Average Error: 12.6 → 13.1
Time: 9.6s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.400785168655679605773741836481995189038 \cdot 10^{-141} \lor \neg \left(b \le -5.616902006401816366798818804296717843031 \cdot 10^{-291}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]
\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 -4.400785168655679605773741836481995189038 \cdot 10^{-141} \lor \neg \left(b \le -5.616902006401816366798818804296717843031 \cdot 10^{-291}\right):\\
\;\;\;\;\left(\left(\left(\sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r117079 = x;
        double r117080 = y;
        double r117081 = z;
        double r117082 = r117080 * r117081;
        double r117083 = t;
        double r117084 = a;
        double r117085 = r117083 * r117084;
        double r117086 = r117082 - r117085;
        double r117087 = r117079 * r117086;
        double r117088 = b;
        double r117089 = c;
        double r117090 = r117089 * r117081;
        double r117091 = i;
        double r117092 = r117091 * r117084;
        double r117093 = r117090 - r117092;
        double r117094 = r117088 * r117093;
        double r117095 = r117087 - r117094;
        double r117096 = j;
        double r117097 = r117089 * r117083;
        double r117098 = r117091 * r117080;
        double r117099 = r117097 - r117098;
        double r117100 = r117096 * r117099;
        double r117101 = r117095 + r117100;
        return r117101;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r117102 = b;
        double r117103 = -4.4007851686556796e-141;
        bool r117104 = r117102 <= r117103;
        double r117105 = -5.6169020064018164e-291;
        bool r117106 = r117102 <= r117105;
        double r117107 = !r117106;
        bool r117108 = r117104 || r117107;
        double r117109 = x;
        double r117110 = y;
        double r117111 = z;
        double r117112 = a;
        double r117113 = t;
        double r117114 = r117112 * r117113;
        double r117115 = -r117114;
        double r117116 = fma(r117110, r117111, r117115);
        double r117117 = r117109 * r117116;
        double r117118 = cbrt(r117117);
        double r117119 = r117118 * r117118;
        double r117120 = cbrt(r117109);
        double r117121 = cbrt(r117116);
        double r117122 = r117120 * r117121;
        double r117123 = r117119 * r117122;
        double r117124 = -r117112;
        double r117125 = fma(r117124, r117113, r117114);
        double r117126 = r117109 * r117125;
        double r117127 = r117123 + r117126;
        double r117128 = c;
        double r117129 = r117128 * r117111;
        double r117130 = i;
        double r117131 = r117130 * r117112;
        double r117132 = r117129 - r117131;
        double r117133 = r117102 * r117132;
        double r117134 = r117127 - r117133;
        double r117135 = j;
        double r117136 = r117128 * r117113;
        double r117137 = r117130 * r117110;
        double r117138 = r117136 - r117137;
        double r117139 = r117135 * r117138;
        double r117140 = r117134 + r117139;
        double r117141 = r117117 + r117126;
        double r117142 = 0.0;
        double r117143 = r117141 - r117142;
        double r117144 = r117143 + r117139;
        double r117145 = r117108 ? r117140 : r117144;
        return r117145;
}

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 < -4.4007851686556796e-141 or -5.6169020064018164e-291 < b

    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. Using strategy rm

    3. Applied prod-diff11.6

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

    4. Applied distribute-lft-in11.6

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

    6. Applied add-cube-cbrt11.8

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

    8. Applied cbrt-prod11.8

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

    if -4.4007851686556796e-141 < b < -5.6169020064018164e-291

    1. Initial program 18.3

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
    2. Using strategy rm

    3. Applied prod-diff18.3

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

    4. Applied distribute-lft-in18.3

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

    5. Taylor expanded around 0 20.1

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.400785168655679605773741836481995189038 \cdot 10^{-141} \lor \neg \left(b \le -5.616902006401816366798818804296717843031 \cdot 10^{-291}\right):\\ \;\;\;\;\left(\left(\left(\sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)} \cdot \sqrt[3]{x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\mathsf{fma}\left(y, z, -a \cdot t\right)}\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \mathsf{fma}\left(y, z, -a \cdot t\right) + x \cdot \mathsf{fma}\left(-a, t, a \cdot t\right)\right) - 0\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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)))))