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

Average Error: 11.9 → 11.1

Time: 20.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -34.94627947654921484854639857076108455658:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 2.275676108404146604755973752542807803943 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \sqrt{j} \cdot \left(\sqrt{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r456690 = x;
        double r456691 = y;
        double r456692 = z;
        double r456693 = r456691 * r456692;
        double r456694 = t;
        double r456695 = a;
        double r456696 = r456694 * r456695;
        double r456697 = r456693 - r456696;
        double r456698 = r456690 * r456697;
        double r456699 = b;
        double r456700 = c;
        double r456701 = r456700 * r456692;
        double r456702 = i;
        double r456703 = r456694 * r456702;
        double r456704 = r456701 - r456703;
        double r456705 = r456699 * r456704;
        double r456706 = r456698 - r456705;
        double r456707 = j;
        double r456708 = r456700 * r456695;
        double r456709 = r456691 * r456702;
        double r456710 = r456708 - r456709;
        double r456711 = r456707 * r456710;
        double r456712 = r456706 + r456711;
        return r456712;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r456713 = j;
        double r456714 = -34.946279476549215;
        bool r456715 = r456713 <= r456714;
        double r456716 = y;
        double r456717 = z;
        double r456718 = r456716 * r456717;
        double r456719 = x;
        double r456720 = r456718 * r456719;
        double r456721 = t;
        double r456722 = a;
        double r456723 = r456719 * r456722;
        double r456724 = r456721 * r456723;
        double r456725 = -r456724;
        double r456726 = r456720 + r456725;
        double r456727 = b;
        double r456728 = c;
        double r456729 = r456727 * r456728;
        double r456730 = r456717 * r456729;
        double r456731 = i;
        double r456732 = r456721 * r456731;
        double r456733 = -r456732;
        double r456734 = r456727 * r456733;
        double r456735 = r456730 + r456734;
        double r456736 = r456726 - r456735;
        double r456737 = r456728 * r456722;
        double r456738 = r456716 * r456731;
        double r456739 = r456737 - r456738;
        double r456740 = r456713 * r456739;
        double r456741 = r456736 + r456740;
        double r456742 = 2.2756761084041466e-122;
        bool r456743 = r456713 <= r456742;
        double r456744 = r456719 * r456721;
        double r456745 = r456722 * r456744;
        double r456746 = -r456745;
        double r456747 = r456720 + r456746;
        double r456748 = r456728 * r456717;
        double r456749 = r456748 - r456732;
        double r456750 = r456727 * r456749;
        double r456751 = r456747 - r456750;
        double r456752 = r456713 * r456728;
        double r456753 = r456722 * r456752;
        double r456754 = -r456738;
        double r456755 = r456754 * r456713;
        double r456756 = r456753 + r456755;
        double r456757 = r456751 + r456756;
        double r456758 = r456717 * r456719;
        double r456759 = r456716 * r456758;
        double r456760 = r456759 + r456725;
        double r456761 = r456760 - r456750;
        double r456762 = sqrt(r456713);
        double r456763 = r456762 * r456739;
        double r456764 = r456762 * r456763;
        double r456765 = r456761 + r456764;
        double r456766 = r456743 ? r456757 : r456765;
        double r456767 = r456715 ? r456741 : r456766;
        return r456767;
}

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	11.1

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

Derivation

1. Split input into 3 regimes

2. if j < -34.946279476549215

   1. Initial program 6.5

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

   3. Applied sub-neg 6.5

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

   4. Applied distribute-lft-in 6.5

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

   5. Simplified 6.5

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

   6. Simplified 7.2

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

   8. Applied *-un-lft-identity 7.2

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

   9. Applied associate-*l* 7.2

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

   10. Simplified 6.6

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

   12. Applied sub-neg 6.6

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

   13. Applied distribute-lft-in 6.6

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

   14. Simplified 7.0

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

   if -34.946279476549215 < j < 2.2756761084041466e-122

   1. Initial program 15.6

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

   3. Applied sub-neg 15.6

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

   4. Applied distribute-lft-in 15.6

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

   5. Simplified 15.6

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

   6. Simplified 15.9

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

   8. Applied sub-neg 15.9

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

   9. Applied distribute-lft-in 15.9

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

   10. Simplified 13.0

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

   11. Simplified 13.0

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

   if 2.2756761084041466e-122 < j

   1. Initial program 9.4

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

   3. Applied sub-neg 9.4

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

   4. Applied distribute-lft-in 9.4

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

   5. Simplified 9.4

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

   6. Simplified 10.5

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

   8. Applied *-un-lft-identity 10.5

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

   9. Applied associate-*l* 10.5

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

   10. Simplified 10.5

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

   12. Applied associate-*l* 10.4

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

   14. Applied add-sqr-sqrt 10.5

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

   15. Applied associate-*l* 10.5

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -34.94627947654921484854639857076108455658:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + b \cdot \left(-t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 2.275676108404146604755973752542807803943 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) + \left(-y \cdot i\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot \left(z \cdot x\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \sqrt{j} \cdot \left(\sqrt{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \end{array}\]

Reproduce

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