Average Error: 12.3 → 13.0
Time: 12.5s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -9.353869575251483128942178976129014511224 \cdot 10^{222} \lor \neg \left(a \le 225904523783003044121493372928\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right)\\ \end{array}\]
\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}\;a \le -9.353869575251483128942178976129014511224 \cdot 10^{222} \lor \neg \left(a \le 225904523783003044121493372928\right):\\
\;\;\;\;\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, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \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 r142134 = x;
        double r142135 = y;
        double r142136 = z;
        double r142137 = r142135 * r142136;
        double r142138 = t;
        double r142139 = a;
        double r142140 = r142138 * r142139;
        double r142141 = r142137 - r142140;
        double r142142 = r142134 * r142141;
        double r142143 = b;
        double r142144 = c;
        double r142145 = r142144 * r142136;
        double r142146 = i;
        double r142147 = r142146 * r142139;
        double r142148 = r142145 - r142147;
        double r142149 = r142143 * r142148;
        double r142150 = r142142 - r142149;
        double r142151 = j;
        double r142152 = r142144 * r142138;
        double r142153 = r142146 * r142135;
        double r142154 = r142152 - r142153;
        double r142155 = r142151 * r142154;
        double r142156 = r142150 + r142155;
        return r142156;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r142157 = a;
        double r142158 = -9.353869575251483e+222;
        bool r142159 = r142157 <= r142158;
        double r142160 = 2.2590452378300304e+29;
        bool r142161 = r142157 <= r142160;
        double r142162 = !r142161;
        bool r142163 = r142159 || r142162;
        double r142164 = i;
        double r142165 = b;
        double r142166 = r142164 * r142165;
        double r142167 = z;
        double r142168 = c;
        double r142169 = r142165 * r142168;
        double r142170 = x;
        double r142171 = t;
        double r142172 = r142170 * r142171;
        double r142173 = r142157 * r142172;
        double r142174 = fma(r142167, r142169, r142173);
        double r142175 = -r142174;
        double r142176 = fma(r142157, r142166, r142175);
        double r142177 = r142168 * r142171;
        double r142178 = y;
        double r142179 = r142164 * r142178;
        double r142180 = r142177 - r142179;
        double r142181 = j;
        double r142182 = r142157 * r142171;
        double r142183 = -r142182;
        double r142184 = fma(r142178, r142167, r142183);
        double r142185 = r142170 * r142184;
        double r142186 = -r142157;
        double r142187 = fma(r142186, r142171, r142182);
        double r142188 = r142170 * r142187;
        double r142189 = r142185 + r142188;
        double r142190 = r142168 * r142167;
        double r142191 = r142164 * r142157;
        double r142192 = r142190 - r142191;
        double r142193 = r142165 * r142192;
        double r142194 = cbrt(r142193);
        double r142195 = r142194 * r142194;
        double r142196 = r142195 * r142194;
        double r142197 = r142189 - r142196;
        double r142198 = fma(r142180, r142181, r142197);
        double r142199 = r142163 ? r142176 : r142198;
        return r142199;
}

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 a < -9.353869575251483e+222 or 2.2590452378300304e+29 < a

    1. Initial program 20.4

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

    2. Simplified20.4

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

    3. Taylor expanded around inf 22.9

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

    4. Simplified22.9

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

    if -9.353869575251483e+222 < a < 2.2590452378300304e+29

    1. Initial program 10.1

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

    2. Simplified10.1

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

    4. Applied prod-diff10.1

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

    5. Applied distribute-lft-in10.1

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

    7. Applied add-cube-cbrt10.4

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

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.353869575251483128942178976129014511224 \cdot 10^{222} \lor \neg \left(a \le 225904523783003044121493372928\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - \left(\sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)} \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right) \cdot \sqrt[3]{b \cdot \left(c \cdot z - i \cdot a\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +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)))))