Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 9.8

Time: 14.8s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -6.83346635036375763 \cdot 10^{180}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;b \le 0.0155641473725553429:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(\left(-i\right) \cdot b\right) \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r96236 = x;
        double r96237 = y;
        double r96238 = z;
        double r96239 = r96237 * r96238;
        double r96240 = t;
        double r96241 = a;
        double r96242 = r96240 * r96241;
        double r96243 = r96239 - r96242;
        double r96244 = r96236 * r96243;
        double r96245 = b;
        double r96246 = c;
        double r96247 = r96246 * r96238;
        double r96248 = i;
        double r96249 = r96248 * r96241;
        double r96250 = r96247 - r96249;
        double r96251 = r96245 * r96250;
        double r96252 = r96244 - r96251;
        double r96253 = j;
        double r96254 = r96246 * r96240;
        double r96255 = r96248 * r96237;
        double r96256 = r96254 - r96255;
        double r96257 = r96253 * r96256;
        double r96258 = r96252 + r96257;
        return r96258;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r96259 = b;
        double r96260 = -6.833466350363758e+180;
        bool r96261 = r96259 <= r96260;
        double r96262 = x;
        double r96263 = y;
        double r96264 = z;
        double r96265 = r96263 * r96264;
        double r96266 = t;
        double r96267 = a;
        double r96268 = r96266 * r96267;
        double r96269 = r96265 - r96268;
        double r96270 = r96262 * r96269;
        double r96271 = c;
        double r96272 = r96271 * r96264;
        double r96273 = i;
        double r96274 = r96273 * r96267;
        double r96275 = r96272 - r96274;
        double r96276 = r96259 * r96275;
        double r96277 = r96270 - r96276;
        double r96278 = j;
        double r96279 = cbrt(r96278);
        double r96280 = r96279 * r96279;
        double r96281 = r96271 * r96266;
        double r96282 = r96273 * r96263;
        double r96283 = r96281 - r96282;
        double r96284 = r96279 * r96283;
        double r96285 = r96280 * r96284;
        double r96286 = r96277 + r96285;
        double r96287 = 0.015564147372555343;
        bool r96288 = r96259 <= r96287;
        double r96289 = r96264 * r96259;
        double r96290 = r96289 * r96271;
        double r96291 = -r96273;
        double r96292 = r96291 * r96259;
        double r96293 = r96292 * r96267;
        double r96294 = r96290 + r96293;
        double r96295 = r96270 - r96294;
        double r96296 = r96281 * r96278;
        double r96297 = r96278 * r96263;
        double r96298 = r96273 * r96297;
        double r96299 = -r96298;
        double r96300 = r96296 + r96299;
        double r96301 = r96295 + r96300;
        double r96302 = r96273 * r96278;
        double r96303 = r96302 * r96263;
        double r96304 = -r96303;
        double r96305 = r96296 + r96304;
        double r96306 = r96277 + r96305;
        double r96307 = r96288 ? r96301 : r96306;
        double r96308 = r96261 ? r96286 : r96307;
        return r96308;
}

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 < -6.833466350363758e+180

   1. Initial program 5.5

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

   3. Applied add-cube-cbrt 5.7

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

   4. Applied associate-*l* 5.7

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

   if -6.833466350363758e+180 < b < 0.015564147372555343

   1. Initial program 14.0

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

   3. Applied sub-neg 14.0

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

   4. Applied distribute-lft-in 14.0

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

   5. Simplified 14.0

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

   6. Simplified 14.0

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

   8. Applied sub-neg 14.0

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

   9. Applied distribute-lft-in 14.0

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

   10. Simplified 12.9

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

   12. Applied distribute-lft-neg-in 12.9

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

   13. Applied associate-*r* 11.0

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

   14. Simplified 11.0

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

   16. Applied associate-*r* 10.8

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

   if 0.015564147372555343 < b

   1. Initial program 6.8

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

   3. Applied sub-neg 6.8

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

   4. Applied distribute-lft-in 6.8

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

   5. Simplified 6.8

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

   6. Simplified 7.9

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

   8. Applied associate-*r* 7.2

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.83346635036375763 \cdot 10^{180}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{elif}\;b \le 0.0155641473725553429:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(z \cdot b\right) \cdot c + \left(\left(-i\right) \cdot b\right) \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(\left(c \cdot t\right) \cdot j + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))))