Average Error: 11.7 → 12.9
Time: 18.8s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.3603574852745576 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \mathbf{elif}\;x \le 6.4460055181613243 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \end{array}\]
\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}\;x \le -2.3603574852745576 \cdot 10^{-289}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\

\mathbf{elif}\;x \le 6.4460055181613243 \cdot 10^{-280}:\\
\;\;\;\;\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, \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\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 r171929 = x;
        double r171930 = y;
        double r171931 = z;
        double r171932 = r171930 * r171931;
        double r171933 = t;
        double r171934 = a;
        double r171935 = r171933 * r171934;
        double r171936 = r171932 - r171935;
        double r171937 = r171929 * r171936;
        double r171938 = b;
        double r171939 = c;
        double r171940 = r171939 * r171931;
        double r171941 = i;
        double r171942 = r171941 * r171934;
        double r171943 = r171940 - r171942;
        double r171944 = r171938 * r171943;
        double r171945 = r171937 - r171944;
        double r171946 = j;
        double r171947 = r171939 * r171933;
        double r171948 = r171941 * r171930;
        double r171949 = r171947 - r171948;
        double r171950 = r171946 * r171949;
        double r171951 = r171945 + r171950;
        return r171951;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r171952 = x;
        double r171953 = -2.3603574852745576e-289;
        bool r171954 = r171952 <= r171953;
        double r171955 = c;
        double r171956 = t;
        double r171957 = r171955 * r171956;
        double r171958 = i;
        double r171959 = y;
        double r171960 = r171958 * r171959;
        double r171961 = r171957 - r171960;
        double r171962 = j;
        double r171963 = z;
        double r171964 = r171959 * r171963;
        double r171965 = r171952 * r171964;
        double r171966 = a;
        double r171967 = r171956 * r171966;
        double r171968 = -r171967;
        double r171969 = r171952 * r171968;
        double r171970 = r171965 + r171969;
        double r171971 = b;
        double r171972 = cbrt(r171971);
        double r171973 = r171972 * r171972;
        double r171974 = r171955 * r171963;
        double r171975 = r171958 * r171966;
        double r171976 = r171974 - r171975;
        double r171977 = r171972 * r171976;
        double r171978 = r171973 * r171977;
        double r171979 = -r171966;
        double r171980 = r171966 * r171958;
        double r171981 = fma(r171979, r171958, r171980);
        double r171982 = r171971 * r171981;
        double r171983 = r171978 + r171982;
        double r171984 = r171970 - r171983;
        double r171985 = fma(r171961, r171962, r171984);
        double r171986 = 6.446005518161324e-280;
        bool r171987 = r171952 <= r171986;
        double r171988 = r171958 * r171971;
        double r171989 = r171971 * r171955;
        double r171990 = r171952 * r171956;
        double r171991 = r171966 * r171990;
        double r171992 = fma(r171963, r171989, r171991);
        double r171993 = -r171992;
        double r171994 = fma(r171966, r171988, r171993);
        double r171995 = sqrt(r171952);
        double r171996 = r171964 - r171967;
        double r171997 = r171995 * r171996;
        double r171998 = r171995 * r171997;
        double r171999 = r171971 * r171976;
        double r172000 = r171999 + r171982;
        double r172001 = r171998 - r172000;
        double r172002 = fma(r171961, r171962, r172001);
        double r172003 = r171987 ? r171994 : r172002;
        double r172004 = r171954 ? r171985 : r172003;
        return r172004;
}

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 x < -2.3603574852745576e-289

    1. Initial program 11.6

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

    2. Simplified11.6

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

    4. Applied prod-diff11.6

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

    5. Applied distribute-lft-in11.6

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

    6. Simplified11.6

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

    8. Applied add-cube-cbrt11.9

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

    9. Applied associate-*l*11.9

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

    11. Applied sub-neg11.9

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

    12. Applied distribute-lft-in11.9

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

    if -2.3603574852745576e-289 < x < 6.446005518161324e-280

    1. Initial program 17.7

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

    2. Simplified17.6

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

    3. Taylor expanded around inf 36.1

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

    4. Simplified36.1

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

    if 6.446005518161324e-280 < x

    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

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

    4. Applied prod-diff11.2

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

    5. Applied distribute-lft-in11.2

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

    6. Simplified11.2

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

    8. Applied add-sqr-sqrt11.3

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

    9. Applied associate-*l*11.3

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.3603574852745576 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \mathbf{elif}\;x \le 6.4460055181613243 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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)))))