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

Average Error: 12.5 → 9.1

Time: 27.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -942100039185.614990234375 \lor \neg \left(x \le 1.377993057330842504113383725556413732388 \cdot 10^{120}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(-i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r622455 = x;
        double r622456 = y;
        double r622457 = z;
        double r622458 = r622456 * r622457;
        double r622459 = t;
        double r622460 = a;
        double r622461 = r622459 * r622460;
        double r622462 = r622458 - r622461;
        double r622463 = r622455 * r622462;
        double r622464 = b;
        double r622465 = c;
        double r622466 = r622465 * r622457;
        double r622467 = i;
        double r622468 = r622459 * r622467;
        double r622469 = r622466 - r622468;
        double r622470 = r622464 * r622469;
        double r622471 = r622463 - r622470;
        double r622472 = j;
        double r622473 = r622465 * r622460;
        double r622474 = r622456 * r622467;
        double r622475 = r622473 - r622474;
        double r622476 = r622472 * r622475;
        double r622477 = r622471 + r622476;
        return r622477;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r622478 = x;
        double r622479 = -942100039185.615;
        bool r622480 = r622478 <= r622479;
        double r622481 = 1.3779930573308425e+120;
        bool r622482 = r622478 <= r622481;
        double r622483 = !r622482;
        bool r622484 = r622480 || r622483;
        double r622485 = y;
        double r622486 = z;
        double r622487 = r622485 * r622486;
        double r622488 = t;
        double r622489 = a;
        double r622490 = r622488 * r622489;
        double r622491 = r622487 - r622490;
        double r622492 = r622478 * r622491;
        double r622493 = b;
        double r622494 = c;
        double r622495 = r622494 * r622486;
        double r622496 = i;
        double r622497 = r622488 * r622496;
        double r622498 = r622495 - r622497;
        double r622499 = r622493 * r622498;
        double r622500 = r622492 - r622499;
        double r622501 = j;
        double r622502 = cbrt(r622501);
        double r622503 = r622502 * r622502;
        double r622504 = r622494 * r622489;
        double r622505 = r622485 * r622496;
        double r622506 = r622504 - r622505;
        double r622507 = r622502 * r622506;
        double r622508 = r622503 * r622507;
        double r622509 = r622500 + r622508;
        double r622510 = r622478 * r622486;
        double r622511 = r622510 * r622485;
        double r622512 = r622478 * r622488;
        double r622513 = r622489 * r622512;
        double r622514 = -r622513;
        double r622515 = r622511 + r622514;
        double r622516 = r622493 * r622494;
        double r622517 = r622486 * r622516;
        double r622518 = r622496 * r622493;
        double r622519 = -r622518;
        double r622520 = r622488 * r622519;
        double r622521 = r622517 + r622520;
        double r622522 = r622515 - r622521;
        double r622523 = r622501 * r622506;
        double r622524 = r622522 + r622523;
        double r622525 = r622484 ? r622509 : r622524;
        return r622525;
}

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	12.5
Target	19.9
Herbie	9.1

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

Derivation

1. Split input into 2 regimes

2. if x < -942100039185.615 or 1.3779930573308425e+120 < x

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 7.6

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

   4. Applied associate-*l* 7.6

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

   if -942100039185.615 < x < 1.3779930573308425e+120

   1. Initial program 14.5

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

   3. Applied sub-neg 14.5

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

   4. Applied distribute-lft-in 14.5

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

   5. Simplified 14.5

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

   6. Simplified 12.5

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

   8. Applied sub-neg 12.5

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

   9. Applied distribute-lft-in 12.5

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

   10. Simplified 12.4

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

   11. Simplified 12.4

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

   13. Applied distribute-rgt-neg-in 12.4

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

   14. Applied associate-*l* 12.0

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

   15. Simplified 12.0

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

   17. Applied associate-*r* 9.7

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -942100039185.614990234375 \lor \neg \left(x \le 1.377993057330842504113383725556413732388 \cdot 10^{120}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(-i \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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 -1.46969429677770502e-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)))))