Average Error: 11.9 → 11.7
Time: 29.6s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.8754562356247043 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(x \cdot \left(y \cdot z - a \cdot t\right) - y \cdot \left(i \cdot j\right)\right)\right)\\ \mathbf{elif}\;y \le 8.602803430088364 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(\mathsf{fma}\left(\left(c \cdot t - i \cdot y\right), j, \left(\left(\sqrt[3]{y \cdot z - a \cdot t} \cdot x\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(x \cdot \left(y \cdot z - a \cdot t\right) - y \cdot \left(i \cdot j\right)\right)\right)\\ \end{array}\]
\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}\;y \le -1.8754562356247043 \cdot 10^{+90}:\\
\;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(x \cdot \left(y \cdot z - a \cdot t\right) - y \cdot \left(i \cdot j\right)\right)\right)\\

\mathbf{elif}\;y \le 8.602803430088364 \cdot 10^{+98}:\\
\;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(\mathsf{fma}\left(\left(c \cdot t - i \cdot y\right), j, \left(\left(\sqrt[3]{y \cdot z - a \cdot t} \cdot x\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(x \cdot \left(y \cdot z - a \cdot t\right) - y \cdot \left(i \cdot j\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 r3034148 = x;
        double r3034149 = y;
        double r3034150 = z;
        double r3034151 = r3034149 * r3034150;
        double r3034152 = t;
        double r3034153 = a;
        double r3034154 = r3034152 * r3034153;
        double r3034155 = r3034151 - r3034154;
        double r3034156 = r3034148 * r3034155;
        double r3034157 = b;
        double r3034158 = c;
        double r3034159 = r3034158 * r3034150;
        double r3034160 = i;
        double r3034161 = r3034160 * r3034153;
        double r3034162 = r3034159 - r3034161;
        double r3034163 = r3034157 * r3034162;
        double r3034164 = r3034156 - r3034163;
        double r3034165 = j;
        double r3034166 = r3034158 * r3034152;
        double r3034167 = r3034160 * r3034149;
        double r3034168 = r3034166 - r3034167;
        double r3034169 = r3034165 * r3034168;
        double r3034170 = r3034164 + r3034169;
        return r3034170;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r3034171 = y;
        double r3034172 = -1.8754562356247043e+90;
        bool r3034173 = r3034171 <= r3034172;
        double r3034174 = i;
        double r3034175 = a;
        double r3034176 = r3034174 * r3034175;
        double r3034177 = c;
        double r3034178 = z;
        double r3034179 = r3034177 * r3034178;
        double r3034180 = r3034176 - r3034179;
        double r3034181 = b;
        double r3034182 = x;
        double r3034183 = r3034171 * r3034178;
        double r3034184 = t;
        double r3034185 = r3034175 * r3034184;
        double r3034186 = r3034183 - r3034185;
        double r3034187 = r3034182 * r3034186;
        double r3034188 = j;
        double r3034189 = r3034174 * r3034188;
        double r3034190 = r3034171 * r3034189;
        double r3034191 = r3034187 - r3034190;
        double r3034192 = fma(r3034180, r3034181, r3034191);
        double r3034193 = 8.602803430088364e+98;
        bool r3034194 = r3034171 <= r3034193;
        double r3034195 = r3034177 * r3034184;
        double r3034196 = r3034174 * r3034171;
        double r3034197 = r3034195 - r3034196;
        double r3034198 = cbrt(r3034186);
        double r3034199 = r3034198 * r3034182;
        double r3034200 = r3034198 * r3034198;
        double r3034201 = r3034199 * r3034200;
        double r3034202 = fma(r3034197, r3034188, r3034201);
        double r3034203 = fma(r3034180, r3034181, r3034202);
        double r3034204 = r3034194 ? r3034203 : r3034192;
        double r3034205 = r3034173 ? r3034192 : r3034204;
        return r3034205;
}

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 y < -1.8754562356247043e+90 or 8.602803430088364e+98 < y

    1. Initial program 20.2

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

    2. Simplified20.2

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

    4. Applied add-cube-cbrt20.5

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

    5. Applied associate-*r*20.5

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

    6. Taylor expanded around -inf 23.9

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

    7. Simplified18.8

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

    if -1.8754562356247043e+90 < y < 8.602803430088364e+98

    1. Initial program 9.3

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

    2. Simplified9.2

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

    4. Applied add-cube-cbrt9.5

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

    5. Applied associate-*l*9.5

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.8754562356247043 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(x \cdot \left(y \cdot z - a \cdot t\right) - y \cdot \left(i \cdot j\right)\right)\right)\\ \mathbf{elif}\;y \le 8.602803430088364 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(\mathsf{fma}\left(\left(c \cdot t - i \cdot y\right), j, \left(\left(\sqrt[3]{y \cdot z - a \cdot t} \cdot x\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(i \cdot a - c \cdot z\right), b, \left(x \cdot \left(y \cdot z - a \cdot t\right) - y \cdot \left(i \cdot j\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +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)))))