Average Error: 12.3 → 11.6
Time: 30.7s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.168636168951270619647828301902041552146 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \sqrt[3]{t \cdot c - y \cdot i}\right), j, \mathsf{fma}\left(b, a \cdot i - z \cdot c, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\ \mathbf{elif}\;b \le 2.834566076854457058584443196390534364264 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(y \cdot z - a \cdot t\right) \cdot x - z \cdot \left(c \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \sqrt[3]{t \cdot c - y \cdot i}\right), j, \mathsf{fma}\left(b, a \cdot i - z \cdot c, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\ \end{array}\]
\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 -2.168636168951270619647828301902041552146 \cdot 10^{-121}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \sqrt[3]{t \cdot c - y \cdot i}\right), j, \mathsf{fma}\left(b, a \cdot i - z \cdot c, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\

\mathbf{elif}\;b \le 2.834566076854457058584443196390534364264 \cdot 10^{-71}:\\
\;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(y \cdot z - a \cdot t\right) \cdot x - z \cdot \left(c \cdot b\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \sqrt[3]{t \cdot c - y \cdot i}\right), j, \mathsf{fma}\left(b, a \cdot i - z \cdot c, \left(y \cdot z - a \cdot t\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 r4174209 = x;
        double r4174210 = y;
        double r4174211 = z;
        double r4174212 = r4174210 * r4174211;
        double r4174213 = t;
        double r4174214 = a;
        double r4174215 = r4174213 * r4174214;
        double r4174216 = r4174212 - r4174215;
        double r4174217 = r4174209 * r4174216;
        double r4174218 = b;
        double r4174219 = c;
        double r4174220 = r4174219 * r4174211;
        double r4174221 = i;
        double r4174222 = r4174221 * r4174214;
        double r4174223 = r4174220 - r4174222;
        double r4174224 = r4174218 * r4174223;
        double r4174225 = r4174217 - r4174224;
        double r4174226 = j;
        double r4174227 = r4174219 * r4174213;
        double r4174228 = r4174221 * r4174210;
        double r4174229 = r4174227 - r4174228;
        double r4174230 = r4174226 * r4174229;
        double r4174231 = r4174225 + r4174230;
        return r4174231;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r4174232 = b;
        double r4174233 = -2.1686361689512706e-121;
        bool r4174234 = r4174232 <= r4174233;
        double r4174235 = t;
        double r4174236 = c;
        double r4174237 = r4174235 * r4174236;
        double r4174238 = y;
        double r4174239 = i;
        double r4174240 = r4174238 * r4174239;
        double r4174241 = r4174237 - r4174240;
        double r4174242 = cbrt(r4174241);
        double r4174243 = r4174242 * r4174242;
        double r4174244 = r4174242 * r4174243;
        double r4174245 = j;
        double r4174246 = a;
        double r4174247 = r4174246 * r4174239;
        double r4174248 = z;
        double r4174249 = r4174248 * r4174236;
        double r4174250 = r4174247 - r4174249;
        double r4174251 = r4174238 * r4174248;
        double r4174252 = r4174246 * r4174235;
        double r4174253 = r4174251 - r4174252;
        double r4174254 = x;
        double r4174255 = r4174253 * r4174254;
        double r4174256 = fma(r4174232, r4174250, r4174255);
        double r4174257 = fma(r4174244, r4174245, r4174256);
        double r4174258 = 2.834566076854457e-71;
        bool r4174259 = r4174232 <= r4174258;
        double r4174260 = r4174236 * r4174232;
        double r4174261 = r4174248 * r4174260;
        double r4174262 = r4174255 - r4174261;
        double r4174263 = fma(r4174241, r4174245, r4174262);
        double r4174264 = r4174259 ? r4174263 : r4174257;
        double r4174265 = r4174234 ? r4174257 : r4174264;
        return r4174265;
}

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 < -2.1686361689512706e-121 or 2.834566076854457e-71 < b

    1. Initial program 9.4

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

    2. Simplified9.4

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

    4. Applied add-cube-cbrt9.6

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

    if -2.1686361689512706e-121 < b < 2.834566076854457e-71

    1. Initial program 16.0

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

    2. Simplified16.0

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

    3. Taylor expanded around inf 14.6

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

    4. Simplified14.1

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.168636168951270619647828301902041552146 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \sqrt[3]{t \cdot c - y \cdot i}\right), j, \mathsf{fma}\left(b, a \cdot i - z \cdot c, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\ \mathbf{elif}\;b \le 2.834566076854457058584443196390534364264 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot c - y \cdot i, j, \left(y \cdot z - a \cdot t\right) \cdot x - z \cdot \left(c \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \left(\sqrt[3]{t \cdot c - y \cdot i} \cdot \sqrt[3]{t \cdot c - y \cdot i}\right), j, \mathsf{fma}\left(b, a \cdot i - z \cdot c, \left(y \cdot z - a \cdot t\right) \cdot x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))))