Average Error: 12.2 → 13.9
Time: 33.8s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -8268102197384482988509626368:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, z \cdot \left(x \cdot y - b \cdot c\right) - x \cdot \left(t \cdot a\right)\right)\\ \mathbf{elif}\;c \le -6.535140165540710269045334632618486765212 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, \mathsf{fma}\left(i \cdot a - c \cdot z, b, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, \mathsf{fma}\left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \sqrt[3]{i \cdot a - c \cdot z}\right), b, x \cdot \left(z \cdot y - t \cdot a\right)\right)\right)\\ \end{array}\]
\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}\;c \le -8268102197384482988509626368:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, z \cdot \left(x \cdot y - b \cdot c\right) - x \cdot \left(t \cdot a\right)\right)\\

\mathbf{elif}\;c \le -6.535140165540710269045334632618486765212 \cdot 10^{-32}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, \mathsf{fma}\left(i \cdot a - c \cdot z, b, 0\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, \mathsf{fma}\left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \sqrt[3]{i \cdot a - c \cdot z}\right), b, x \cdot \left(z \cdot y - t \cdot a\right)\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 r4352886 = x;
        double r4352887 = y;
        double r4352888 = z;
        double r4352889 = r4352887 * r4352888;
        double r4352890 = t;
        double r4352891 = a;
        double r4352892 = r4352890 * r4352891;
        double r4352893 = r4352889 - r4352892;
        double r4352894 = r4352886 * r4352893;
        double r4352895 = b;
        double r4352896 = c;
        double r4352897 = r4352896 * r4352888;
        double r4352898 = i;
        double r4352899 = r4352898 * r4352891;
        double r4352900 = r4352897 - r4352899;
        double r4352901 = r4352895 * r4352900;
        double r4352902 = r4352894 - r4352901;
        double r4352903 = j;
        double r4352904 = r4352896 * r4352890;
        double r4352905 = r4352898 * r4352887;
        double r4352906 = r4352904 - r4352905;
        double r4352907 = r4352903 * r4352906;
        double r4352908 = r4352902 + r4352907;
        return r4352908;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r4352909 = c;
        double r4352910 = -8.268102197384483e+27;
        bool r4352911 = r4352909 <= r4352910;
        double r4352912 = t;
        double r4352913 = r4352912 * r4352909;
        double r4352914 = i;
        double r4352915 = y;
        double r4352916 = r4352914 * r4352915;
        double r4352917 = r4352913 - r4352916;
        double r4352918 = j;
        double r4352919 = z;
        double r4352920 = x;
        double r4352921 = r4352920 * r4352915;
        double r4352922 = b;
        double r4352923 = r4352922 * r4352909;
        double r4352924 = r4352921 - r4352923;
        double r4352925 = r4352919 * r4352924;
        double r4352926 = a;
        double r4352927 = r4352912 * r4352926;
        double r4352928 = r4352920 * r4352927;
        double r4352929 = r4352925 - r4352928;
        double r4352930 = fma(r4352917, r4352918, r4352929);
        double r4352931 = -6.53514016554071e-32;
        bool r4352932 = r4352909 <= r4352931;
        double r4352933 = r4352914 * r4352926;
        double r4352934 = r4352909 * r4352919;
        double r4352935 = r4352933 - r4352934;
        double r4352936 = 0.0;
        double r4352937 = fma(r4352935, r4352922, r4352936);
        double r4352938 = fma(r4352917, r4352918, r4352937);
        double r4352939 = cbrt(r4352935);
        double r4352940 = r4352939 * r4352939;
        double r4352941 = r4352939 * r4352940;
        double r4352942 = r4352919 * r4352915;
        double r4352943 = r4352942 - r4352927;
        double r4352944 = r4352920 * r4352943;
        double r4352945 = fma(r4352941, r4352922, r4352944);
        double r4352946 = fma(r4352917, r4352918, r4352945);
        double r4352947 = r4352932 ? r4352938 : r4352946;
        double r4352948 = r4352911 ? r4352930 : r4352947;
        return r4352948;
}

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 3 regimes

  2. if c < -8.268102197384483e+27

    1. Initial program 16.8

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

    2. Simplified16.8

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

    3. Taylor expanded around inf 22.1

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

    4. Simplified21.8

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

    if -8.268102197384483e+27 < c < -6.53514016554071e-32

    1. Initial program 10.6

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

    2. Simplified10.6

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

    3. Taylor expanded around inf 10.6

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

    4. Taylor expanded around 0 23.2

      \[\leadsto \mathsf{fma}\left(t \cdot c - i \cdot y, j, \mathsf{fma}\left(a \cdot i - z \cdot c, b, \color{blue}{0}\right)\right)\]

    if -6.53514016554071e-32 < c

    1. Initial program 11.2

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

    2. Simplified11.2

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

    3. Taylor expanded around inf 11.2

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

    5. Applied add-cube-cbrt11.5

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -8268102197384482988509626368:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, z \cdot \left(x \cdot y - b \cdot c\right) - x \cdot \left(t \cdot a\right)\right)\\ \mathbf{elif}\;c \le -6.535140165540710269045334632618486765212 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, \mathsf{fma}\left(i \cdot a - c \cdot z, b, 0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - i \cdot y, j, \mathsf{fma}\left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \left(\sqrt[3]{i \cdot a - c \cdot z} \cdot \sqrt[3]{i \cdot a - c \cdot z}\right), b, x \cdot \left(z \cdot y - t \cdot a\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))