Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 12.4

Time: 15.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -3.06881634392199504 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(i \cdot a - c \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b} + \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;j \le 1.143058614838747 \cdot 10^{-175}:\\ \;\;\;\;\left(a \cdot \left(i \cdot b\right) - z \cdot \left(b \cdot c\right)\right) + \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(i \cdot b\right) - \left(z \cdot b\right) \cdot c\right) + \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r120771 = x;
        double r120772 = y;
        double r120773 = z;
        double r120774 = r120772 * r120773;
        double r120775 = t;
        double r120776 = a;
        double r120777 = r120775 * r120776;
        double r120778 = r120774 - r120777;
        double r120779 = r120771 * r120778;
        double r120780 = b;
        double r120781 = c;
        double r120782 = r120781 * r120773;
        double r120783 = i;
        double r120784 = r120783 * r120776;
        double r120785 = r120782 - r120784;
        double r120786 = r120780 * r120785;
        double r120787 = r120779 - r120786;
        double r120788 = j;
        double r120789 = r120781 * r120775;
        double r120790 = r120783 * r120772;
        double r120791 = r120789 - r120790;
        double r120792 = r120788 * r120791;
        double r120793 = r120787 + r120792;
        return r120793;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r120794 = j;
        double r120795 = -3.068816343921995e-18;
        bool r120796 = r120794 <= r120795;
        double r120797 = i;
        double r120798 = a;
        double r120799 = r120797 * r120798;
        double r120800 = c;
        double r120801 = z;
        double r120802 = r120800 * r120801;
        double r120803 = r120799 - r120802;
        double r120804 = b;
        double r120805 = cbrt(r120804);
        double r120806 = r120805 * r120805;
        double r120807 = r120803 * r120806;
        double r120808 = r120807 * r120805;
        double r120809 = t;
        double r120810 = r120800 * r120809;
        double r120811 = y;
        double r120812 = r120797 * r120811;
        double r120813 = r120810 - r120812;
        double r120814 = x;
        double r120815 = r120811 * r120801;
        double r120816 = r120809 * r120798;
        double r120817 = r120815 - r120816;
        double r120818 = r120814 * r120817;
        double r120819 = fma(r120794, r120813, r120818);
        double r120820 = r120808 + r120819;
        double r120821 = 1.143058614838747e-175;
        bool r120822 = r120794 <= r120821;
        double r120823 = r120797 * r120804;
        double r120824 = r120798 * r120823;
        double r120825 = r120804 * r120800;
        double r120826 = r120801 * r120825;
        double r120827 = r120824 - r120826;
        double r120828 = r120798 * r120809;
        double r120829 = -r120828;
        double r120830 = fma(r120811, r120801, r120829);
        double r120831 = r120814 * r120830;
        double r120832 = r120794 * r120811;
        double r120833 = r120797 * r120832;
        double r120834 = r120831 - r120833;
        double r120835 = r120827 + r120834;
        double r120836 = r120801 * r120804;
        double r120837 = r120836 * r120800;
        double r120838 = r120824 - r120837;
        double r120839 = r120838 + r120819;
        double r120840 = r120822 ? r120835 : r120839;
        double r120841 = r120796 ? r120820 : r120840;
        return r120841;
}

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 j < -3.068816343921995e-18

   1. Initial program 7.4

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

   2. Simplified 7.4

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

   4. Applied fma-udef 7.4

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

   6. Applied add-cube-cbrt 7.6

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

   7. Applied associate-*r* 7.6

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

   if -3.068816343921995e-18 < j < 1.143058614838747e-175

   1. Initial program 16.2

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

   2. Simplified 16.2

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

   4. Applied fma-udef 16.2

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

   6. Applied add-cube-cbrt 16.5

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

   7. Applied associate-*r* 16.5

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

   8. Taylor expanded around inf 16.9

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

   9. Taylor expanded around inf 16.1

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

   10. Simplified 16.2

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

   if 1.143058614838747e-175 < j

   1. Initial program 9.9

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

   2. Simplified 9.9

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

   4. Applied fma-udef 9.9

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

   6. Applied add-cube-cbrt 10.2

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

   7. Applied associate-*r* 10.2

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

   8. Taylor expanded around inf 11.1

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

   10. Applied associate-*r* 10.7

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -3.06881634392199504 \cdot 10^{-18}:\\ \;\;\;\;\left(\left(i \cdot a - c \cdot z\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right) \cdot \sqrt[3]{b} + \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;j \le 1.143058614838747 \cdot 10^{-175}:\\ \;\;\;\;\left(a \cdot \left(i \cdot b\right) - z \cdot \left(b \cdot c\right)\right) + \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) - i \cdot \left(j \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(i \cdot b\right) - \left(z \cdot b\right) \cdot c\right) + \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))))