Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 12.3

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 r81168 = x;
        double r81169 = y;
        double r81170 = z;
        double r81171 = r81169 * r81170;
        double r81172 = t;
        double r81173 = a;
        double r81174 = r81172 * r81173;
        double r81175 = r81171 - r81174;
        double r81176 = r81168 * r81175;
        double r81177 = b;
        double r81178 = c;
        double r81179 = r81178 * r81170;
        double r81180 = i;
        double r81181 = r81180 * r81173;
        double r81182 = r81179 - r81181;
        double r81183 = r81177 * r81182;
        double r81184 = r81176 - r81183;
        double r81185 = j;
        double r81186 = r81178 * r81172;
        double r81187 = r81180 * r81169;
        double r81188 = r81186 - r81187;
        double r81189 = r81185 * r81188;
        double r81190 = r81184 + r81189;
        return r81190;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r81191 = x;
        double r81192 = y;
        double r81193 = z;
        double r81194 = r81192 * r81193;
        double r81195 = t;
        double r81196 = a;
        double r81197 = r81195 * r81196;
        double r81198 = r81194 - r81197;
        double r81199 = r81191 * r81198;
        double r81200 = b;
        double r81201 = c;
        double r81202 = r81201 * r81193;
        double r81203 = i;
        double r81204 = r81203 * r81196;
        double r81205 = r81202 - r81204;
        double r81206 = r81200 * r81205;
        double r81207 = r81199 - r81206;
        double r81208 = j;
        double r81209 = r81201 * r81195;
        double r81210 = r81203 * r81192;
        double r81211 = r81209 - r81210;
        double r81212 = r81208 * r81211;
        double r81213 = r81207 + r81212;
        return r81213;
}

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 4 regimes

2. if z < -9.210072683042902e+197 or 112.17173264432692 < z

   1. Initial program 19.0

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

   3. Applied sub-neg 19.0

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

   4. Applied distribute-lft-in 19.0

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

   5. Simplified 18.0

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

   7. Applied associate-*r* 13.5

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

   if -9.210072683042902e+197 < z < -2.363984397184064e-225

   1. Initial program 11.3

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

   3. Applied sub-neg 11.3

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

   4. Applied distribute-lft-in 11.3

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

   5. Simplified 11.6

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

   7. Applied sub-neg 11.6

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

   8. Applied distribute-rgt-in 11.6

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

   9. Simplified 12.1

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

   10. Simplified 11.6

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

   if -2.363984397184064e-225 < z < 1.2726446353115491e-186

   1. Initial program 9.3

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

   3. Applied sub-neg 9.3

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

   4. Applied distribute-lft-in 9.3

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

   5. Simplified 10.2

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

   7. Applied add-cube-cbrt 10.3

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

   if 1.2726446353115491e-186 < z < 112.17173264432692

   1. Initial program 8.5

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

   3. Applied sub-neg 8.5

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

   4. Applied distribute-lft-in 8.5

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

   5. Simplified 8.4

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(j \cdot \left(c \cdot t\right) + \color{blue}{\left(-j \cdot i\right) \cdot y}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.3

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

Reproduce

herbie shell --seed 2019291 
(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)))))