Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.5 → 11.7

Time: 25.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 4.868355865026179074986914961335065973696 \cdot 10^{-307}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(\left(y \cdot z\right) \cdot x + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-i \cdot a\right)\right)\right)\\ \mathbf{elif}\;i \le 5.137418068683974589048388994643372248735 \cdot 10^{83}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r107222 = x;
        double r107223 = y;
        double r107224 = z;
        double r107225 = r107223 * r107224;
        double r107226 = t;
        double r107227 = a;
        double r107228 = r107226 * r107227;
        double r107229 = r107225 - r107228;
        double r107230 = r107222 * r107229;
        double r107231 = b;
        double r107232 = c;
        double r107233 = r107232 * r107224;
        double r107234 = i;
        double r107235 = r107234 * r107227;
        double r107236 = r107233 - r107235;
        double r107237 = r107231 * r107236;
        double r107238 = r107230 - r107237;
        double r107239 = j;
        double r107240 = r107232 * r107226;
        double r107241 = r107234 * r107223;
        double r107242 = r107240 - r107241;
        double r107243 = r107239 * r107242;
        double r107244 = r107238 + r107243;
        return r107244;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r107245 = i;
        double r107246 = 4.868355865026179e-307;
        bool r107247 = r107245 <= r107246;
        double r107248 = j;
        double r107249 = c;
        double r107250 = t;
        double r107251 = r107249 * r107250;
        double r107252 = y;
        double r107253 = r107245 * r107252;
        double r107254 = r107251 - r107253;
        double r107255 = r107248 * r107254;
        double r107256 = z;
        double r107257 = r107252 * r107256;
        double r107258 = x;
        double r107259 = r107257 * r107258;
        double r107260 = a;
        double r107261 = r107258 * r107260;
        double r107262 = r107250 * r107261;
        double r107263 = -r107262;
        double r107264 = r107259 + r107263;
        double r107265 = b;
        double r107266 = r107265 * r107249;
        double r107267 = r107256 * r107266;
        double r107268 = r107245 * r107260;
        double r107269 = -r107268;
        double r107270 = r107265 * r107269;
        double r107271 = r107267 + r107270;
        double r107272 = r107264 - r107271;
        double r107273 = r107255 + r107272;
        double r107274 = 5.137418068683975e+83;
        bool r107275 = r107245 <= r107274;
        double r107276 = r107256 * r107258;
        double r107277 = r107252 * r107276;
        double r107278 = r107258 * r107250;
        double r107279 = r107260 * r107278;
        double r107280 = -r107279;
        double r107281 = r107277 + r107280;
        double r107282 = r107281 - r107271;
        double r107283 = r107282 + r107255;
        double r107284 = r107250 * r107260;
        double r107285 = r107257 - r107284;
        double r107286 = r107258 * r107285;
        double r107287 = r107249 * r107256;
        double r107288 = r107287 - r107268;
        double r107289 = r107265 * r107288;
        double r107290 = r107286 - r107289;
        double r107291 = r107248 * r107249;
        double r107292 = r107250 * r107291;
        double r107293 = r107248 * r107252;
        double r107294 = r107245 * r107293;
        double r107295 = -r107294;
        double r107296 = r107292 + r107295;
        double r107297 = r107290 + r107296;
        double r107298 = r107275 ? r107283 : r107297;
        double r107299 = r107247 ? r107273 : r107298;
        return r107299;
}

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 i < 4.868355865026179e-307

   1. Initial program 12.2

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

   3. Applied sub-neg 12.2

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

   4. Applied distribute-lft-in 12.2

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

   5. Simplified 12.2

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

   6. Simplified 12.2

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

   8. Applied sub-neg 12.2

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

   9. Applied distribute-lft-in 12.2

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

   10. Simplified 13.0

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

   12. Applied pow1 13.0

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

   13. Applied pow1 13.0

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

   14. Applied pow-prod-down 13.0

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

   15. Applied pow1 13.0

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

   16. Applied pow-prod-down 13.0

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

   17. Simplified 13.2

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

   if 4.868355865026179e-307 < i < 5.137418068683975e+83

   1. Initial program 10.0

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

   3. Applied sub-neg 10.0

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

   4. Applied distribute-lft-in 10.0

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

   5. Simplified 10.0

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

   6. Simplified 9.6

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

   8. Applied sub-neg 9.6

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

   9. Applied distribute-lft-in 9.6

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

   10. Simplified 9.9

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

   12. Applied associate-*l* 8.7

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

   if 5.137418068683975e+83 < i

   1. Initial program 20.9

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

   3. Applied sub-neg 20.9

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

   4. Applied distribute-lft-in 20.9

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

   5. Simplified 20.9

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

   6. Simplified 14.3

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

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.868355865026179074986914961335065973696 \cdot 10^{-307}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(\left(y \cdot z\right) \cdot x + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-i \cdot a\right)\right)\right)\\ \mathbf{elif}\;i \le 5.137418068683974589048388994643372248735 \cdot 10^{83}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))))