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

Average Error: 12.2 → 12.2

Time: 9.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -7.590287677643962220134055107004784190296 \cdot 10^{-240}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{c \cdot z - t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 3.850850789495975179857330160448870508298 \cdot 10^{-202}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + 0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \sqrt{j} \cdot \left(\sqrt{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r907690 = x;
        double r907691 = y;
        double r907692 = z;
        double r907693 = r907691 * r907692;
        double r907694 = t;
        double r907695 = a;
        double r907696 = r907694 * r907695;
        double r907697 = r907693 - r907696;
        double r907698 = r907690 * r907697;
        double r907699 = b;
        double r907700 = c;
        double r907701 = r907700 * r907692;
        double r907702 = i;
        double r907703 = r907694 * r907702;
        double r907704 = r907701 - r907703;
        double r907705 = r907699 * r907704;
        double r907706 = r907698 - r907705;
        double r907707 = j;
        double r907708 = r907700 * r907695;
        double r907709 = r907691 * r907702;
        double r907710 = r907708 - r907709;
        double r907711 = r907707 * r907710;
        double r907712 = r907706 + r907711;
        return r907712;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r907713 = j;
        double r907714 = -7.590287677643962e-240;
        bool r907715 = r907713 <= r907714;
        double r907716 = x;
        double r907717 = y;
        double r907718 = z;
        double r907719 = r907717 * r907718;
        double r907720 = t;
        double r907721 = a;
        double r907722 = r907720 * r907721;
        double r907723 = r907719 - r907722;
        double r907724 = r907716 * r907723;
        double r907725 = b;
        double r907726 = c;
        double r907727 = r907726 * r907718;
        double r907728 = i;
        double r907729 = r907720 * r907728;
        double r907730 = r907727 - r907729;
        double r907731 = r907725 * r907730;
        double r907732 = cbrt(r907731);
        double r907733 = r907732 * r907732;
        double r907734 = cbrt(r907725);
        double r907735 = cbrt(r907730);
        double r907736 = r907734 * r907735;
        double r907737 = r907733 * r907736;
        double r907738 = r907724 - r907737;
        double r907739 = r907726 * r907721;
        double r907740 = r907717 * r907728;
        double r907741 = r907739 - r907740;
        double r907742 = r907713 * r907741;
        double r907743 = r907738 + r907742;
        double r907744 = 3.850850789495975e-202;
        bool r907745 = r907713 <= r907744;
        double r907746 = r907733 * r907732;
        double r907747 = r907724 - r907746;
        double r907748 = 0.0;
        double r907749 = r907747 + r907748;
        double r907750 = r907724 - r907731;
        double r907751 = sqrt(r907713);
        double r907752 = r907751 * r907741;
        double r907753 = r907751 * r907752;
        double r907754 = r907750 + r907753;
        double r907755 = r907745 ? r907749 : r907754;
        double r907756 = r907715 ? r907743 : r907755;
        return r907756;
}

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	12.2
Target	20.2
Herbie	12.2

\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if j < -7.590287677643962e-240

   1. Initial program 11.5

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

   3. Applied add-cube-cbrt 11.8

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

   5. Applied cbrt-prod 11.8

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

   if -7.590287677643962e-240 < j < 3.850850789495975e-202

   1. Initial program 17.3

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

   3. Applied add-cube-cbrt 17.6

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

   4. Taylor expanded around 0 16.3

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + \color{blue}{0}\]

   if 3.850850789495975e-202 < j

   1. Initial program 10.7

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

   3. Applied add-sqr-sqrt 10.8

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

   4. Applied associate-*l* 10.8

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

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -7.590287677643962220134055107004784190296 \cdot 10^{-240}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{c \cdot z - t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 3.850850789495975179857330160448870508298 \cdot 10^{-202}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + 0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \sqrt{j} \cdot \left(\sqrt{j} \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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)))))