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

Average Error: 12.3 → 12.5

Time: 8.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.0888365422142503 \cdot 10^{-114}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + \left(a \cdot \left(j \cdot c\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;x \le 2.6424334011412275 \cdot 10^{-227}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-j\right) \cdot \left(y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1027684 = x;
        double r1027685 = y;
        double r1027686 = z;
        double r1027687 = r1027685 * r1027686;
        double r1027688 = t;
        double r1027689 = a;
        double r1027690 = r1027688 * r1027689;
        double r1027691 = r1027687 - r1027690;
        double r1027692 = r1027684 * r1027691;
        double r1027693 = b;
        double r1027694 = c;
        double r1027695 = r1027694 * r1027686;
        double r1027696 = i;
        double r1027697 = r1027688 * r1027696;
        double r1027698 = r1027695 - r1027697;
        double r1027699 = r1027693 * r1027698;
        double r1027700 = r1027692 - r1027699;
        double r1027701 = j;
        double r1027702 = r1027694 * r1027689;
        double r1027703 = r1027685 * r1027696;
        double r1027704 = r1027702 - r1027703;
        double r1027705 = r1027701 * r1027704;
        double r1027706 = r1027700 + r1027705;
        return r1027706;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1027707 = x;
        double r1027708 = -2.0888365422142503e-114;
        bool r1027709 = r1027707 <= r1027708;
        double r1027710 = y;
        double r1027711 = z;
        double r1027712 = r1027710 * r1027711;
        double r1027713 = t;
        double r1027714 = a;
        double r1027715 = r1027713 * r1027714;
        double r1027716 = r1027712 - r1027715;
        double r1027717 = r1027707 * r1027716;
        double r1027718 = b;
        double r1027719 = cbrt(r1027718);
        double r1027720 = r1027719 * r1027719;
        double r1027721 = c;
        double r1027722 = r1027721 * r1027711;
        double r1027723 = i;
        double r1027724 = r1027713 * r1027723;
        double r1027725 = r1027722 - r1027724;
        double r1027726 = r1027719 * r1027725;
        double r1027727 = r1027720 * r1027726;
        double r1027728 = r1027717 - r1027727;
        double r1027729 = j;
        double r1027730 = r1027729 * r1027721;
        double r1027731 = r1027714 * r1027730;
        double r1027732 = -1.0;
        double r1027733 = r1027729 * r1027710;
        double r1027734 = r1027723 * r1027733;
        double r1027735 = r1027732 * r1027734;
        double r1027736 = r1027731 + r1027735;
        double r1027737 = r1027728 + r1027736;
        double r1027738 = 2.6424334011412275e-227;
        bool r1027739 = r1027707 <= r1027738;
        double r1027740 = 0.0;
        double r1027741 = r1027718 * r1027725;
        double r1027742 = r1027740 - r1027741;
        double r1027743 = r1027721 * r1027714;
        double r1027744 = r1027710 * r1027723;
        double r1027745 = r1027743 - r1027744;
        double r1027746 = r1027729 * r1027745;
        double r1027747 = r1027742 + r1027746;
        double r1027748 = r1027717 - r1027741;
        double r1027749 = r1027714 * r1027729;
        double r1027750 = r1027749 * r1027721;
        double r1027751 = -r1027729;
        double r1027752 = r1027751 * r1027744;
        double r1027753 = r1027750 + r1027752;
        double r1027754 = r1027748 + r1027753;
        double r1027755 = r1027739 ? r1027747 : r1027754;
        double r1027756 = r1027709 ? r1027737 : r1027755;
        return r1027756;
}

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	19.9
Herbie	12.5

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

Derivation

1. Split input into 3 regimes

2. if x < -2.0888365422142503e-114

   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 9.3

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

   4. Applied associate-*l* 9.3

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

   6. Applied sub-neg 9.3

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

   7. Applied distribute-lft-in 9.3

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

   8. Applied distribute-lft-in 9.4

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

   9. Simplified 9.8

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

   10. Simplified 9.7

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

   11. Taylor expanded around inf 9.3

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

   13. Applied add-cube-cbrt 9.6

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

   14. Applied associate-*l* 9.6

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

   if -2.0888365422142503e-114 < x < 2.6424334011412275e-227

   1. Initial program 16.3

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

   2. Taylor expanded around 0 17.0

      \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]

   if 2.6424334011412275e-227 < x

   1. Initial program 11.7

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

   3. Applied add-cube-cbrt 12.0

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

   4. Applied associate-*l* 12.0

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

   6. Applied sub-neg 12.0

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

   7. Applied distribute-lft-in 12.0

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

   8. Applied distribute-lft-in 12.0

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

   9. Simplified 11.8

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

   10. Simplified 11.7

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

   12. Applied associate-*r* 11.5

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.0888365422142503 \cdot 10^{-114}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + \left(a \cdot \left(j \cdot c\right) + -1 \cdot \left(i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{elif}\;x \le 2.6424334011412275 \cdot 10^{-227}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\left(a \cdot j\right) \cdot c + \left(-j\right) \cdot \left(y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))))