Average Error: 11.5 → 11.6
Time: 51.1s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.643183985893601 \cdot 10^{-145}:\\ \;\;\;\;(j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\sqrt[3]{z \cdot y - t \cdot a} \cdot x\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right))_*\\ \mathbf{elif}\;b \le 7.229565060411778 \cdot 10^{-202}:\\ \;\;\;\;(j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(z \cdot y - t \cdot a\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;(j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\sqrt[3]{z \cdot y - t \cdot a} \cdot x\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right))_*\\ \end{array}\]
\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}\;b \le -9.643183985893601 \cdot 10^{-145}:\\
\;\;\;\;(j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\sqrt[3]{z \cdot y - t \cdot a} \cdot x\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right))_*\\

\mathbf{elif}\;b \le 7.229565060411778 \cdot 10^{-202}:\\
\;\;\;\;(j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(z \cdot y - t \cdot a\right)\right))_*\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r10925908 = x;
        double r10925909 = y;
        double r10925910 = z;
        double r10925911 = r10925909 * r10925910;
        double r10925912 = t;
        double r10925913 = a;
        double r10925914 = r10925912 * r10925913;
        double r10925915 = r10925911 - r10925914;
        double r10925916 = r10925908 * r10925915;
        double r10925917 = b;
        double r10925918 = c;
        double r10925919 = r10925918 * r10925910;
        double r10925920 = i;
        double r10925921 = r10925920 * r10925913;
        double r10925922 = r10925919 - r10925921;
        double r10925923 = r10925917 * r10925922;
        double r10925924 = r10925916 - r10925923;
        double r10925925 = j;
        double r10925926 = r10925918 * r10925912;
        double r10925927 = r10925920 * r10925909;
        double r10925928 = r10925926 - r10925927;
        double r10925929 = r10925925 * r10925928;
        double r10925930 = r10925924 + r10925929;
        return r10925930;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r10925931 = b;
        double r10925932 = -9.643183985893601e-145;
        bool r10925933 = r10925931 <= r10925932;
        double r10925934 = j;
        double r10925935 = t;
        double r10925936 = c;
        double r10925937 = r10925935 * r10925936;
        double r10925938 = y;
        double r10925939 = i;
        double r10925940 = r10925938 * r10925939;
        double r10925941 = r10925937 - r10925940;
        double r10925942 = z;
        double r10925943 = r10925942 * r10925938;
        double r10925944 = a;
        double r10925945 = r10925935 * r10925944;
        double r10925946 = r10925943 - r10925945;
        double r10925947 = cbrt(r10925946);
        double r10925948 = r10925947 * r10925947;
        double r10925949 = x;
        double r10925950 = r10925947 * r10925949;
        double r10925951 = r10925948 * r10925950;
        double r10925952 = r10925936 * r10925942;
        double r10925953 = r10925944 * r10925939;
        double r10925954 = r10925952 - r10925953;
        double r10925955 = r10925954 * r10925931;
        double r10925956 = r10925951 - r10925955;
        double r10925957 = fma(r10925934, r10925941, r10925956);
        double r10925958 = 7.229565060411778e-202;
        bool r10925959 = r10925931 <= r10925958;
        double r10925960 = r10925949 * r10925946;
        double r10925961 = fma(r10925934, r10925941, r10925960);
        double r10925962 = r10925959 ? r10925961 : r10925957;
        double r10925963 = r10925933 ? r10925957 : r10925962;
        return r10925963;
}

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 b < -9.643183985893601e-145 or 7.229565060411778e-202 < b

    1. Initial program 10.0

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

    2. Simplified10.0

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

    4. Applied add-cube-cbrt10.2

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

    5. Applied associate-*l*10.2

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

    if -9.643183985893601e-145 < b < 7.229565060411778e-202

    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. Simplified15.4

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

    3. Taylor expanded around 0 15.3

      \[\leadsto (j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(z \cdot y - t \cdot a\right) \cdot x - \color{blue}{0}\right))_*\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.643183985893601 \cdot 10^{-145}:\\ \;\;\;\;(j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\sqrt[3]{z \cdot y - t \cdot a} \cdot x\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right))_*\\ \mathbf{elif}\;b \le 7.229565060411778 \cdot 10^{-202}:\\ \;\;\;\;(j \cdot \left(t \cdot c - y \cdot i\right) + \left(x \cdot \left(z \cdot y - t \cdot a\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;(j \cdot \left(t \cdot c - y \cdot i\right) + \left(\left(\sqrt[3]{z \cdot y - t \cdot a} \cdot \sqrt[3]{z \cdot y - t \cdot a}\right) \cdot \left(\sqrt[3]{z \cdot y - t \cdot a} \cdot x\right) - \left(c \cdot z - a \cdot i\right) \cdot b\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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)))))