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

Average Error: 12.3 → 11.2

Time: 20.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -757396.01299941004253923892974853515625:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \le -2.220774384724803129038396915875251468025 \cdot 10^{-289}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;t \le 1.95019227045322671408130666215434279292 \cdot 10^{164}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r692581 = x;
        double r692582 = y;
        double r692583 = z;
        double r692584 = r692582 * r692583;
        double r692585 = t;
        double r692586 = a;
        double r692587 = r692585 * r692586;
        double r692588 = r692584 - r692587;
        double r692589 = r692581 * r692588;
        double r692590 = b;
        double r692591 = c;
        double r692592 = r692591 * r692583;
        double r692593 = i;
        double r692594 = r692585 * r692593;
        double r692595 = r692592 - r692594;
        double r692596 = r692590 * r692595;
        double r692597 = r692589 - r692596;
        double r692598 = j;
        double r692599 = r692591 * r692586;
        double r692600 = r692582 * r692593;
        double r692601 = r692599 - r692600;
        double r692602 = r692598 * r692601;
        double r692603 = r692597 + r692602;
        return r692603;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r692604 = t;
        double r692605 = -757396.01299941;
        bool r692606 = r692604 <= r692605;
        double r692607 = x;
        double r692608 = y;
        double r692609 = z;
        double r692610 = r692608 * r692609;
        double r692611 = a;
        double r692612 = r692604 * r692611;
        double r692613 = r692610 - r692612;
        double r692614 = r692607 * r692613;
        double r692615 = b;
        double r692616 = c;
        double r692617 = r692615 * r692616;
        double r692618 = r692609 * r692617;
        double r692619 = i;
        double r692620 = r692619 * r692615;
        double r692621 = r692604 * r692620;
        double r692622 = -r692621;
        double r692623 = r692618 + r692622;
        double r692624 = r692614 - r692623;
        double r692625 = j;
        double r692626 = r692611 * r692625;
        double r692627 = r692626 * r692616;
        double r692628 = r692608 * r692619;
        double r692629 = -r692628;
        double r692630 = r692629 * r692625;
        double r692631 = r692627 + r692630;
        double r692632 = r692624 + r692631;
        double r692633 = -2.220774384724803e-289;
        bool r692634 = r692604 <= r692633;
        double r692635 = r692616 * r692609;
        double r692636 = r692604 * r692619;
        double r692637 = r692635 - r692636;
        double r692638 = r692615 * r692637;
        double r692639 = r692614 - r692638;
        double r692640 = r692625 * r692616;
        double r692641 = r692611 * r692640;
        double r692642 = r692625 * r692608;
        double r692643 = r692619 * r692642;
        double r692644 = -r692643;
        double r692645 = r692641 + r692644;
        double r692646 = r692639 + r692645;
        double r692647 = 1.9501922704532267e+164;
        bool r692648 = r692604 <= r692647;
        double r692649 = r692609 * r692608;
        double r692650 = r692607 * r692649;
        double r692651 = r692607 * r692604;
        double r692652 = r692611 * r692651;
        double r692653 = r692650 - r692652;
        double r692654 = r692653 - r692638;
        double r692655 = r692654 + r692631;
        double r692656 = r692648 ? r692655 : r692632;
        double r692657 = r692634 ? r692646 : r692656;
        double r692658 = r692606 ? r692632 : r692657;
        return r692658;
}

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	12.3
Target	19.5
Herbie	11.2

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

Derivation

1. Split input into 3 regimes

2. if t < -757396.01299941 or 1.9501922704532267e+164 < t

   1. Initial program 17.6

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

   3. Applied sub-neg 17.6

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

   4. Applied distribute-lft-in 17.6

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

   5. Simplified 19.6

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

   6. Simplified 19.6

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

   8. Applied associate-*r* 17.9

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

   10. Applied sub-neg 17.9

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

   11. Applied distribute-lft-in 17.9

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

   12. Simplified 18.2

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

   13. Simplified 13.8

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

   if -757396.01299941 < t < -2.220774384724803e-289

   1. Initial program 10.1

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

   3. Applied sub-neg 10.1

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

   4. Applied distribute-lft-in 10.1

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

   5. Simplified 9.8

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

   6. Simplified 9.8

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

   8. Applied distribute-lft-neg-out 9.8

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

   9. Simplified 8.8

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

   if -2.220774384724803e-289 < t < 1.9501922704532267e+164

   1. Initial program 10.7

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

   3. Applied sub-neg 10.7

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

   4. Applied distribute-lft-in 10.7

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

   5. Simplified 10.4

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

   6. Simplified 10.4

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

   8. Applied associate-*r* 11.0

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

   9. Taylor expanded around inf 11.2

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -757396.01299941004253923892974853515625:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \le -2.220774384724803129038396915875251468025 \cdot 10^{-289}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;t \le 1.95019227045322671408130666215434279292 \cdot 10^{164}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) - a \cdot \left(x \cdot t\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-y \cdot i\right) \cdot j\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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)))))