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

Average Error: 11.7 → 12.0

Time: 10.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.6042623790340836 \cdot 10^{-142} \lor \neg \left(x \le 8.47609977682779403 \cdot 10^{-187}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\left(\sqrt[3]{\sqrt[3]{c \cdot z - t \cdot i} \cdot \sqrt[3]{c \cdot z - t \cdot i}} \cdot \sqrt[3]{\sqrt[3]{c \cdot z - t \cdot i}}\right) \cdot \sqrt[3]{c \cdot z - t \cdot i}\right)\right) \cdot \sqrt[3]{c \cdot z - t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r741846 = x;
        double r741847 = y;
        double r741848 = z;
        double r741849 = r741847 * r741848;
        double r741850 = t;
        double r741851 = a;
        double r741852 = r741850 * r741851;
        double r741853 = r741849 - r741852;
        double r741854 = r741846 * r741853;
        double r741855 = b;
        double r741856 = c;
        double r741857 = r741856 * r741848;
        double r741858 = i;
        double r741859 = r741850 * r741858;
        double r741860 = r741857 - r741859;
        double r741861 = r741855 * r741860;
        double r741862 = r741854 - r741861;
        double r741863 = j;
        double r741864 = r741856 * r741851;
        double r741865 = r741847 * r741858;
        double r741866 = r741864 - r741865;
        double r741867 = r741863 * r741866;
        double r741868 = r741862 + r741867;
        return r741868;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r741869 = x;
        double r741870 = -9.604262379034084e-142;
        bool r741871 = r741869 <= r741870;
        double r741872 = 8.476099776827794e-187;
        bool r741873 = r741869 <= r741872;
        double r741874 = !r741873;
        bool r741875 = r741871 || r741874;
        double r741876 = y;
        double r741877 = z;
        double r741878 = r741876 * r741877;
        double r741879 = t;
        double r741880 = a;
        double r741881 = r741879 * r741880;
        double r741882 = r741878 - r741881;
        double r741883 = r741869 * r741882;
        double r741884 = b;
        double r741885 = c;
        double r741886 = r741885 * r741877;
        double r741887 = i;
        double r741888 = r741879 * r741887;
        double r741889 = r741886 - r741888;
        double r741890 = cbrt(r741889);
        double r741891 = r741890 * r741890;
        double r741892 = cbrt(r741891);
        double r741893 = cbrt(r741890);
        double r741894 = r741892 * r741893;
        double r741895 = r741894 * r741890;
        double r741896 = r741884 * r741895;
        double r741897 = r741896 * r741890;
        double r741898 = r741883 - r741897;
        double r741899 = j;
        double r741900 = r741885 * r741880;
        double r741901 = r741876 * r741887;
        double r741902 = r741900 - r741901;
        double r741903 = r741899 * r741902;
        double r741904 = r741898 + r741903;
        double r741905 = 0.0;
        double r741906 = r741869 * r741905;
        double r741907 = r741884 * r741889;
        double r741908 = r741906 - r741907;
        double r741909 = r741908 + r741903;
        double r741910 = r741875 ? r741904 : r741909;
        return r741910;
}

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	11.7
Target	19.0
Herbie	12.0

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

Derivation

1. Split input into 2 regimes

2. if x < -9.604262379034084e-142 or 8.476099776827794e-187 < x

   1. Initial program 9.5

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

   3. Applied add-cube-cbrt 9.8

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

   4. Applied associate-*r* 9.8

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

   6. Applied add-cube-cbrt 9.8

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

   7. Applied cbrt-prod 9.8

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

   if -9.604262379034084e-142 < x < 8.476099776827794e-187

   1. Initial program 16.8

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

   2. Taylor expanded around 0 17.3

      \[\leadsto \left(x \cdot \color{blue}{0} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.6042623790340836 \cdot 10^{-142} \lor \neg \left(x \le 8.47609977682779403 \cdot 10^{-187}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(b \cdot \left(\left(\sqrt[3]{\sqrt[3]{c \cdot z - t \cdot i} \cdot \sqrt[3]{c \cdot z - t \cdot i}} \cdot \sqrt[3]{\sqrt[3]{c \cdot z - t \cdot i}}\right) \cdot \sqrt[3]{c \cdot z - t \cdot i}\right)\right) \cdot \sqrt[3]{c \cdot z - t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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)))))