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

Average Error: 12.1 → 10.6

Time: 8.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;j \le -7.5553708004463452 \cdot 10^{121}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 1.0086390222625704 \cdot 10^{124}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) - i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r803657 = x;
        double r803658 = y;
        double r803659 = z;
        double r803660 = r803658 * r803659;
        double r803661 = t;
        double r803662 = a;
        double r803663 = r803661 * r803662;
        double r803664 = r803660 - r803663;
        double r803665 = r803657 * r803664;
        double r803666 = b;
        double r803667 = c;
        double r803668 = r803667 * r803659;
        double r803669 = i;
        double r803670 = r803661 * r803669;
        double r803671 = r803668 - r803670;
        double r803672 = r803666 * r803671;
        double r803673 = r803665 - r803672;
        double r803674 = j;
        double r803675 = r803667 * r803662;
        double r803676 = r803658 * r803669;
        double r803677 = r803675 - r803676;
        double r803678 = r803674 * r803677;
        double r803679 = r803673 + r803678;
        return r803679;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r803680 = j;
        double r803681 = -7.555370800446345e+121;
        bool r803682 = r803680 <= r803681;
        double r803683 = x;
        double r803684 = y;
        double r803685 = z;
        double r803686 = r803684 * r803685;
        double r803687 = t;
        double r803688 = a;
        double r803689 = r803687 * r803688;
        double r803690 = r803686 - r803689;
        double r803691 = cbrt(r803690);
        double r803692 = r803691 * r803691;
        double r803693 = r803683 * r803692;
        double r803694 = r803693 * r803691;
        double r803695 = b;
        double r803696 = c;
        double r803697 = r803696 * r803685;
        double r803698 = i;
        double r803699 = r803687 * r803698;
        double r803700 = r803697 - r803699;
        double r803701 = r803695 * r803700;
        double r803702 = r803694 - r803701;
        double r803703 = r803696 * r803688;
        double r803704 = r803684 * r803698;
        double r803705 = r803703 - r803704;
        double r803706 = r803680 * r803705;
        double r803707 = r803702 + r803706;
        double r803708 = 1.0086390222625704e+124;
        bool r803709 = r803680 <= r803708;
        double r803710 = r803683 * r803690;
        double r803711 = r803710 - r803701;
        double r803712 = r803680 * r803696;
        double r803713 = r803688 * r803712;
        double r803714 = r803684 * r803680;
        double r803715 = r803698 * r803714;
        double r803716 = r803713 - r803715;
        double r803717 = r803711 + r803716;
        double r803718 = 0.0;
        double r803719 = r803718 - r803701;
        double r803720 = r803719 + r803706;
        double r803721 = r803709 ? r803717 : r803720;
        double r803722 = r803682 ? r803707 : r803721;
        return r803722;
}

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	12.1
Target	20.5
Herbie	10.6

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

Derivation

1. Split input into 3 regimes

2. if j < -7.555370800446345e+121

   1. Initial program 7.0

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

   3. Applied add-cube-cbrt 7.2

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

   4. Applied associate-*r* 7.2

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

   if -7.555370800446345e+121 < j < 1.0086390222625704e+124

   1. Initial program 13.4

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

   3. Applied add-cube-cbrt 13.6

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

   4. Applied associate-*l* 13.6

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

   6. Applied add-cube-cbrt 13.7

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

   7. Applied cbrt-prod 13.7

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

   8. Applied associate-*l* 13.7

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

   9. Taylor expanded around inf 10.5

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

   if 1.0086390222625704e+124 < j

   1. Initial program 6.6

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

   2. Taylor expanded around 0 14.9

      \[\leadsto \left(\color{blue}{0} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -7.5553708004463452 \cdot 10^{121}:\\ \;\;\;\;\left(\left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;j \le 1.0086390222625704 \cdot 10^{124}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + \left(a \cdot \left(j \cdot c\right) - i \cdot \left(y \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0 - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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)))))