Average Error: 11.9 → 10.4
Time: 29.7s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.601442109879438909334021118328802127187 \cdot 10^{51}:\\ \;\;\;\;\left(\left(x \cdot z\right) \cdot y + \left(-\left(x \cdot a\right) \cdot t\right)\right) + \mathsf{fma}\left(b, i \cdot a - c \cdot z, j \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right) + \mathsf{fma}\left(b, i \cdot a - c \cdot z, \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\\ \end{array}\]
\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}\;x \le 2.601442109879438909334021118328802127187 \cdot 10^{51}:\\
\;\;\;\;\left(\left(x \cdot z\right) \cdot y + \left(-\left(x \cdot a\right) \cdot t\right)\right) + \mathsf{fma}\left(b, i \cdot a - c \cdot z, j \cdot \left(c \cdot t - i \cdot y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right) + \mathsf{fma}\left(b, i \cdot a - c \cdot z, \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{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 r113211 = x;
        double r113212 = y;
        double r113213 = z;
        double r113214 = r113212 * r113213;
        double r113215 = t;
        double r113216 = a;
        double r113217 = r113215 * r113216;
        double r113218 = r113214 - r113217;
        double r113219 = r113211 * r113218;
        double r113220 = b;
        double r113221 = c;
        double r113222 = r113221 * r113213;
        double r113223 = i;
        double r113224 = r113223 * r113216;
        double r113225 = r113222 - r113224;
        double r113226 = r113220 * r113225;
        double r113227 = r113219 - r113226;
        double r113228 = j;
        double r113229 = r113221 * r113215;
        double r113230 = r113223 * r113212;
        double r113231 = r113229 - r113230;
        double r113232 = r113228 * r113231;
        double r113233 = r113227 + r113232;
        return r113233;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r113234 = x;
        double r113235 = 2.601442109879439e+51;
        bool r113236 = r113234 <= r113235;
        double r113237 = z;
        double r113238 = r113234 * r113237;
        double r113239 = y;
        double r113240 = r113238 * r113239;
        double r113241 = a;
        double r113242 = r113234 * r113241;
        double r113243 = t;
        double r113244 = r113242 * r113243;
        double r113245 = -r113244;
        double r113246 = r113240 + r113245;
        double r113247 = b;
        double r113248 = i;
        double r113249 = r113248 * r113241;
        double r113250 = c;
        double r113251 = r113250 * r113237;
        double r113252 = r113249 - r113251;
        double r113253 = j;
        double r113254 = r113250 * r113243;
        double r113255 = r113248 * r113239;
        double r113256 = r113254 - r113255;
        double r113257 = r113253 * r113256;
        double r113258 = fma(r113247, r113252, r113257);
        double r113259 = r113246 + r113258;
        double r113260 = r113239 * r113237;
        double r113261 = r113243 * r113241;
        double r113262 = r113260 - r113261;
        double r113263 = r113234 * r113262;
        double r113264 = cbrt(r113256);
        double r113265 = r113264 * r113264;
        double r113266 = r113253 * r113265;
        double r113267 = r113266 * r113264;
        double r113268 = fma(r113247, r113252, r113267);
        double r113269 = r113263 + r113268;
        double r113270 = r113236 ? r113259 : r113269;
        return r113270;
}

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 < 2.601442109879439e+51

    1. Initial program 12.7

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

    2. Simplified12.7

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

    4. Applied fma-udef12.7

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

    6. Applied sub-neg12.7

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

    7. Applied distribute-lft-in12.7

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

    8. Simplified12.7

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

    9. Simplified12.3

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

    11. Applied associate-*r*12.2

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

    12. Simplified12.2

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

    14. Applied associate-*r*10.9

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

    if 2.601442109879439e+51 < x

    1. Initial program 7.1

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

    2. Simplified7.1

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

    4. Applied fma-udef7.1

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

    6. Applied add-cube-cbrt7.3

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

    7. Applied associate-*r*7.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.601442109879438909334021118328802127187 \cdot 10^{51}:\\ \;\;\;\;\left(\left(x \cdot z\right) \cdot y + \left(-\left(x \cdot a\right) \cdot t\right)\right) + \mathsf{fma}\left(b, i \cdot a - c \cdot z, j \cdot \left(c \cdot t - i \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z - t \cdot a\right) + \mathsf{fma}\left(b, i \cdot a - c \cdot z, \left(j \cdot \left(\sqrt[3]{c \cdot t - i \cdot y} \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))))