Average Error: 11.7 → 11.7
Time: 33.8s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.109685036748723 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\\ \mathbf{elif}\;b \le 3.5516828515061712 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(y \cdot z - t \cdot a\right) \cdot x - c \cdot \left(z \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\\ \end{array}\]
\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}\;b \le -3.109685036748723 \cdot 10^{-15}:\\
\;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\\

\mathbf{elif}\;b \le 3.5516828515061712 \cdot 10^{-109}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(y \cdot z - t \cdot a\right) \cdot x - c \cdot \left(z \cdot b\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\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 r3943153 = x;
        double r3943154 = y;
        double r3943155 = z;
        double r3943156 = r3943154 * r3943155;
        double r3943157 = t;
        double r3943158 = a;
        double r3943159 = r3943157 * r3943158;
        double r3943160 = r3943156 - r3943159;
        double r3943161 = r3943153 * r3943160;
        double r3943162 = b;
        double r3943163 = c;
        double r3943164 = r3943163 * r3943155;
        double r3943165 = i;
        double r3943166 = r3943165 * r3943158;
        double r3943167 = r3943164 - r3943166;
        double r3943168 = r3943162 * r3943167;
        double r3943169 = r3943161 - r3943168;
        double r3943170 = j;
        double r3943171 = r3943163 * r3943157;
        double r3943172 = r3943165 * r3943154;
        double r3943173 = r3943171 - r3943172;
        double r3943174 = r3943170 * r3943173;
        double r3943175 = r3943169 + r3943174;
        return r3943175;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r3943176 = b;
        double r3943177 = -3.109685036748723e-15;
        bool r3943178 = r3943176 <= r3943177;
        double r3943179 = t;
        double r3943180 = c;
        double r3943181 = r3943179 * r3943180;
        double r3943182 = y;
        double r3943183 = i;
        double r3943184 = r3943182 * r3943183;
        double r3943185 = r3943181 - r3943184;
        double r3943186 = j;
        double r3943187 = a;
        double r3943188 = r3943187 * r3943183;
        double r3943189 = z;
        double r3943190 = r3943180 * r3943189;
        double r3943191 = r3943188 - r3943190;
        double r3943192 = r3943182 * r3943189;
        double r3943193 = r3943179 * r3943187;
        double r3943194 = r3943192 - r3943193;
        double r3943195 = x;
        double r3943196 = r3943194 * r3943195;
        double r3943197 = fma(r3943191, r3943176, r3943196);
        double r3943198 = fma(r3943185, r3943186, r3943197);
        double r3943199 = cbrt(r3943198);
        double r3943200 = r3943199 * r3943199;
        double r3943201 = r3943200 * r3943199;
        double r3943202 = 3.5516828515061712e-109;
        bool r3943203 = r3943176 <= r3943202;
        double r3943204 = r3943189 * r3943176;
        double r3943205 = r3943180 * r3943204;
        double r3943206 = r3943196 - r3943205;
        double r3943207 = fma(r3943185, r3943186, r3943206);
        double r3943208 = r3943203 ? r3943207 : r3943201;
        double r3943209 = r3943178 ? r3943201 : r3943208;
        return r3943209;
}

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 < -3.109685036748723e-15 or 3.5516828515061712e-109 < b

    1. Initial program 8.0

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

    2. Simplified7.9

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

    4. Applied add-cube-cbrt8.9

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

    if -3.109685036748723e-15 < b < 3.5516828515061712e-109

    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

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

    3. Taylor expanded around inf 14.8

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

    4. Simplified14.6

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.109685036748723 \cdot 10^{-15}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\\ \mathbf{elif}\;b \le 3.5516828515061712 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(y \cdot z - t \cdot a\right) \cdot x - c \cdot \left(z \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(t \cdot c - y \cdot i, j, \mathsf{fma}\left(a \cdot i - c \cdot z, b, \left(y \cdot z - t \cdot a\right) \cdot x\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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)))))