1

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

   4. Applied distribute-lft-in 14.1

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

   6. Applied distribute-rgt-neg-in 14.1

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

   7. Applied associate-*r* 12.1

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

   9. Applied associate-*r* 9.9

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

   11. Applied add-cube-cbrt 10.0

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

   if 4.90678942617142e-20 < b

   1. Initial program 7.7

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

   3. Applied sub-neg 7.7

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

   4. Applied distribute-lft-in 7.7

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

   6. Applied add-cube-cbrt 7.9

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

   7. Applied associate-*r* 7.9

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0454813922804694 \cdot 10^{83}:\\ \;\;\;\;\left(0 - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;b \le 4.9067894261714203 \cdot 10^{-20}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{\left(b \cdot c\right) \cdot z} \cdot \sqrt[3]{\left(b \cdot c\right) \cdot z}\right) \cdot \sqrt[3]{\left(b \cdot c\right) \cdot z} + \left(b \cdot t\right) \cdot \left(-i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - \left(b \cdot \left(c \cdot z\right) + b \cdot \left(-t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))