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

Average Error: 11.7 → 8.8

Time: 28.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -7.292924892323122 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(-i \cdot \left(y \cdot j\right)\right) + \left(c \cdot j\right) \cdot a\right)\\ \mathbf{elif}\;a \le 1.3675312070107937 \cdot 10^{-118}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt[3]{c \cdot a - i \cdot y} \cdot \left(\left(\sqrt[3]{c \cdot a - i \cdot y} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(j \cdot \left(y \cdot \left(-i\right)\right) + \left(c \cdot j\right) \cdot a\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r37523750 = x;
        double r37523751 = y;
        double r37523752 = z;
        double r37523753 = r37523751 * r37523752;
        double r37523754 = t;
        double r37523755 = a;
        double r37523756 = r37523754 * r37523755;
        double r37523757 = r37523753 - r37523756;
        double r37523758 = r37523750 * r37523757;
        double r37523759 = b;
        double r37523760 = c;
        double r37523761 = r37523760 * r37523752;
        double r37523762 = i;
        double r37523763 = r37523754 * r37523762;
        double r37523764 = r37523761 - r37523763;
        double r37523765 = r37523759 * r37523764;
        double r37523766 = r37523758 - r37523765;
        double r37523767 = j;
        double r37523768 = r37523760 * r37523755;
        double r37523769 = r37523751 * r37523762;
        double r37523770 = r37523768 - r37523769;
        double r37523771 = r37523767 * r37523770;
        double r37523772 = r37523766 + r37523771;
        return r37523772;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r37523773 = a;
        double r37523774 = -7.292924892323122e-25;
        bool r37523775 = r37523773 <= r37523774;
        double r37523776 = x;
        double r37523777 = z;
        double r37523778 = r37523776 * r37523777;
        double r37523779 = y;
        double r37523780 = r37523778 * r37523779;
        double r37523781 = t;
        double r37523782 = r37523781 * r37523776;
        double r37523783 = r37523782 * r37523773;
        double r37523784 = r37523780 - r37523783;
        double r37523785 = b;
        double r37523786 = c;
        double r37523787 = r37523777 * r37523786;
        double r37523788 = i;
        double r37523789 = r37523788 * r37523781;
        double r37523790 = r37523787 - r37523789;
        double r37523791 = r37523785 * r37523790;
        double r37523792 = r37523784 - r37523791;
        double r37523793 = j;
        double r37523794 = r37523779 * r37523793;
        double r37523795 = r37523788 * r37523794;
        double r37523796 = -r37523795;
        double r37523797 = r37523786 * r37523793;
        double r37523798 = r37523797 * r37523773;
        double r37523799 = r37523796 + r37523798;
        double r37523800 = r37523792 + r37523799;
        double r37523801 = 1.3675312070107937e-118;
        bool r37523802 = r37523773 <= r37523801;
        double r37523803 = r37523779 * r37523777;
        double r37523804 = r37523773 * r37523781;
        double r37523805 = r37523803 - r37523804;
        double r37523806 = r37523776 * r37523805;
        double r37523807 = r37523806 - r37523791;
        double r37523808 = r37523786 * r37523773;
        double r37523809 = r37523788 * r37523779;
        double r37523810 = r37523808 - r37523809;
        double r37523811 = cbrt(r37523810);
        double r37523812 = r37523811 * r37523811;
        double r37523813 = r37523812 * r37523793;
        double r37523814 = r37523811 * r37523813;
        double r37523815 = r37523807 + r37523814;
        double r37523816 = r37523803 * r37523776;
        double r37523817 = r37523816 - r37523783;
        double r37523818 = r37523817 - r37523791;
        double r37523819 = -r37523788;
        double r37523820 = r37523779 * r37523819;
        double r37523821 = r37523793 * r37523820;
        double r37523822 = r37523821 + r37523798;
        double r37523823 = r37523818 + r37523822;
        double r37523824 = r37523802 ? r37523815 : r37523823;
        double r37523825 = r37523775 ? r37523800 : r37523824;
        return r37523825;
}

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	8.8

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

Derivation

1. Split input into 3 regimes

2. if a < -7.292924892323122e-25

   1. Initial program 15.2

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

   3. Applied add-cube-cbrt 15.5

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

   4. Applied associate-*l* 15.5

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

   5. Taylor expanded around inf 12.0

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

   7. Applied sub-neg 12.0

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

   8. Applied distribute-lft-in 12.0

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

   9. Applied distribute-lft-in 12.0

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

   10. Simplified 8.1

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

   11. Simplified 7.1

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

   13. Applied associate-*r* 7.0

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

   if -7.292924892323122e-25 < a < 1.3675312070107937e-118

   1. Initial program 9.1

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

   3. Applied add-cube-cbrt 9.4

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

   4. Applied associate-*r* 9.4

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

   if 1.3675312070107937e-118 < a

   1. Initial program 13.4

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

   3. Applied add-cube-cbrt 13.7

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

   4. Applied associate-*l* 13.7

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

   5. Taylor expanded around inf 11.8

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

   7. Applied sub-neg 11.8

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

   8. Applied distribute-lft-in 11.8

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

   9. Applied distribute-lft-in 11.8

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

   10. Simplified 9.3

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

   11. Simplified 8.8

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

   13. Applied associate-*l* 9.2

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.292924892323122 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(-i \cdot \left(y \cdot j\right)\right) + \left(c \cdot j\right) \cdot a\right)\\ \mathbf{elif}\;a \le 1.3675312070107937 \cdot 10^{-118}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt[3]{c \cdot a - i \cdot y} \cdot \left(\left(\sqrt[3]{c \cdot a - i \cdot y} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(j \cdot \left(y \cdot \left(-i\right)\right) + \left(c \cdot j\right) \cdot a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))))