Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.9 → 9.2

Time: 14.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -797.53682869769:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;x \le 1.00452736762116509 \cdot 10^{-41}:\\ \;\;\;\;\left(\left(\left(-t \cdot \left(x \cdot a\right)\right) + y \cdot \left(z \cdot x\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r104730 = x;
        double r104731 = y;
        double r104732 = z;
        double r104733 = r104731 * r104732;
        double r104734 = t;
        double r104735 = a;
        double r104736 = r104734 * r104735;
        double r104737 = r104733 - r104736;
        double r104738 = r104730 * r104737;
        double r104739 = b;
        double r104740 = c;
        double r104741 = r104740 * r104732;
        double r104742 = i;
        double r104743 = r104742 * r104735;
        double r104744 = r104741 - r104743;
        double r104745 = r104739 * r104744;
        double r104746 = r104738 - r104745;
        double r104747 = j;
        double r104748 = r104740 * r104734;
        double r104749 = r104742 * r104731;
        double r104750 = r104748 - r104749;
        double r104751 = r104747 * r104750;
        double r104752 = r104746 + r104751;
        return r104752;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r104753 = x;
        double r104754 = -797.53682869769;
        bool r104755 = r104753 <= r104754;
        double r104756 = y;
        double r104757 = z;
        double r104758 = r104756 * r104757;
        double r104759 = t;
        double r104760 = a;
        double r104761 = r104759 * r104760;
        double r104762 = r104758 - r104761;
        double r104763 = r104753 * r104762;
        double r104764 = b;
        double r104765 = c;
        double r104766 = r104765 * r104757;
        double r104767 = i;
        double r104768 = r104767 * r104760;
        double r104769 = r104766 - r104768;
        double r104770 = r104764 * r104769;
        double r104771 = r104763 - r104770;
        double r104772 = j;
        double r104773 = r104772 * r104765;
        double r104774 = r104759 * r104773;
        double r104775 = r104772 * r104756;
        double r104776 = r104767 * r104775;
        double r104777 = -r104776;
        double r104778 = r104774 + r104777;
        double r104779 = r104771 + r104778;
        double r104780 = 1.0045273676211651e-41;
        bool r104781 = r104753 <= r104780;
        double r104782 = r104753 * r104760;
        double r104783 = r104759 * r104782;
        double r104784 = -r104783;
        double r104785 = r104757 * r104753;
        double r104786 = r104756 * r104785;
        double r104787 = r104784 + r104786;
        double r104788 = r104764 * r104765;
        double r104789 = r104757 * r104788;
        double r104790 = -r104768;
        double r104791 = r104790 * r104764;
        double r104792 = r104789 + r104791;
        double r104793 = r104787 - r104792;
        double r104794 = r104765 * r104759;
        double r104795 = r104767 * r104756;
        double r104796 = r104794 - r104795;
        double r104797 = r104772 * r104796;
        double r104798 = r104793 + r104797;
        double r104799 = r104757 * r104764;
        double r104800 = r104799 * r104765;
        double r104801 = r104800 + r104791;
        double r104802 = r104763 - r104801;
        double r104803 = r104802 + r104797;
        double r104804 = r104781 ? r104798 : r104803;
        double r104805 = r104755 ? r104779 : r104804;
        return r104805;
}

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

Derivation

1. Split input into 3 regimes

2. if x < -797.53682869769

   1. Initial program 6.6

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

   3. Applied sub-neg 6.6

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

   4. Applied distribute-lft-in 6.6

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

   5. Simplified 7.7

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

   6. Simplified 7.7

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

   if -797.53682869769 < x < 1.0045273676211651e-41

   1. Initial program 15.1

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

   3. Applied sub-neg 15.1

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

   4. Applied distribute-lft-in 15.1

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

   5. Simplified 15.4

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

   6. Simplified 15.4

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

   8. Applied sub-neg 15.4

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

   9. Applied distribute-lft-in 15.4

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

   10. Simplified 15.4

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

   11. Simplified 13.1

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

   13. Applied *-un-lft-identity 13.1

       \[\leadsto \left(\left(\left(y \cdot z\right) \cdot x + \left(-\color{blue}{\left(1 \cdot a\right)} \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   14. Applied associate-*l* 13.1

       \[\leadsto \left(\left(\left(y \cdot z\right) \cdot x + \left(-\color{blue}{1 \cdot \left(a \cdot \left(x \cdot t\right)\right)}\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   15. Simplified 13.1

       \[\leadsto \left(\left(\left(y \cdot z\right) \cdot x + \left(-1 \cdot \color{blue}{\left(t \cdot \left(x \cdot a\right)\right)}\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
   16. Using strategy rm

   17. Applied associate-*l* 9.8

       \[\leadsto \left(\left(\color{blue}{y \cdot \left(z \cdot x\right)} + \left(-1 \cdot \left(t \cdot \left(x \cdot a\right)\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   if 1.0045273676211651e-41 < x

   1. Initial program 8.0

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

   3. Applied sub-neg 8.0

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

   4. Applied distribute-lft-in 8.0

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

   5. Simplified 8.3

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

   6. Simplified 8.3

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

   8. Applied associate-*r* 8.8

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -797.53682869769:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;x \le 1.00452736762116509 \cdot 10^{-41}:\\ \;\;\;\;\left(\left(\left(-t \cdot \left(x \cdot a\right)\right) + y \cdot \left(z \cdot x\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(-i \cdot a\right) \cdot b\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))