Data.Colour.Matrix:determinant from colour-2.3.3, A

Average Error: 11.9 → 12.2

Time: 9.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -8.311437424410602968393294732626591381699 \cdot 10^{-187}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + 1 \cdot \left(-1 \cdot \left(\left(i \cdot j\right) \cdot y\right)\right)\right)\\ \mathbf{elif}\;b \le -6.469196241460224510801417511235181480356 \cdot 10^{-289}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + \left(a \cdot \left(j \cdot c\right) + 1 \cdot \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;b \le 2.701853472416222263113483854524259515545 \cdot 10^{-61}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + 1 \cdot \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + 1 \cdot \left(-1 \cdot \left(\left(i \cdot j\right) \cdot y\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r821164 = x;
        double r821165 = y;
        double r821166 = z;
        double r821167 = r821165 * r821166;
        double r821168 = t;
        double r821169 = a;
        double r821170 = r821168 * r821169;
        double r821171 = r821167 - r821170;
        double r821172 = r821164 * r821171;
        double r821173 = b;
        double r821174 = c;
        double r821175 = r821174 * r821166;
        double r821176 = i;
        double r821177 = r821168 * r821176;
        double r821178 = r821175 - r821177;
        double r821179 = r821173 * r821178;
        double r821180 = r821172 - r821179;
        double r821181 = j;
        double r821182 = r821174 * r821169;
        double r821183 = r821165 * r821176;
        double r821184 = r821182 - r821183;
        double r821185 = r821181 * r821184;
        double r821186 = r821180 + r821185;
        return r821186;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r821187 = b;
        double r821188 = -8.311437424410603e-187;
        bool r821189 = r821187 <= r821188;
        double r821190 = x;
        double r821191 = y;
        double r821192 = z;
        double r821193 = r821191 * r821192;
        double r821194 = t;
        double r821195 = a;
        double r821196 = r821194 * r821195;
        double r821197 = r821193 - r821196;
        double r821198 = r821190 * r821197;
        double r821199 = c;
        double r821200 = r821199 * r821192;
        double r821201 = i;
        double r821202 = r821194 * r821201;
        double r821203 = r821200 - r821202;
        double r821204 = r821187 * r821203;
        double r821205 = r821198 - r821204;
        double r821206 = j;
        double r821207 = r821206 * r821199;
        double r821208 = r821195 * r821207;
        double r821209 = 1.0;
        double r821210 = -1.0;
        double r821211 = r821201 * r821206;
        double r821212 = r821211 * r821191;
        double r821213 = r821210 * r821212;
        double r821214 = r821209 * r821213;
        double r821215 = r821208 + r821214;
        double r821216 = r821205 + r821215;
        double r821217 = -6.4691962414602245e-289;
        bool r821218 = r821187 <= r821217;
        double r821219 = 0.0;
        double r821220 = r821198 - r821219;
        double r821221 = r821206 * r821191;
        double r821222 = r821201 * r821221;
        double r821223 = r821210 * r821222;
        double r821224 = r821209 * r821223;
        double r821225 = r821208 + r821224;
        double r821226 = r821220 + r821225;
        double r821227 = 2.7018534724162223e-61;
        bool r821228 = r821187 <= r821227;
        double r821229 = r821195 * r821206;
        double r821230 = r821229 * r821199;
        double r821231 = r821230 + r821224;
        double r821232 = r821205 + r821231;
        double r821233 = r821228 ? r821232 : r821216;
        double r821234 = r821218 ? r821226 : r821233;
        double r821235 = r821189 ? r821216 : r821234;
        return r821235;
}

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

Target

Original	11.9
Target	20.0
Herbie	12.2

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if b < -8.311437424410603e-187 or 2.7018534724162223e-61 < b

   1. Initial program 9.1

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

   3. Applied add-cube-cbrt 9.3

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

   4. Applied associate-*l* 9.3

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

   6. Applied sub-neg 9.3

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

   7. Applied distribute-lft-in 9.3

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

   8. Applied distribute-lft-in 9.3

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

   9. Simplified 9.7

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

   10. Simplified 9.6

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

   12. Applied *-un-lft-identity 9.6

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

   13. Applied associate-*l* 9.6

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

   14. Simplified 9.3

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

   16. Applied associate-*r* 9.5

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

   if -8.311437424410603e-187 < b < -6.4691962414602245e-289

   1. Initial program 17.1

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

   3. Applied add-cube-cbrt 17.5

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

   4. Applied associate-*l* 17.5

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

   6. Applied sub-neg 17.5

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

   7. Applied distribute-lft-in 17.5

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

   8. Applied distribute-lft-in 17.5

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

   9. Simplified 18.3

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

   10. Simplified 18.1

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

   12. Applied *-un-lft-identity 18.1

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

   13. Applied associate-*l* 18.1

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

   14. Simplified 17.5

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

   15. Taylor expanded around 0 17.1

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{0}\right) + \left(a \cdot \left(j \cdot c\right) + 1 \cdot \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\right)\]

   if -6.4691962414602245e-289 < b < 2.7018534724162223e-61

   1. Initial program 16.5

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

   3. Applied add-cube-cbrt 16.9

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

   4. Applied associate-*l* 16.9

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

   6. Applied sub-neg 16.9

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

   7. Applied distribute-lft-in 16.9

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

   8. Applied distribute-lft-in 16.9

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

   9. Simplified 17.7

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

   10. Simplified 17.5

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

   12. Applied *-un-lft-identity 17.5

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

   13. Applied associate-*l* 17.5

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

   14. Simplified 17.2

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

   16. Applied associate-*r* 16.6

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\color{blue}{\left(a \cdot j\right) \cdot c} + 1 \cdot \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.311437424410602968393294732626591381699 \cdot 10^{-187}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + 1 \cdot \left(-1 \cdot \left(\left(i \cdot j\right) \cdot y\right)\right)\right)\\ \mathbf{elif}\;b \le -6.469196241460224510801417511235181480356 \cdot 10^{-289}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - 0\right) + \left(a \cdot \left(j \cdot c\right) + 1 \cdot \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\right)\\ \mathbf{elif}\;b \le 2.701853472416222263113483854524259515545 \cdot 10^{-61}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + 1 \cdot \left(-1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + 1 \cdot \left(-1 \cdot \left(\left(i \cdot j\right) \cdot y\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))