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

Average Error: 11.7 → 8.8

Time: 26.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -7.292924892323122 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(-i \cdot \left(y \cdot j\right)\right) + \left(c \cdot j\right) \cdot a\right)\\ \mathbf{elif}\;a \le 1.3675312070107937 \cdot 10^{-118}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt[3]{c \cdot a - i \cdot y} \cdot \left(\left(\sqrt[3]{c \cdot a - i \cdot y} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(j \cdot \left(y \cdot \left(-i\right)\right) + \left(c \cdot j\right) \cdot a\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r36768710 = x;
        double r36768711 = y;
        double r36768712 = z;
        double r36768713 = r36768711 * r36768712;
        double r36768714 = t;
        double r36768715 = a;
        double r36768716 = r36768714 * r36768715;
        double r36768717 = r36768713 - r36768716;
        double r36768718 = r36768710 * r36768717;
        double r36768719 = b;
        double r36768720 = c;
        double r36768721 = r36768720 * r36768712;
        double r36768722 = i;
        double r36768723 = r36768714 * r36768722;
        double r36768724 = r36768721 - r36768723;
        double r36768725 = r36768719 * r36768724;
        double r36768726 = r36768718 - r36768725;
        double r36768727 = j;
        double r36768728 = r36768720 * r36768715;
        double r36768729 = r36768711 * r36768722;
        double r36768730 = r36768728 - r36768729;
        double r36768731 = r36768727 * r36768730;
        double r36768732 = r36768726 + r36768731;
        return r36768732;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r36768733 = a;
        double r36768734 = -7.292924892323122e-25;
        bool r36768735 = r36768733 <= r36768734;
        double r36768736 = x;
        double r36768737 = z;
        double r36768738 = r36768736 * r36768737;
        double r36768739 = y;
        double r36768740 = r36768738 * r36768739;
        double r36768741 = t;
        double r36768742 = r36768741 * r36768736;
        double r36768743 = r36768742 * r36768733;
        double r36768744 = r36768740 - r36768743;
        double r36768745 = b;
        double r36768746 = c;
        double r36768747 = r36768737 * r36768746;
        double r36768748 = i;
        double r36768749 = r36768748 * r36768741;
        double r36768750 = r36768747 - r36768749;
        double r36768751 = r36768745 * r36768750;
        double r36768752 = r36768744 - r36768751;
        double r36768753 = j;
        double r36768754 = r36768739 * r36768753;
        double r36768755 = r36768748 * r36768754;
        double r36768756 = -r36768755;
        double r36768757 = r36768746 * r36768753;
        double r36768758 = r36768757 * r36768733;
        double r36768759 = r36768756 + r36768758;
        double r36768760 = r36768752 + r36768759;
        double r36768761 = 1.3675312070107937e-118;
        bool r36768762 = r36768733 <= r36768761;
        double r36768763 = r36768739 * r36768737;
        double r36768764 = r36768733 * r36768741;
        double r36768765 = r36768763 - r36768764;
        double r36768766 = r36768736 * r36768765;
        double r36768767 = r36768766 - r36768751;
        double r36768768 = r36768746 * r36768733;
        double r36768769 = r36768748 * r36768739;
        double r36768770 = r36768768 - r36768769;
        double r36768771 = cbrt(r36768770);
        double r36768772 = r36768771 * r36768771;
        double r36768773 = r36768772 * r36768753;
        double r36768774 = r36768771 * r36768773;
        double r36768775 = r36768767 + r36768774;
        double r36768776 = r36768763 * r36768736;
        double r36768777 = r36768776 - r36768743;
        double r36768778 = r36768777 - r36768751;
        double r36768779 = -r36768748;
        double r36768780 = r36768739 * r36768779;
        double r36768781 = r36768753 * r36768780;
        double r36768782 = r36768781 + r36768758;
        double r36768783 = r36768778 + r36768782;
        double r36768784 = r36768762 ? r36768775 : r36768783;
        double r36768785 = r36768735 ? r36768760 : r36768784;
        return r36768785;
}

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	11.7
Target	18.4
Herbie	8.8

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -7.292924892323122e-25

   1. Initial program 15.2

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

   3. Applied add-cube-cbrt 15.5

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

   4. Applied associate-*l* 15.5

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

   5. Taylor expanded around inf 12.0

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

   7. Applied sub-neg 12.0

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

   8. Applied distribute-lft-in 12.0

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

   9. Applied distribute-lft-in 12.0

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

   10. Simplified 8.1

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

   11. Simplified 7.1

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

   13. Applied associate-*r* 7.0

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

   if -7.292924892323122e-25 < a < 1.3675312070107937e-118

   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 9.4

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

   4. Applied associate-*r* 9.4

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

   if 1.3675312070107937e-118 < a

   1. Initial program 13.4

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

   3. Applied add-cube-cbrt 13.7

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

   4. Applied associate-*l* 13.7

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

   5. Taylor expanded around inf 11.8

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

   7. Applied sub-neg 11.8

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

   8. Applied distribute-lft-in 11.8

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

   9. Applied distribute-lft-in 11.8

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

   10. Simplified 9.3

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

   11. Simplified 8.8

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

   13. Applied associate-*l* 9.2

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.292924892323122 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(\left(x \cdot z\right) \cdot y - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(\left(-i \cdot \left(y \cdot j\right)\right) + \left(c \cdot j\right) \cdot a\right)\\ \mathbf{elif}\;a \le 1.3675312070107937 \cdot 10^{-118}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \sqrt[3]{c \cdot a - i \cdot y} \cdot \left(\left(\sqrt[3]{c \cdot a - i \cdot y} \cdot \sqrt[3]{c \cdot a - i \cdot y}\right) \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x - \left(t \cdot x\right) \cdot a\right) - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(j \cdot \left(y \cdot \left(-i\right)\right) + \left(c \cdot j\right) \cdot a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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) (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)))))