Average Error: 12.0 → 10.3
Time: 31.9s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.049590576885732440987301511602082461698 \cdot 10^{94}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(x \cdot a\right)\right)\\ \mathbf{elif}\;z \le 3.061671691665755733973394415758569553163 \cdot 10^{144}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(b, i \cdot a - c \cdot z, \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right) \cdot \sqrt[3]{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(x \cdot a\right)\right)\\ \end{array}\]
\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}\;z \le -4.049590576885732440987301511602082461698 \cdot 10^{94}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(x \cdot a\right)\right)\\

\mathbf{elif}\;z \le 3.061671691665755733973394415758569553163 \cdot 10^{144}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(b, i \cdot a - c \cdot z, \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right) \cdot \sqrt[3]{x}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(x \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 r4354932 = x;
        double r4354933 = y;
        double r4354934 = z;
        double r4354935 = r4354933 * r4354934;
        double r4354936 = t;
        double r4354937 = a;
        double r4354938 = r4354936 * r4354937;
        double r4354939 = r4354935 - r4354938;
        double r4354940 = r4354932 * r4354939;
        double r4354941 = b;
        double r4354942 = c;
        double r4354943 = r4354942 * r4354934;
        double r4354944 = i;
        double r4354945 = r4354944 * r4354937;
        double r4354946 = r4354943 - r4354945;
        double r4354947 = r4354941 * r4354946;
        double r4354948 = r4354940 - r4354947;
        double r4354949 = j;
        double r4354950 = r4354942 * r4354936;
        double r4354951 = r4354944 * r4354933;
        double r4354952 = r4354950 - r4354951;
        double r4354953 = r4354949 * r4354952;
        double r4354954 = r4354948 + r4354953;
        return r4354954;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r4354955 = z;
        double r4354956 = -4.0495905768857324e+94;
        bool r4354957 = r4354955 <= r4354956;
        double r4354958 = t;
        double r4354959 = c;
        double r4354960 = r4354958 * r4354959;
        double r4354961 = y;
        double r4354962 = i;
        double r4354963 = r4354961 * r4354962;
        double r4354964 = r4354960 - r4354963;
        double r4354965 = j;
        double r4354966 = x;
        double r4354967 = r4354966 * r4354961;
        double r4354968 = b;
        double r4354969 = r4354959 * r4354968;
        double r4354970 = r4354967 - r4354969;
        double r4354971 = r4354970 * r4354955;
        double r4354972 = a;
        double r4354973 = r4354966 * r4354972;
        double r4354974 = r4354958 * r4354973;
        double r4354975 = r4354971 - r4354974;
        double r4354976 = fma(r4354964, r4354965, r4354975);
        double r4354977 = 3.0616716916657557e+144;
        bool r4354978 = r4354955 <= r4354977;
        double r4354979 = r4354962 * r4354972;
        double r4354980 = r4354959 * r4354955;
        double r4354981 = r4354979 - r4354980;
        double r4354982 = cbrt(r4354966);
        double r4354983 = r4354961 * r4354955;
        double r4354984 = r4354958 * r4354972;
        double r4354985 = r4354983 - r4354984;
        double r4354986 = r4354982 * r4354985;
        double r4354987 = r4354982 * r4354986;
        double r4354988 = r4354987 * r4354982;
        double r4354989 = fma(r4354968, r4354981, r4354988);
        double r4354990 = fma(r4354964, r4354965, r4354989);
        double r4354991 = r4354978 ? r4354990 : r4354976;
        double r4354992 = r4354957 ? r4354976 : r4354991;
        return r4354992;
}

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 < -4.0495905768857324e+94 or 3.0616716916657557e+144 < z

    1. Initial program 21.1

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

    2. Simplified21.1

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

    4. Applied add-cube-cbrt21.4

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

    5. Applied associate-*r*21.4

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

    6. Taylor expanded around inf 19.0

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

    7. Simplified11.7

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

    if -4.0495905768857324e+94 < z < 3.0616716916657557e+144

    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. Simplified9.7

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

    4. Applied add-cube-cbrt9.9

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

    5. Applied associate-*r*9.9

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

    7. Applied associate-*r*9.9

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.049590576885732440987301511602082461698 \cdot 10^{94}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(x \cdot a\right)\right)\\ \mathbf{elif}\;z \le 3.061671691665755733973394415758569553163 \cdot 10^{144}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(b, i \cdot a - c \cdot z, \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right) \cdot \sqrt[3]{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(x \cdot y - c \cdot b\right) \cdot z - t \cdot \left(x \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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)))))