Average Error: 11.9 → 9.1
Time: 22.5s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.62427126302633396 \cdot 10^{52} \lor \neg \left(x \le 7.24120035060947113 \cdot 10^{55}\right):\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(-t \cdot \left(x \cdot a\right)\right) + \left(x \cdot z\right) \cdot y\right)\right)\\ \end{array}\]
\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}\;x \le -9.62427126302633396 \cdot 10^{52} \lor \neg \left(x \le 7.24120035060947113 \cdot 10^{55}\right):\\
\;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(-t \cdot \left(x \cdot a\right)\right) + \left(x \cdot z\right) \cdot y\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 r131154 = x;
        double r131155 = y;
        double r131156 = z;
        double r131157 = r131155 * r131156;
        double r131158 = t;
        double r131159 = a;
        double r131160 = r131158 * r131159;
        double r131161 = r131157 - r131160;
        double r131162 = r131154 * r131161;
        double r131163 = b;
        double r131164 = c;
        double r131165 = r131164 * r131156;
        double r131166 = i;
        double r131167 = r131166 * r131159;
        double r131168 = r131165 - r131167;
        double r131169 = r131163 * r131168;
        double r131170 = r131162 - r131169;
        double r131171 = j;
        double r131172 = r131164 * r131158;
        double r131173 = r131166 * r131155;
        double r131174 = r131172 - r131173;
        double r131175 = r131171 * r131174;
        double r131176 = r131170 + r131175;
        return r131176;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r131177 = x;
        double r131178 = -9.624271263026334e+52;
        bool r131179 = r131177 <= r131178;
        double r131180 = 7.241200350609471e+55;
        bool r131181 = r131177 <= r131180;
        double r131182 = !r131181;
        bool r131183 = r131179 || r131182;
        double r131184 = i;
        double r131185 = a;
        double r131186 = r131184 * r131185;
        double r131187 = c;
        double r131188 = z;
        double r131189 = r131187 * r131188;
        double r131190 = r131186 - r131189;
        double r131191 = b;
        double r131192 = j;
        double r131193 = t;
        double r131194 = r131187 * r131193;
        double r131195 = y;
        double r131196 = r131184 * r131195;
        double r131197 = r131194 - r131196;
        double r131198 = r131195 * r131188;
        double r131199 = r131193 * r131185;
        double r131200 = r131198 - r131199;
        double r131201 = cbrt(r131200);
        double r131202 = r131201 * r131201;
        double r131203 = r131177 * r131202;
        double r131204 = r131203 * r131201;
        double r131205 = fma(r131192, r131197, r131204);
        double r131206 = fma(r131190, r131191, r131205);
        double r131207 = r131177 * r131185;
        double r131208 = r131193 * r131207;
        double r131209 = -r131208;
        double r131210 = r131177 * r131188;
        double r131211 = r131210 * r131195;
        double r131212 = r131209 + r131211;
        double r131213 = fma(r131192, r131197, r131212);
        double r131214 = fma(r131190, r131191, r131213);
        double r131215 = r131183 ? r131206 : r131214;
        return r131215;
}

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 x < -9.624271263026334e+52 or 7.241200350609471e+55 < x

    1. Initial program 6.6

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

    2. Simplified6.6

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

    4. Applied add-cube-cbrt7.2

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

    5. Applied associate-*r*7.2

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

    if -9.624271263026334e+52 < x < 7.241200350609471e+55

    1. Initial program 14.1

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

    2. Simplified14.1

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

    4. Applied sub-neg14.1

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

    5. Applied distribute-lft-in14.1

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

    6. Simplified14.1

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

    7. Simplified12.1

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

    9. Applied *-un-lft-identity12.1

      \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(z \cdot y\right) + \left(-\color{blue}{\left(1 \cdot a\right)} \cdot \left(x \cdot t\right)\right)\right)\right)\]

    10. Applied associate-*l*12.1

      \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(z \cdot y\right) + \left(-\color{blue}{1 \cdot \left(a \cdot \left(x \cdot t\right)\right)}\right)\right)\right)\]

    11. Simplified12.2

      \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, x \cdot \left(z \cdot y\right) + \left(-1 \cdot \color{blue}{\left(t \cdot \left(x \cdot a\right)\right)}\right)\right)\right)\]
    12. Using strategy rm

    13. Applied associate-*r*9.9

      \[\leadsto \mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \color{blue}{\left(x \cdot z\right) \cdot y} + \left(-1 \cdot \left(t \cdot \left(x \cdot a\right)\right)\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.62427126302633396 \cdot 10^{52} \lor \neg \left(x \le 7.24120035060947113 \cdot 10^{55}\right):\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(x \cdot \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right) \cdot \sqrt[3]{y \cdot z - t \cdot a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(-t \cdot \left(x \cdot a\right)\right) + \left(x \cdot z\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))))