Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.1 → 11.8

Time: 30.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -9.427248904452578 \cdot 10^{-129}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + \left(-x\right) \cdot \left(a \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - a \cdot i\right) \cdot \sqrt[3]{b}\right)\right)\\ \mathbf{elif}\;b \le 1.4812126748566952 \cdot 10^{-249}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z\right) + \left(-x\right) \cdot \left(a \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-x\right) \cdot \left(a \cdot t\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r3301731 = x;
        double r3301732 = y;
        double r3301733 = z;
        double r3301734 = r3301732 * r3301733;
        double r3301735 = t;
        double r3301736 = a;
        double r3301737 = r3301735 * r3301736;
        double r3301738 = r3301734 - r3301737;
        double r3301739 = r3301731 * r3301738;
        double r3301740 = b;
        double r3301741 = c;
        double r3301742 = r3301741 * r3301733;
        double r3301743 = i;
        double r3301744 = r3301743 * r3301736;
        double r3301745 = r3301742 - r3301744;
        double r3301746 = r3301740 * r3301745;
        double r3301747 = r3301739 - r3301746;
        double r3301748 = j;
        double r3301749 = r3301741 * r3301735;
        double r3301750 = r3301743 * r3301732;
        double r3301751 = r3301749 - r3301750;
        double r3301752 = r3301748 * r3301751;
        double r3301753 = r3301747 + r3301752;
        return r3301753;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r3301754 = b;
        double r3301755 = -9.427248904452578e-129;
        bool r3301756 = r3301754 <= r3301755;
        double r3301757 = j;
        double r3301758 = t;
        double r3301759 = c;
        double r3301760 = r3301758 * r3301759;
        double r3301761 = y;
        double r3301762 = i;
        double r3301763 = r3301761 * r3301762;
        double r3301764 = r3301760 - r3301763;
        double r3301765 = r3301757 * r3301764;
        double r3301766 = x;
        double r3301767 = z;
        double r3301768 = r3301761 * r3301767;
        double r3301769 = r3301766 * r3301768;
        double r3301770 = -r3301766;
        double r3301771 = a;
        double r3301772 = r3301771 * r3301758;
        double r3301773 = r3301770 * r3301772;
        double r3301774 = r3301769 + r3301773;
        double r3301775 = cbrt(r3301754);
        double r3301776 = r3301775 * r3301775;
        double r3301777 = r3301759 * r3301767;
        double r3301778 = r3301771 * r3301762;
        double r3301779 = r3301777 - r3301778;
        double r3301780 = r3301779 * r3301775;
        double r3301781 = r3301776 * r3301780;
        double r3301782 = r3301774 - r3301781;
        double r3301783 = r3301765 + r3301782;
        double r3301784 = 1.4812126748566952e-249;
        bool r3301785 = r3301754 <= r3301784;
        double r3301786 = r3301765 + r3301774;
        double r3301787 = r3301766 * r3301767;
        double r3301788 = r3301787 * r3301761;
        double r3301789 = r3301788 + r3301773;
        double r3301790 = r3301754 * r3301779;
        double r3301791 = r3301789 - r3301790;
        double r3301792 = r3301791 + r3301765;
        double r3301793 = r3301785 ? r3301786 : r3301792;
        double r3301794 = r3301756 ? r3301783 : r3301793;
        return r3301794;
}

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 b < -9.427248904452578e-129

   1. Initial program 8.1

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

   3. Applied sub-neg 8.1

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

   4. Applied distribute-lft-in 8.1

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

   6. Applied add-cube-cbrt 8.5

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

   7. Applied associate-*l* 8.5

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

   if -9.427248904452578e-129 < b < 1.4812126748566952e-249

   1. Initial program 16.0

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

   3. Applied sub-neg 16.0

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

   4. Applied distribute-lft-in 16.0

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

   5. Taylor expanded around 0 17.9

      \[\leadsto \left(\left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \color{blue}{0}\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   if 1.4812126748566952e-249 < b

   1. Initial program 10.4

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

   3. Applied sub-neg 10.4

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

   4. Applied distribute-lft-in 10.4

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

   6. Applied add-cube-cbrt 10.6

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

   7. Applied associate-*l* 10.6

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

   9. Applied pow1 10.6

      \[\leadsto \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot \color{blue}{{z}^{1}}\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   10. Applied pow1 10.6

       \[\leadsto \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(\color{blue}{{y}^{1}} \cdot {z}^{1}\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   11. Applied pow-prod-down 10.6

       \[\leadsto \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \color{blue}{{\left(y \cdot z\right)}^{1}}\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   12. Applied pow1 10.6

       \[\leadsto \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{1}} \cdot {\left(y \cdot z\right)}^{1}\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   13. Applied pow-prod-down 10.6

       \[\leadsto \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{{\left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right)}^{1}} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   14. Applied pow1 10.6

       \[\leadsto \left(\left(\left(\sqrt[3]{x} \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right)}^{1} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   15. Applied pow1 10.6

       \[\leadsto \left(\left(\left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{1}} \cdot {\left(\sqrt[3]{x}\right)}^{1}\right) \cdot {\left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right)}^{1} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   16. Applied pow-prod-down 10.6

       \[\leadsto \left(\left(\color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{1}} \cdot {\left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right)}^{1} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   17. Applied pow-prod-down 10.6

       \[\leadsto \left(\left(\color{blue}{{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right)\right)}^{1}} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   18. Simplified 10.7

       \[\leadsto \left(\left({\color{blue}{\left(y \cdot \left(x \cdot z\right)\right)}}^{1} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.427248904452578 \cdot 10^{-129}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(x \cdot \left(y \cdot z\right) + \left(-x\right) \cdot \left(a \cdot t\right)\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - a \cdot i\right) \cdot \sqrt[3]{b}\right)\right)\\ \mathbf{elif}\;b \le 1.4812126748566952 \cdot 10^{-249}:\\ \;\;\;\;j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(y \cdot z\right) + \left(-x\right) \cdot \left(a \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-x\right) \cdot \left(a \cdot t\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))