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

Average Error: 11.3 → 9.3

Time: 28.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.4675246779790518 \cdot 10^{+72}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right)\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)}\\ \mathbf{elif}\;b \le 1.6011043676188824 \cdot 10^{-149}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(i \cdot \left(t \cdot \left(-b\right)\right) + \left(c \cdot b\right) \cdot z\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x - a \cdot \left(x \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r39531816 = x;
        double r39531817 = y;
        double r39531818 = z;
        double r39531819 = r39531817 * r39531818;
        double r39531820 = t;
        double r39531821 = a;
        double r39531822 = r39531820 * r39531821;
        double r39531823 = r39531819 - r39531822;
        double r39531824 = r39531816 * r39531823;
        double r39531825 = b;
        double r39531826 = c;
        double r39531827 = r39531826 * r39531818;
        double r39531828 = i;
        double r39531829 = r39531820 * r39531828;
        double r39531830 = r39531827 - r39531829;
        double r39531831 = r39531825 * r39531830;
        double r39531832 = r39531824 - r39531831;
        double r39531833 = j;
        double r39531834 = r39531826 * r39531821;
        double r39531835 = r39531817 * r39531828;
        double r39531836 = r39531834 - r39531835;
        double r39531837 = r39531833 * r39531836;
        double r39531838 = r39531832 + r39531837;
        return r39531838;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r39531839 = b;
        double r39531840 = -3.4675246779790518e+72;
        bool r39531841 = r39531839 <= r39531840;
        double r39531842 = y;
        double r39531843 = z;
        double r39531844 = r39531842 * r39531843;
        double r39531845 = t;
        double r39531846 = a;
        double r39531847 = r39531845 * r39531846;
        double r39531848 = r39531844 - r39531847;
        double r39531849 = x;
        double r39531850 = r39531848 * r39531849;
        double r39531851 = c;
        double r39531852 = r39531851 * r39531843;
        double r39531853 = i;
        double r39531854 = r39531853 * r39531845;
        double r39531855 = r39531852 - r39531854;
        double r39531856 = r39531839 * r39531855;
        double r39531857 = r39531850 - r39531856;
        double r39531858 = j;
        double r39531859 = r39531851 * r39531846;
        double r39531860 = r39531853 * r39531842;
        double r39531861 = r39531859 - r39531860;
        double r39531862 = r39531858 * r39531861;
        double r39531863 = cbrt(r39531862);
        double r39531864 = cbrt(r39531858);
        double r39531865 = cbrt(r39531861);
        double r39531866 = r39531864 * r39531865;
        double r39531867 = r39531863 * r39531866;
        double r39531868 = r39531867 * r39531863;
        double r39531869 = r39531857 + r39531868;
        double r39531870 = 1.6011043676188824e-149;
        bool r39531871 = r39531839 <= r39531870;
        double r39531872 = -r39531839;
        double r39531873 = r39531845 * r39531872;
        double r39531874 = r39531853 * r39531873;
        double r39531875 = r39531851 * r39531839;
        double r39531876 = r39531875 * r39531843;
        double r39531877 = r39531874 + r39531876;
        double r39531878 = r39531850 - r39531877;
        double r39531879 = r39531878 + r39531862;
        double r39531880 = r39531844 * r39531849;
        double r39531881 = r39531849 * r39531845;
        double r39531882 = r39531846 * r39531881;
        double r39531883 = r39531880 - r39531882;
        double r39531884 = cbrt(r39531839);
        double r39531885 = r39531884 * r39531855;
        double r39531886 = r39531884 * r39531884;
        double r39531887 = r39531885 * r39531886;
        double r39531888 = r39531883 - r39531887;
        double r39531889 = r39531888 + r39531862;
        double r39531890 = r39531871 ? r39531879 : r39531889;
        double r39531891 = r39531841 ? r39531869 : r39531890;
        return r39531891;
}

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	11.3
Target	19.3
Herbie	9.3

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

Derivation

1. Split input into 3 regimes

2. if b < -3.4675246779790518e+72

   1. Initial program 6.8

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

   3. Applied add-cube-cbrt 7.0

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

   5. Applied cbrt-prod 7.0

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

   if -3.4675246779790518e+72 < b < 1.6011043676188824e-149

   1. Initial program 13.5

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

   3. Applied add-cube-cbrt 13.6

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

   4. Applied associate-*l* 13.6

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

   6. Applied sub-neg 13.6

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

   7. Applied distribute-lft-in 13.6

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

   8. Applied distribute-lft-in 13.6

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

   9. Simplified 11.7

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

   10. Simplified 9.3

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

   if 1.6011043676188824e-149 < b

   1. Initial program 9.6

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

   3. Applied add-cube-cbrt 10.0

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

   4. Applied associate-*l* 10.0

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

   5. Taylor expanded around inf 10.2

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.4675246779790518 \cdot 10^{+72}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right)\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)}\\ \mathbf{elif}\;b \le 1.6011043676188824 \cdot 10^{-149}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(i \cdot \left(t \cdot \left(-b\right)\right) + \left(c \cdot b\right) \cdot z\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x - a \cdot \left(x \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot t\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))))