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

Average Error: 11.9 → 10.7

Time: 7.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6.38829755749127996 \cdot 10^{-164} \lor \neg \left(y \le 8.34153219746767868 \cdot 10^{46}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(j \cdot c\right) \cdot a + -1 \cdot \left(\left(i \cdot j\right) \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a\right)\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r800667 = x;
        double r800668 = y;
        double r800669 = z;
        double r800670 = r800668 * r800669;
        double r800671 = t;
        double r800672 = a;
        double r800673 = r800671 * r800672;
        double r800674 = r800670 - r800673;
        double r800675 = r800667 * r800674;
        double r800676 = b;
        double r800677 = c;
        double r800678 = r800677 * r800669;
        double r800679 = i;
        double r800680 = r800671 * r800679;
        double r800681 = r800678 - r800680;
        double r800682 = r800676 * r800681;
        double r800683 = r800675 - r800682;
        double r800684 = j;
        double r800685 = r800677 * r800672;
        double r800686 = r800668 * r800679;
        double r800687 = r800685 - r800686;
        double r800688 = r800684 * r800687;
        double r800689 = r800683 + r800688;
        return r800689;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r800690 = y;
        double r800691 = -6.38829755749128e-164;
        bool r800692 = r800690 <= r800691;
        double r800693 = 8.341532197467679e+46;
        bool r800694 = r800690 <= r800693;
        double r800695 = !r800694;
        bool r800696 = r800692 || r800695;
        double r800697 = x;
        double r800698 = z;
        double r800699 = r800690 * r800698;
        double r800700 = t;
        double r800701 = a;
        double r800702 = r800700 * r800701;
        double r800703 = r800699 - r800702;
        double r800704 = r800697 * r800703;
        double r800705 = b;
        double r800706 = c;
        double r800707 = r800706 * r800698;
        double r800708 = i;
        double r800709 = r800700 * r800708;
        double r800710 = r800707 - r800709;
        double r800711 = r800705 * r800710;
        double r800712 = r800704 - r800711;
        double r800713 = j;
        double r800714 = r800713 * r800706;
        double r800715 = r800714 * r800701;
        double r800716 = -1.0;
        double r800717 = r800708 * r800713;
        double r800718 = r800717 * r800690;
        double r800719 = r800716 * r800718;
        double r800720 = r800715 + r800719;
        double r800721 = r800712 + r800720;
        double r800722 = cbrt(r800713);
        double r800723 = r800722 * r800722;
        double r800724 = r800706 * r800701;
        double r800725 = r800722 * r800724;
        double r800726 = r800723 * r800725;
        double r800727 = r800713 * r800690;
        double r800728 = r800708 * r800727;
        double r800729 = r800716 * r800728;
        double r800730 = r800726 + r800729;
        double r800731 = r800712 + r800730;
        double r800732 = r800696 ? r800721 : r800731;
        return r800732;
}

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	19.8
Herbie	10.7

\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -6.38829755749128e-164 or 8.341532197467679e+46 < y

   1. Initial program 14.6

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

   3. Applied sub-neg 14.6

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

   4. Applied distribute-lft-in 14.6

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

   5. Taylor expanded around inf 15.2

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

   7. Applied associate-*r* 14.5

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

   9. Applied associate-*r* 12.3

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

   if -6.38829755749128e-164 < y < 8.341532197467679e+46

   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 sub-neg 9.1

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

   4. Applied distribute-lft-in 9.1

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

   5. Taylor expanded around inf 8.9

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

   7. Applied add-cube-cbrt 9.1

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

   8. Applied associate-*l* 9.1

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\color{blue}{\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a\right)\right)} + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.38829755749127996 \cdot 10^{-164} \lor \neg \left(y \le 8.34153219746767868 \cdot 10^{46}\right):\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(j \cdot c\right) \cdot a + -1 \cdot \left(\left(i \cdot j\right) \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(c \cdot a\right)\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))