Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.2 → 12.2

Time: 20.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r111696 = x;
        double r111697 = y;
        double r111698 = z;
        double r111699 = r111697 * r111698;
        double r111700 = t;
        double r111701 = a;
        double r111702 = r111700 * r111701;
        double r111703 = r111699 - r111702;
        double r111704 = r111696 * r111703;
        double r111705 = b;
        double r111706 = c;
        double r111707 = r111706 * r111698;
        double r111708 = i;
        double r111709 = r111708 * r111701;
        double r111710 = r111707 - r111709;
        double r111711 = r111705 * r111710;
        double r111712 = r111704 - r111711;
        double r111713 = j;
        double r111714 = r111706 * r111700;
        double r111715 = r111708 * r111697;
        double r111716 = r111714 - r111715;
        double r111717 = r111713 * r111716;
        double r111718 = r111712 + r111717;
        return r111718;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r111719 = x;
        double r111720 = y;
        double r111721 = z;
        double r111722 = r111720 * r111721;
        double r111723 = t;
        double r111724 = a;
        double r111725 = r111723 * r111724;
        double r111726 = r111722 - r111725;
        double r111727 = r111719 * r111726;
        double r111728 = b;
        double r111729 = c;
        double r111730 = r111729 * r111721;
        double r111731 = i;
        double r111732 = r111731 * r111724;
        double r111733 = r111730 - r111732;
        double r111734 = r111728 * r111733;
        double r111735 = r111727 - r111734;
        double r111736 = j;
        double r111737 = r111729 * r111723;
        double r111738 = r111731 * r111720;
        double r111739 = r111737 - r111738;
        double r111740 = r111736 * r111739;
        double r111741 = r111735 + r111740;
        return r111741;
}

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 2 regimes

2. if j < -2.2048212568945363e-249 or 0.00588003701470001 < j

   1. Initial program 10.6

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

   3. Applied add-cube-cbrt 10.9

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

   4. Applied associate-*l* 10.9

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

   6. Applied sub-neg 10.9

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

   7. Applied distribute-lft-in 10.9

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

   8. Applied distribute-lft-in 10.9

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

   9. Simplified 11.4

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

   10. Simplified 11.3

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

   12. Applied distribute-lft-neg-out 11.3

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

   13. Simplified 11.2

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

   15. Applied add-cube-cbrt 11.3

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

   16. Applied associate-*l* 11.3

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

   if -2.2048212568945363e-249 < j < 0.00588003701470001

   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 add-cube-cbrt 15.2

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

   4. Applied associate-*l* 15.2

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.2

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

Reproduce

herbie shell --seed 2019298 
(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)))))