Average Error: 12.5 → 12.5
Time: 14.7s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.48709846260866168 \cdot 10^{119}:\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right)\\ \end{array}\]
\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}\;a \le -1.48709846260866168 \cdot 10^{119}:\\
\;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r128873 = x;
        double r128874 = y;
        double r128875 = z;
        double r128876 = r128874 * r128875;
        double r128877 = t;
        double r128878 = a;
        double r128879 = r128877 * r128878;
        double r128880 = r128876 - r128879;
        double r128881 = r128873 * r128880;
        double r128882 = b;
        double r128883 = c;
        double r128884 = r128883 * r128875;
        double r128885 = i;
        double r128886 = r128885 * r128878;
        double r128887 = r128884 - r128886;
        double r128888 = r128882 * r128887;
        double r128889 = r128881 - r128888;
        double r128890 = j;
        double r128891 = r128883 * r128877;
        double r128892 = r128885 * r128874;
        double r128893 = r128891 - r128892;
        double r128894 = r128890 * r128893;
        double r128895 = r128889 + r128894;
        return r128895;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r128896 = a;
        double r128897 = -1.4870984626086617e+119;
        bool r128898 = r128896 <= r128897;
        double r128899 = i;
        double r128900 = b;
        double r128901 = r128899 * r128900;
        double r128902 = z;
        double r128903 = c;
        double r128904 = r128900 * r128903;
        double r128905 = x;
        double r128906 = t;
        double r128907 = r128905 * r128906;
        double r128908 = r128896 * r128907;
        double r128909 = fma(r128902, r128904, r128908);
        double r128910 = -r128909;
        double r128911 = fma(r128896, r128901, r128910);
        double r128912 = r128903 * r128906;
        double r128913 = y;
        double r128914 = r128899 * r128913;
        double r128915 = r128912 - r128914;
        double r128916 = j;
        double r128917 = r128913 * r128902;
        double r128918 = r128906 * r128896;
        double r128919 = r128917 - r128918;
        double r128920 = r128905 * r128919;
        double r128921 = r128903 * r128902;
        double r128922 = r128899 * r128896;
        double r128923 = r128921 - r128922;
        double r128924 = cbrt(r128923);
        double r128925 = r128924 * r128924;
        double r128926 = r128925 * r128924;
        double r128927 = r128900 * r128926;
        double r128928 = r128920 - r128927;
        double r128929 = fma(r128915, r128916, r128928);
        double r128930 = r128898 ? r128911 : r128929;
        return r128930;
}

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 a < -1.4870984626086617e+119

    1. Initial program 23.1

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

    2. Simplified23.1

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

    3. Taylor expanded around inf 20.3

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

    4. Simplified20.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)}\]

    if -1.4870984626086617e+119 < a

    1. Initial program 11.3

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

    2. Simplified11.3

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

    4. Applied add-cube-cbrt11.6

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

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.48709846260866168 \cdot 10^{119}:\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\left(\sqrt[3]{c \cdot z - i \cdot a} \cdot \sqrt[3]{c \cdot z - i \cdot a}\right) \cdot \sqrt[3]{c \cdot z - i \cdot a}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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)))))