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

Average Error: 12.5 → 9.2

Time: 27.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.557156540206759491060099266665758475148 \cdot 10^{77}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{elif}\;b \le 6.288387807050133384086894370081620679201 \cdot 10^{-14}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\right) \cdot \sqrt[3]{c \cdot a - y \cdot i}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r510673 = x;
        double r510674 = y;
        double r510675 = z;
        double r510676 = r510674 * r510675;
        double r510677 = t;
        double r510678 = a;
        double r510679 = r510677 * r510678;
        double r510680 = r510676 - r510679;
        double r510681 = r510673 * r510680;
        double r510682 = b;
        double r510683 = c;
        double r510684 = r510683 * r510675;
        double r510685 = i;
        double r510686 = r510677 * r510685;
        double r510687 = r510684 - r510686;
        double r510688 = r510682 * r510687;
        double r510689 = r510681 - r510688;
        double r510690 = j;
        double r510691 = r510683 * r510678;
        double r510692 = r510674 * r510685;
        double r510693 = r510691 - r510692;
        double r510694 = r510690 * r510693;
        double r510695 = r510689 + r510694;
        return r510695;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r510696 = b;
        double r510697 = -2.5571565402067595e+77;
        bool r510698 = r510696 <= r510697;
        double r510699 = x;
        double r510700 = y;
        double r510701 = z;
        double r510702 = r510700 * r510701;
        double r510703 = t;
        double r510704 = a;
        double r510705 = r510703 * r510704;
        double r510706 = r510702 - r510705;
        double r510707 = r510699 * r510706;
        double r510708 = c;
        double r510709 = r510708 * r510701;
        double r510710 = i;
        double r510711 = r510703 * r510710;
        double r510712 = r510709 - r510711;
        double r510713 = r510696 * r510712;
        double r510714 = r510707 - r510713;
        double r510715 = j;
        double r510716 = r510708 * r510704;
        double r510717 = r510700 * r510710;
        double r510718 = r510716 - r510717;
        double r510719 = r510715 * r510718;
        double r510720 = cbrt(r510719);
        double r510721 = r510720 * r510720;
        double r510722 = r510721 * r510720;
        double r510723 = r510714 + r510722;
        double r510724 = 6.288387807050133e-14;
        bool r510725 = r510696 <= r510724;
        double r510726 = r510696 * r510708;
        double r510727 = r510701 * r510726;
        double r510728 = r510710 * r510696;
        double r510729 = r510703 * r510728;
        double r510730 = -r510729;
        double r510731 = r510727 + r510730;
        double r510732 = r510707 - r510731;
        double r510733 = r510719 + r510732;
        double r510734 = cbrt(r510718);
        double r510735 = r510734 * r510734;
        double r510736 = r510715 * r510735;
        double r510737 = r510736 * r510734;
        double r510738 = r510714 + r510737;
        double r510739 = r510725 ? r510733 : r510738;
        double r510740 = r510698 ? r510723 : r510739;
        return r510740;
}

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	12.5
Target	20.4
Herbie	9.2

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

Derivation

1. Split input into 3 regimes

2. if b < -2.5571565402067595e+77

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 7.6

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

   if -2.5571565402067595e+77 < b < 6.288387807050133e-14

   1. Initial program 15.3

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

   3. Applied sub-neg 15.3

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

   4. Applied distribute-lft-in 15.3

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

   5. Simplified 12.8

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

   7. Applied pow1 12.8

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

   8. Applied pow1 12.8

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

   9. Applied pow-prod-down 12.8

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

   10. Simplified 10.1

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

   if 6.288387807050133e-14 < b

   1. Initial program 7.4

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

   3. Applied add-cube-cbrt 7.5

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

   4. Applied associate-*r* 7.5

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.557156540206759491060099266665758475148 \cdot 10^{77}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(\sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\right) \cdot \sqrt[3]{j \cdot \left(c \cdot a - y \cdot i\right)}\\ \mathbf{elif}\;b \le 6.288387807050133384086894370081620679201 \cdot 10^{-14}:\\ \;\;\;\;j \cdot \left(c \cdot a - y \cdot i\right) + \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \left(-t \cdot \left(i \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(j \cdot \left(\sqrt[3]{c \cdot a - y \cdot i} \cdot \sqrt[3]{c \cdot a - y \cdot i}\right)\right) \cdot \sqrt[3]{c \cdot a - y \cdot i}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))))