Average Error: 11.8 → 12.2
Time: 13.0s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.38118333135362335 \cdot 10^{92} \lor \neg \left(a \le 3.66852259421464815 \cdot 10^{132}\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(1 \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]
\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}\;a \le -2.38118333135362335 \cdot 10^{92} \lor \neg \left(a \le 3.66852259421464815 \cdot 10^{132}\right):\\
\;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\left(\left(1 \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\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 r163317 = x;
        double r163318 = y;
        double r163319 = z;
        double r163320 = r163318 * r163319;
        double r163321 = t;
        double r163322 = a;
        double r163323 = r163321 * r163322;
        double r163324 = r163320 - r163323;
        double r163325 = r163317 * r163324;
        double r163326 = b;
        double r163327 = c;
        double r163328 = r163327 * r163319;
        double r163329 = i;
        double r163330 = r163329 * r163322;
        double r163331 = r163328 - r163330;
        double r163332 = r163326 * r163331;
        double r163333 = r163325 - r163332;
        double r163334 = j;
        double r163335 = r163327 * r163321;
        double r163336 = r163329 * r163318;
        double r163337 = r163335 - r163336;
        double r163338 = r163334 * r163337;
        double r163339 = r163333 + r163338;
        return r163339;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r163340 = a;
        double r163341 = -2.3811833313536233e+92;
        bool r163342 = r163340 <= r163341;
        double r163343 = 3.668522594214648e+132;
        bool r163344 = r163340 <= r163343;
        double r163345 = !r163344;
        bool r163346 = r163342 || r163345;
        double r163347 = i;
        double r163348 = b;
        double r163349 = r163347 * r163348;
        double r163350 = z;
        double r163351 = c;
        double r163352 = r163348 * r163351;
        double r163353 = x;
        double r163354 = t;
        double r163355 = r163353 * r163354;
        double r163356 = r163340 * r163355;
        double r163357 = fma(r163350, r163352, r163356);
        double r163358 = -r163357;
        double r163359 = fma(r163340, r163349, r163358);
        double r163360 = 1.0;
        double r163361 = r163351 * r163354;
        double r163362 = y;
        double r163363 = r163347 * r163362;
        double r163364 = r163361 - r163363;
        double r163365 = cbrt(r163364);
        double r163366 = r163360 * r163365;
        double r163367 = r163366 * r163365;
        double r163368 = r163367 * r163365;
        double r163369 = j;
        double r163370 = r163362 * r163350;
        double r163371 = r163354 * r163340;
        double r163372 = r163370 - r163371;
        double r163373 = r163353 * r163372;
        double r163374 = r163351 * r163350;
        double r163375 = r163347 * r163340;
        double r163376 = r163374 - r163375;
        double r163377 = r163348 * r163376;
        double r163378 = r163373 - r163377;
        double r163379 = fma(r163368, r163369, r163378);
        double r163380 = r163346 ? r163359 : r163379;
        return r163380;
}

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 a < -2.3811833313536233e+92 or 3.668522594214648e+132 < a

    1. Initial program 21.0

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

    2. Simplified21.0

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

    3. Taylor expanded around inf 21.3

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

    4. Simplified21.3

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

    if -2.3811833313536233e+92 < a < 3.668522594214648e+132

    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

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

    4. Applied add-cube-cbrt9.7

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

    6. Applied *-un-lft-identity9.7

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

  4. Final simplification12.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.38118333135362335 \cdot 10^{92} \lor \neg \left(a \le 3.66852259421464815 \cdot 10^{132}\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(1 \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}\right) \cdot \sqrt[3]{c \cdot t - i \cdot y}, j, x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))))