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

Average Error: 12.3 → 11.8

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.936819268186205650838255402135178279408 \cdot 10^{-268} \lor \neg \left(a \le 1.284975181144422199319550481563476668578 \cdot 10^{121}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z} + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r855752 = x;
        double r855753 = y;
        double r855754 = z;
        double r855755 = r855753 * r855754;
        double r855756 = t;
        double r855757 = a;
        double r855758 = r855756 * r855757;
        double r855759 = r855755 - r855758;
        double r855760 = r855752 * r855759;
        double r855761 = b;
        double r855762 = c;
        double r855763 = r855762 * r855754;
        double r855764 = i;
        double r855765 = r855756 * r855764;
        double r855766 = r855763 - r855765;
        double r855767 = r855761 * r855766;
        double r855768 = r855760 - r855767;
        double r855769 = j;
        double r855770 = r855762 * r855757;
        double r855771 = r855753 * r855764;
        double r855772 = r855770 - r855771;
        double r855773 = r855769 * r855772;
        double r855774 = r855768 + r855773;
        return r855774;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r855775 = a;
        double r855776 = -2.9368192681862057e-268;
        bool r855777 = r855775 <= r855776;
        double r855778 = 1.2849751811444222e+121;
        bool r855779 = r855775 <= r855778;
        double r855780 = !r855779;
        bool r855781 = r855777 || r855780;
        double r855782 = x;
        double r855783 = y;
        double r855784 = r855782 * r855783;
        double r855785 = z;
        double r855786 = r855784 * r855785;
        double r855787 = cbrt(r855786);
        double r855788 = r855787 * r855787;
        double r855789 = r855788 * r855787;
        double r855790 = -1.0;
        double r855791 = t;
        double r855792 = r855782 * r855791;
        double r855793 = r855775 * r855792;
        double r855794 = r855790 * r855793;
        double r855795 = 1.0;
        double r855796 = pow(r855794, r855795);
        double r855797 = r855789 + r855796;
        double r855798 = b;
        double r855799 = c;
        double r855800 = r855799 * r855785;
        double r855801 = i;
        double r855802 = r855791 * r855801;
        double r855803 = r855800 - r855802;
        double r855804 = r855798 * r855803;
        double r855805 = r855797 - r855804;
        double r855806 = j;
        double r855807 = r855799 * r855775;
        double r855808 = r855783 * r855801;
        double r855809 = r855807 - r855808;
        double r855810 = r855806 * r855809;
        double r855811 = r855805 + r855810;
        double r855812 = cbrt(r855782);
        double r855813 = r855812 * r855812;
        double r855814 = r855783 * r855785;
        double r855815 = r855812 * r855814;
        double r855816 = r855813 * r855815;
        double r855817 = r855791 * r855775;
        double r855818 = -r855817;
        double r855819 = r855782 * r855818;
        double r855820 = r855816 + r855819;
        double r855821 = r855820 - r855804;
        double r855822 = r855821 + r855810;
        double r855823 = r855781 ? r855811 : r855822;
        return r855823;
}

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	20.3
Herbie	11.8

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

Derivation

1. Split input into 2 regimes

2. if a < -2.9368192681862057e-268 or 1.2849751811444222e+121 < a

   1. Initial program 14.0

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

   3. Applied sub-neg 14.0

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

   4. Applied distribute-lft-in 14.0

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

   6. Applied associate-*r* 14.1

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

   8. Applied pow1 14.1

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

   9. Applied pow1 14.1

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

   10. Applied pow-prod-down 14.1

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

   11. Simplified 12.9

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

   13. Applied add-cube-cbrt 13.0

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

   if -2.9368192681862057e-268 < a < 1.2849751811444222e+121

   1. Initial program 10.1

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

   3. Applied sub-neg 10.1

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

   4. Applied distribute-lft-in 10.1

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

   6. Applied add-cube-cbrt 10.3

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

   7. Applied associate-*l* 10.3

      \[\leadsto \left(\left(\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right)} + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.936819268186205650838255402135178279408 \cdot 10^{-268} \lor \neg \left(a \le 1.284975181144422199319550481563476668578 \cdot 10^{121}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\left(x \cdot y\right) \cdot z} \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z}\right) \cdot \sqrt[3]{\left(x \cdot y\right) \cdot z} + {\left(-1 \cdot \left(a \cdot \left(x \cdot t\right)\right)\right)}^{1}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z\right)\right) + x \cdot \left(-t \cdot a\right)\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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)))))