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

Average Error: 11.9 → 9.1

Time: 33.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.734842912893802954211285640305587494427 \cdot 10^{65}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \left(\sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)}\right)\\ \mathbf{elif}\;x \le 4.989813711427708520463647735774083823355 \cdot 10^{50}:\\ \;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y + \left(-t\right) \cdot \left(a \cdot x\right)\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)} \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)}\right) \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)}\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r36477672 = x;
        double r36477673 = y;
        double r36477674 = z;
        double r36477675 = r36477673 * r36477674;
        double r36477676 = t;
        double r36477677 = a;
        double r36477678 = r36477676 * r36477677;
        double r36477679 = r36477675 - r36477678;
        double r36477680 = r36477672 * r36477679;
        double r36477681 = b;
        double r36477682 = c;
        double r36477683 = r36477682 * r36477674;
        double r36477684 = i;
        double r36477685 = r36477676 * r36477684;
        double r36477686 = r36477683 - r36477685;
        double r36477687 = r36477681 * r36477686;
        double r36477688 = r36477680 - r36477687;
        double r36477689 = j;
        double r36477690 = r36477682 * r36477677;
        double r36477691 = r36477673 * r36477684;
        double r36477692 = r36477690 - r36477691;
        double r36477693 = r36477689 * r36477692;
        double r36477694 = r36477688 + r36477693;
        return r36477694;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r36477695 = x;
        double r36477696 = -8.734842912893803e+65;
        bool r36477697 = r36477695 <= r36477696;
        double r36477698 = y;
        double r36477699 = z;
        double r36477700 = r36477698 * r36477699;
        double r36477701 = a;
        double r36477702 = t;
        double r36477703 = r36477701 * r36477702;
        double r36477704 = r36477700 - r36477703;
        double r36477705 = r36477704 * r36477695;
        double r36477706 = b;
        double r36477707 = c;
        double r36477708 = r36477699 * r36477707;
        double r36477709 = i;
        double r36477710 = r36477709 * r36477702;
        double r36477711 = r36477708 - r36477710;
        double r36477712 = r36477706 * r36477711;
        double r36477713 = r36477705 - r36477712;
        double r36477714 = j;
        double r36477715 = r36477707 * r36477701;
        double r36477716 = r36477709 * r36477698;
        double r36477717 = r36477715 - r36477716;
        double r36477718 = r36477714 * r36477717;
        double r36477719 = cbrt(r36477718);
        double r36477720 = r36477719 * r36477719;
        double r36477721 = r36477719 * r36477720;
        double r36477722 = r36477713 + r36477721;
        double r36477723 = 4.9898137114277085e+50;
        bool r36477724 = r36477695 <= r36477723;
        double r36477725 = r36477699 * r36477695;
        double r36477726 = r36477725 * r36477698;
        double r36477727 = -r36477702;
        double r36477728 = r36477701 * r36477695;
        double r36477729 = r36477727 * r36477728;
        double r36477730 = r36477726 + r36477729;
        double r36477731 = r36477730 - r36477712;
        double r36477732 = r36477731 + r36477718;
        double r36477733 = cbrt(r36477712);
        double r36477734 = r36477733 * r36477733;
        double r36477735 = r36477734 * r36477733;
        double r36477736 = r36477705 - r36477735;
        double r36477737 = r36477736 + r36477718;
        double r36477738 = r36477724 ? r36477732 : r36477737;
        double r36477739 = r36477697 ? r36477722 : r36477738;
        return r36477739;
}

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	19.7
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 3 regimes

2. if x < -8.734842912893803e+65

   1. Initial program 6.2

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

   3. Applied add-cube-cbrt 6.4

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

   if -8.734842912893803e+65 < x < 4.9898137114277085e+50

   1. Initial program 14.0

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

   3. Applied add-cube-cbrt 14.2

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

   4. Applied associate-*l* 14.2

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

   6. Applied sub-neg 14.2

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

   7. Applied distribute-lft-in 14.2

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

   8. Applied distribute-lft-in 14.2

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

   9. Simplified 12.1

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

   10. Simplified 10.0

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

   if 4.9898137114277085e+50 < x

   1. Initial program 7.3

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

   3. Applied add-cube-cbrt 7.5

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

4. Final simplification 9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.734842912893802954211285640305587494427 \cdot 10^{65}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \left(\sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - i \cdot y\right)}\right)\\ \mathbf{elif}\;x \le 4.989813711427708520463647735774083823355 \cdot 10^{50}:\\ \;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y + \left(-t\right) \cdot \left(a \cdot x\right)\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(\sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)} \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)}\right) \cdot \sqrt[3]{b \cdot \left(z \cdot c - i \cdot t\right)}\right) + j \cdot \left(c \cdot a - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* 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) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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