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

Average Error: 11.7 → 9.8

Time: 29.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.075828090768532 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(a \cdot x\right)\right)\\ \mathbf{elif}\;z \le 9.741177042073514 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(t \cdot i - c \cdot z, b, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r37342893 = x;
        double r37342894 = y;
        double r37342895 = z;
        double r37342896 = r37342894 * r37342895;
        double r37342897 = t;
        double r37342898 = a;
        double r37342899 = r37342897 * r37342898;
        double r37342900 = r37342896 - r37342899;
        double r37342901 = r37342893 * r37342900;
        double r37342902 = b;
        double r37342903 = c;
        double r37342904 = r37342903 * r37342895;
        double r37342905 = i;
        double r37342906 = r37342897 * r37342905;
        double r37342907 = r37342904 - r37342906;
        double r37342908 = r37342902 * r37342907;
        double r37342909 = r37342901 - r37342908;
        double r37342910 = j;
        double r37342911 = r37342903 * r37342898;
        double r37342912 = r37342894 * r37342905;
        double r37342913 = r37342911 - r37342912;
        double r37342914 = r37342910 * r37342913;
        double r37342915 = r37342909 + r37342914;
        return r37342915;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r37342916 = z;
        double r37342917 = -8.075828090768532e+82;
        bool r37342918 = r37342916 <= r37342917;
        double r37342919 = a;
        double r37342920 = c;
        double r37342921 = r37342919 * r37342920;
        double r37342922 = y;
        double r37342923 = i;
        double r37342924 = r37342922 * r37342923;
        double r37342925 = r37342921 - r37342924;
        double r37342926 = j;
        double r37342927 = x;
        double r37342928 = r37342927 * r37342922;
        double r37342929 = b;
        double r37342930 = r37342920 * r37342929;
        double r37342931 = r37342928 - r37342930;
        double r37342932 = r37342931 * r37342916;
        double r37342933 = t;
        double r37342934 = r37342919 * r37342927;
        double r37342935 = r37342933 * r37342934;
        double r37342936 = r37342932 - r37342935;
        double r37342937 = fma(r37342925, r37342926, r37342936);
        double r37342938 = 9.741177042073514e+85;
        bool r37342939 = r37342916 <= r37342938;
        double r37342940 = r37342933 * r37342923;
        double r37342941 = r37342920 * r37342916;
        double r37342942 = r37342940 - r37342941;
        double r37342943 = r37342922 * r37342916;
        double r37342944 = r37342919 * r37342933;
        double r37342945 = r37342943 - r37342944;
        double r37342946 = r37342945 * r37342927;
        double r37342947 = fma(r37342942, r37342929, r37342946);
        double r37342948 = fma(r37342925, r37342926, r37342947);
        double r37342949 = r37342939 ? r37342948 : r37342937;
        double r37342950 = r37342918 ? r37342937 : r37342949;
        return r37342950;
}

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	18.4
Herbie	9.8

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -8.075828090768532e+82 or 9.741177042073514e+85 < z

   1. Initial program 18.8

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

   2. Simplified 18.8

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

   3. Taylor expanded around inf 18.8

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

   4. Taylor expanded around inf 17.9

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

   5. Simplified 11.2

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

   if -8.075828090768532e+82 < z < 9.741177042073514e+85

   1. Initial program 9.3

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

   2. Simplified 9.3

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

   3. Taylor expanded around inf 9.3

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.075828090768532 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(a \cdot x\right)\right)\\ \mathbf{elif}\;z \le 9.741177042073514 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(t \cdot i - c \cdot z, b, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(a \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :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)))))