Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.3 → 11.4

Time: 56.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -3.993137182585706 \cdot 10^{-167}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(c \cdot b\right) \cdot z + \left(-\sqrt[3]{i \cdot \left(b \cdot a\right)} \cdot \sqrt[3]{i \cdot \left(b \cdot a\right)}\right) \cdot \sqrt[3]{i \cdot \left(b \cdot a\right)}\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\ \mathbf{elif}\;j \le 6.745927926481541 \cdot 10^{-246}:\\ \;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-i\right) \cdot \left(b \cdot a\right) + \left(c \cdot b\right) \cdot z\right)\\ \mathbf{elif}\;j \le 1.716355565378914 \cdot 10^{+27}:\\ \;\;\;\;\left(t \cdot c - i \cdot y\right) \cdot j + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-i\right) \cdot \left(b \cdot a\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(c \cdot b\right) \cdot z + \left(-\sqrt[3]{i \cdot \left(b \cdot a\right)} \cdot \sqrt[3]{i \cdot \left(b \cdot a\right)}\right) \cdot \sqrt[3]{i \cdot \left(b \cdot a\right)}\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r9337284 = x;
        double r9337285 = y;
        double r9337286 = z;
        double r9337287 = r9337285 * r9337286;
        double r9337288 = t;
        double r9337289 = a;
        double r9337290 = r9337288 * r9337289;
        double r9337291 = r9337287 - r9337290;
        double r9337292 = r9337284 * r9337291;
        double r9337293 = b;
        double r9337294 = c;
        double r9337295 = r9337294 * r9337286;
        double r9337296 = i;
        double r9337297 = r9337296 * r9337289;
        double r9337298 = r9337295 - r9337297;
        double r9337299 = r9337293 * r9337298;
        double r9337300 = r9337292 - r9337299;
        double r9337301 = j;
        double r9337302 = r9337294 * r9337288;
        double r9337303 = r9337296 * r9337285;
        double r9337304 = r9337302 - r9337303;
        double r9337305 = r9337301 * r9337304;
        double r9337306 = r9337300 + r9337305;
        return r9337306;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r9337307 = j;
        double r9337308 = -3.993137182585706e-167;
        bool r9337309 = r9337307 <= r9337308;
        double r9337310 = y;
        double r9337311 = z;
        double r9337312 = r9337310 * r9337311;
        double r9337313 = t;
        double r9337314 = a;
        double r9337315 = r9337313 * r9337314;
        double r9337316 = r9337312 - r9337315;
        double r9337317 = x;
        double r9337318 = r9337316 * r9337317;
        double r9337319 = c;
        double r9337320 = b;
        double r9337321 = r9337319 * r9337320;
        double r9337322 = r9337321 * r9337311;
        double r9337323 = i;
        double r9337324 = r9337320 * r9337314;
        double r9337325 = r9337323 * r9337324;
        double r9337326 = cbrt(r9337325);
        double r9337327 = r9337326 * r9337326;
        double r9337328 = -r9337327;
        double r9337329 = r9337328 * r9337326;
        double r9337330 = r9337322 + r9337329;
        double r9337331 = r9337318 - r9337330;
        double r9337332 = r9337313 * r9337319;
        double r9337333 = r9337323 * r9337310;
        double r9337334 = r9337332 - r9337333;
        double r9337335 = r9337334 * r9337307;
        double r9337336 = r9337331 + r9337335;
        double r9337337 = 6.745927926481541e-246;
        bool r9337338 = r9337307 <= r9337337;
        double r9337339 = -r9337323;
        double r9337340 = r9337339 * r9337324;
        double r9337341 = r9337340 + r9337322;
        double r9337342 = r9337318 - r9337341;
        double r9337343 = 1.716355565378914e+27;
        bool r9337344 = r9337307 <= r9337343;
        double r9337345 = r9337320 * r9337311;
        double r9337346 = r9337319 * r9337345;
        double r9337347 = r9337340 + r9337346;
        double r9337348 = r9337318 - r9337347;
        double r9337349 = r9337335 + r9337348;
        double r9337350 = r9337344 ? r9337349 : r9337336;
        double r9337351 = r9337338 ? r9337342 : r9337350;
        double r9337352 = r9337309 ? r9337336 : r9337351;
        return r9337352;
}

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 < -3.993137182585706e-167 or 1.716355565378914e+27 < j

   1. Initial program 8.4

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

   3. Applied add-cube-cbrt 8.7

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

   4. Applied associate-*l* 8.7

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

   6. Applied sub-neg 8.7

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

   7. Applied distribute-lft-in 8.7

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

   8. Applied distribute-lft-in 8.7

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

   9. Simplified 9.0

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

   10. Taylor expanded around -inf 8.8

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

   12. Applied add-cube-cbrt 8.9

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

   if -3.993137182585706e-167 < j < 6.745927926481541e-246

   1. Initial program 16.7

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

   3. Applied add-cube-cbrt 17.0

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

   4. Applied associate-*l* 17.0

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

   6. Applied sub-neg 17.0

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

   7. Applied distribute-lft-in 17.0

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

   8. Applied distribute-lft-in 17.0

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

   9. Simplified 17.7

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

   10. Taylor expanded around -inf 17.6

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

   11. Taylor expanded around 0 17.5

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

   if 6.745927926481541e-246 < j < 1.716355565378914e+27

   1. Initial program 12.9

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

   3. Applied add-cube-cbrt 13.2

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

   4. Applied associate-*l* 13.2

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

   6. Applied sub-neg 13.2

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

   7. Applied distribute-lft-in 13.2

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

   8. Applied distribute-lft-in 13.2

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

   9. Simplified 12.6

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

   10. Taylor expanded around -inf 12.1

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

   12. Applied associate-*r* 11.7

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

4. Final simplification 11.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -3.993137182585706 \cdot 10^{-167}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(c \cdot b\right) \cdot z + \left(-\sqrt[3]{i \cdot \left(b \cdot a\right)} \cdot \sqrt[3]{i \cdot \left(b \cdot a\right)}\right) \cdot \sqrt[3]{i \cdot \left(b \cdot a\right)}\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\ \mathbf{elif}\;j \le 6.745927926481541 \cdot 10^{-246}:\\ \;\;\;\;\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-i\right) \cdot \left(b \cdot a\right) + \left(c \cdot b\right) \cdot z\right)\\ \mathbf{elif}\;j \le 1.716355565378914 \cdot 10^{+27}:\\ \;\;\;\;\left(t \cdot c - i \cdot y\right) \cdot j + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-i\right) \cdot \left(b \cdot a\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(c \cdot b\right) \cdot z + \left(-\sqrt[3]{i \cdot \left(b \cdot a\right)} \cdot \sqrt[3]{i \cdot \left(b \cdot a\right)}\right) \cdot \sqrt[3]{i \cdot \left(b \cdot a\right)}\right)\right) + \left(t \cdot c - i \cdot y\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(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)))))