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

Average Error: 11.8 → 9.4

Time: 15.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.1538663063516028 \cdot 10^{27}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{elif}\;b \le 1.1956400865377228 \cdot 10^{24}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(\left(-i\right) \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1072680 = x;
        double r1072681 = y;
        double r1072682 = z;
        double r1072683 = r1072681 * r1072682;
        double r1072684 = t;
        double r1072685 = a;
        double r1072686 = r1072684 * r1072685;
        double r1072687 = r1072683 - r1072686;
        double r1072688 = r1072680 * r1072687;
        double r1072689 = b;
        double r1072690 = c;
        double r1072691 = r1072690 * r1072682;
        double r1072692 = i;
        double r1072693 = r1072684 * r1072692;
        double r1072694 = r1072691 - r1072693;
        double r1072695 = r1072689 * r1072694;
        double r1072696 = r1072688 - r1072695;
        double r1072697 = j;
        double r1072698 = r1072690 * r1072685;
        double r1072699 = r1072681 * r1072692;
        double r1072700 = r1072698 - r1072699;
        double r1072701 = r1072697 * r1072700;
        double r1072702 = r1072696 + r1072701;
        return r1072702;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1072703 = b;
        double r1072704 = -1.1538663063516028e+27;
        bool r1072705 = r1072703 <= r1072704;
        double r1072706 = x;
        double r1072707 = y;
        double r1072708 = z;
        double r1072709 = r1072707 * r1072708;
        double r1072710 = t;
        double r1072711 = a;
        double r1072712 = r1072710 * r1072711;
        double r1072713 = r1072709 - r1072712;
        double r1072714 = r1072706 * r1072713;
        double r1072715 = c;
        double r1072716 = r1072715 * r1072708;
        double r1072717 = i;
        double r1072718 = r1072710 * r1072717;
        double r1072719 = r1072716 - r1072718;
        double r1072720 = r1072703 * r1072719;
        double r1072721 = r1072714 - r1072720;
        double r1072722 = j;
        double r1072723 = r1072715 * r1072711;
        double r1072724 = r1072707 * r1072717;
        double r1072725 = r1072723 - r1072724;
        double r1072726 = r1072722 * r1072725;
        double r1072727 = cbrt(r1072726);
        double r1072728 = r1072727 * r1072727;
        double r1072729 = r1072728 * r1072727;
        double r1072730 = r1072721 + r1072729;
        double r1072731 = 1.1956400865377228e+24;
        bool r1072732 = r1072703 <= r1072731;
        double r1072733 = r1072709 * r1072706;
        double r1072734 = r1072706 * r1072710;
        double r1072735 = r1072711 * r1072734;
        double r1072736 = -r1072735;
        double r1072737 = r1072733 + r1072736;
        double r1072738 = r1072703 * r1072715;
        double r1072739 = r1072708 * r1072738;
        double r1072740 = -r1072717;
        double r1072741 = r1072740 * r1072703;
        double r1072742 = r1072710 * r1072741;
        double r1072743 = r1072739 + r1072742;
        double r1072744 = r1072737 - r1072743;
        double r1072745 = r1072744 + r1072726;
        double r1072746 = sqrt(r1072703);
        double r1072747 = r1072746 * r1072719;
        double r1072748 = r1072746 * r1072747;
        double r1072749 = r1072714 - r1072748;
        double r1072750 = r1072749 + r1072726;
        double r1072751 = r1072732 ? r1072745 : r1072750;
        double r1072752 = r1072705 ? r1072730 : r1072751;
        return r1072752;
}

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	11.8
Target	19.8
Herbie	9.4

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

Derivation

1. Split input into 3 regimes

2. if b < -1.1538663063516028e+27

   1. Initial program 7.2

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

   3. Applied add-cube-cbrt 7.4

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

   if -1.1538663063516028e+27 < b < 1.1956400865377228e+24

   1. Initial program 14.1

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

   3. Applied sub-neg 14.1

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

   4. Applied distribute-lft-in 14.1

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

   5. Simplified 12.3

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

   6. Simplified 12.3

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

   8. Applied distribute-rgt-neg-in 12.3

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

   9. Applied associate-*l* 9.9

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

   11. Applied sub-neg 9.9

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

   12. Applied distribute-lft-in 9.9

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

   13. Simplified 9.9

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

   14. Simplified 10.4

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

   if 1.1956400865377228e+24 < b

   1. Initial program 7.6

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

   3. Applied add-sqr-sqrt 7.7

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

   4. Applied associate-*l* 7.7

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1538663063516028 \cdot 10^{27}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{elif}\;b \le 1.1956400865377228 \cdot 10^{24}:\\ \;\;\;\;\left(\left(\left(y \cdot z\right) \cdot x + \left(-a \cdot \left(x \cdot t\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + t \cdot \left(\left(-i\right) \cdot b\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \sqrt{b} \cdot \left(\sqrt{b} \cdot \left(c \cdot z - t \cdot i\right)\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))))