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

Average Error: 12.3 → 11.8

Time: 8.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r893839 = x;
        double r893840 = y;
        double r893841 = z;
        double r893842 = r893840 * r893841;
        double r893843 = t;
        double r893844 = a;
        double r893845 = r893843 * r893844;
        double r893846 = r893842 - r893845;
        double r893847 = r893839 * r893846;
        double r893848 = b;
        double r893849 = c;
        double r893850 = r893849 * r893841;
        double r893851 = i;
        double r893852 = r893843 * r893851;
        double r893853 = r893850 - r893852;
        double r893854 = r893848 * r893853;
        double r893855 = r893847 - r893854;
        double r893856 = j;
        double r893857 = r893849 * r893844;
        double r893858 = r893840 * r893851;
        double r893859 = r893857 - r893858;
        double r893860 = r893856 * r893859;
        double r893861 = r893855 + r893860;
        return r893861;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r893862 = a;
        double r893863 = -2.9368192681862057e-268;
        bool r893864 = r893862 <= r893863;
        double r893865 = 1.2849751811444222e+121;
        bool r893866 = r893862 <= r893865;
        double r893867 = !r893866;
        bool r893868 = r893864 || r893867;
        double r893869 = x;
        double r893870 = y;
        double r893871 = r893869 * r893870;
        double r893872 = z;
        double r893873 = r893871 * r893872;
        double r893874 = cbrt(r893873);
        double r893875 = r893874 * r893874;
        double r893876 = r893875 * r893874;
        double r893877 = -1.0;
        double r893878 = t;
        double r893879 = r893869 * r893878;
        double r893880 = r893862 * r893879;
        double r893881 = r893877 * r893880;
        double r893882 = r893876 + r893881;
        double r893883 = b;
        double r893884 = c;
        double r893885 = r893884 * r893872;
        double r893886 = i;
        double r893887 = r893878 * r893886;
        double r893888 = r893885 - r893887;
        double r893889 = r893883 * r893888;
        double r893890 = r893882 - r893889;
        double r893891 = j;
        double r893892 = r893884 * r893862;
        double r893893 = r893870 * r893886;
        double r893894 = r893892 - r893893;
        double r893895 = r893891 * r893894;
        double r893896 = r893890 + r893895;
        double r893897 = cbrt(r893869);
        double r893898 = r893897 * r893897;
        double r893899 = r893870 * r893872;
        double r893900 = r893897 * r893899;
        double r893901 = r893898 * r893900;
        double r893902 = r893878 * r893862;
        double r893903 = -r893902;
        double r893904 = r893869 * r893903;
        double r893905 = r893901 + r893904;
        double r893906 = r893905 - r893889;
        double r893907 = r893906 + r893895;
        double r893908 = r893868 ? r893896 : r893907;
        return r893908;
}

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. Taylor expanded around inf 12.9

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

   9. Applied add-cube-cbrt 13.0

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

   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

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

Reproduce

herbie shell --seed 2020001 
(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)))))