Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.6 → 10.8

Time: 22.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.615542034760807155180373229086399078369 \lor \neg \left(z \le 5.605088386505130723897825567204211548534 \cdot 10^{-117}\right):\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-\left(x \cdot a\right) \cdot t\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-\left(x \cdot a\right) \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r82259 = x;
        double r82260 = y;
        double r82261 = z;
        double r82262 = r82260 * r82261;
        double r82263 = t;
        double r82264 = a;
        double r82265 = r82263 * r82264;
        double r82266 = r82262 - r82265;
        double r82267 = r82259 * r82266;
        double r82268 = b;
        double r82269 = c;
        double r82270 = r82269 * r82261;
        double r82271 = i;
        double r82272 = r82271 * r82264;
        double r82273 = r82270 - r82272;
        double r82274 = r82268 * r82273;
        double r82275 = r82267 - r82274;
        double r82276 = j;
        double r82277 = r82269 * r82263;
        double r82278 = r82271 * r82260;
        double r82279 = r82277 - r82278;
        double r82280 = r82276 * r82279;
        double r82281 = r82275 + r82280;
        return r82281;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r82282 = z;
        double r82283 = -7.615542034760807;
        bool r82284 = r82282 <= r82283;
        double r82285 = 5.605088386505131e-117;
        bool r82286 = r82282 <= r82285;
        double r82287 = !r82286;
        bool r82288 = r82284 || r82287;
        double r82289 = x;
        double r82290 = y;
        double r82291 = r82282 * r82290;
        double r82292 = r82289 * r82291;
        double r82293 = a;
        double r82294 = r82289 * r82293;
        double r82295 = t;
        double r82296 = r82294 * r82295;
        double r82297 = -r82296;
        double r82298 = r82292 + r82297;
        double r82299 = b;
        double r82300 = c;
        double r82301 = r82299 * r82300;
        double r82302 = r82282 * r82301;
        double r82303 = i;
        double r82304 = r82303 * r82293;
        double r82305 = -r82304;
        double r82306 = r82305 * r82299;
        double r82307 = r82302 + r82306;
        double r82308 = r82298 - r82307;
        double r82309 = j;
        double r82310 = r82309 * r82300;
        double r82311 = r82295 * r82310;
        double r82312 = r82309 * r82290;
        double r82313 = r82303 * r82312;
        double r82314 = -r82313;
        double r82315 = r82311 + r82314;
        double r82316 = r82308 + r82315;
        double r82317 = r82300 * r82282;
        double r82318 = r82317 - r82304;
        double r82319 = r82299 * r82318;
        double r82320 = r82298 - r82319;
        double r82321 = r82303 * r82309;
        double r82322 = r82321 * r82290;
        double r82323 = -r82322;
        double r82324 = r82311 + r82323;
        double r82325 = r82320 + r82324;
        double r82326 = r82288 ? r82316 : r82325;
        return r82326;
}

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 z < -7.615542034760807 or 5.605088386505131e-117 < z

   1. Initial program 15.4

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

   3. Applied sub-neg 15.4

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

   4. Applied distribute-lft-in 15.4

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

   5. Simplified 15.6

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

   6. Simplified 15.6

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

   8. Applied sub-neg 15.6

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

   9. Applied distribute-lft-in 15.6

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

   10. Simplified 15.6

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

   11. Simplified 16.0

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

   13. Applied associate-*r* 15.7

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

   14. Simplified 15.7

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

   16. Applied sub-neg 15.7

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

   17. Applied distribute-lft-in 15.7

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

   18. Simplified 12.8

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

   19. Simplified 12.8

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

   if -7.615542034760807 < z < 5.605088386505131e-117

   1. Initial program 9.7

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

   3. Applied sub-neg 9.7

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

   4. Applied distribute-lft-in 9.7

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

   5. Simplified 9.7

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

   6. Simplified 9.4

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

   8. Applied sub-neg 9.4

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

   9. Applied distribute-lft-in 9.4

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

   10. Simplified 9.4

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

   11. Simplified 10.4

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

   13. Applied associate-*r* 9.7

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

   14. Simplified 9.7

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

   16. Applied associate-*r* 8.7

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

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.615542034760807155180373229086399078369 \lor \neg \left(z \le 5.605088386505130723897825567204211548534 \cdot 10^{-117}\right):\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-\left(x \cdot a\right) \cdot t\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-\left(x \cdot a\right) \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(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)))))