Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.6 → 10.8

Time: 26.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.615542034760807155180373229086399078369 \lor \neg \left(z \le 5.605088386505130723897825567204211548534 \cdot 10^{-117}\right):\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r85916 = x;
        double r85917 = y;
        double r85918 = z;
        double r85919 = r85917 * r85918;
        double r85920 = t;
        double r85921 = a;
        double r85922 = r85920 * r85921;
        double r85923 = r85919 - r85922;
        double r85924 = r85916 * r85923;
        double r85925 = b;
        double r85926 = c;
        double r85927 = r85926 * r85918;
        double r85928 = i;
        double r85929 = r85928 * r85921;
        double r85930 = r85927 - r85929;
        double r85931 = r85925 * r85930;
        double r85932 = r85924 - r85931;
        double r85933 = j;
        double r85934 = r85926 * r85920;
        double r85935 = r85928 * r85917;
        double r85936 = r85934 - r85935;
        double r85937 = r85933 * r85936;
        double r85938 = r85932 + r85937;
        return r85938;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r85939 = z;
        double r85940 = -7.615542034760807;
        bool r85941 = r85939 <= r85940;
        double r85942 = 5.605088386505131e-117;
        bool r85943 = r85939 <= r85942;
        double r85944 = !r85943;
        bool r85945 = r85941 || r85944;
        double r85946 = x;
        double r85947 = y;
        double r85948 = r85939 * r85947;
        double r85949 = r85946 * r85948;
        double r85950 = t;
        double r85951 = a;
        double r85952 = r85946 * r85951;
        double r85953 = r85950 * r85952;
        double r85954 = -r85953;
        double r85955 = r85949 + r85954;
        double r85956 = b;
        double r85957 = c;
        double r85958 = r85956 * r85957;
        double r85959 = r85939 * r85958;
        double r85960 = i;
        double r85961 = r85960 * r85951;
        double r85962 = -r85961;
        double r85963 = r85962 * r85956;
        double r85964 = r85959 + r85963;
        double r85965 = r85955 - r85964;
        double r85966 = j;
        double r85967 = r85966 * r85957;
        double r85968 = r85950 * r85967;
        double r85969 = r85966 * r85947;
        double r85970 = r85960 * r85969;
        double r85971 = -r85970;
        double r85972 = r85968 + r85971;
        double r85973 = r85965 + r85972;
        double r85974 = r85957 * r85939;
        double r85975 = r85974 - r85961;
        double r85976 = r85956 * r85975;
        double r85977 = r85955 - r85976;
        double r85978 = r85960 * r85966;
        double r85979 = r85978 * r85947;
        double r85980 = -r85979;
        double r85981 = r85968 + r85980;
        double r85982 = r85977 + r85981;
        double r85983 = r85945 ? r85973 : r85982;
        return r85983;
}

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

Derivation

1. Split input into 2 regimes

2. if z < -7.615542034760807 or 5.605088386505131e-117 < z

   1. Initial program 15.4

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

   3. Applied sub-neg 15.4

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

   4. Applied distribute-lft-in 15.4

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

   5. Simplified 15.6

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

   6. Simplified 15.6

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

   8. Applied sub-neg 15.6

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

   9. Applied distribute-lft-in 15.6

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

   10. Simplified 15.6

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

   11. Simplified 16.0

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

   12. Taylor expanded around inf 15.7

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

   14. Applied sub-neg 15.7

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

   15. Applied distribute-lft-in 15.7

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

   16. Simplified 12.8

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

   17. Simplified 12.8

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

   if -7.615542034760807 < z < 5.605088386505131e-117

   1. Initial program 9.7

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

   3. Applied sub-neg 9.7

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

   4. Applied distribute-lft-in 9.7

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

   5. Simplified 9.7

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

   6. Simplified 9.4

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

   8. Applied sub-neg 9.4

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

   9. Applied distribute-lft-in 9.4

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

   10. Simplified 9.4

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

   11. Simplified 10.4

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

   12. Taylor expanded around inf 9.7

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

   14. Applied associate-*r* 8.7

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

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.615542034760807155180373229086399078369 \lor \neg \left(z \le 5.605088386505130723897825567204211548534 \cdot 10^{-117}\right):\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - \left(z \cdot \left(b \cdot c\right) + \left(-i \cdot a\right) \cdot b\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-i \cdot \left(j \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(z \cdot y\right) + \left(-t \cdot \left(x \cdot a\right)\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \left(t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))