Average Error: 11.2 → 10.2
Time: 1.7m
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -3.1388759839959296 \cdot 10^{+83}:\\ \;\;\;\;\left(j \cdot \left(c \cdot t - i \cdot y\right) - \mathsf{fma}\left(t, \left(a \cdot x\right), \left(\left(c \cdot z - i \cdot a\right) \cdot b\right)\right)\right) + \left(y \cdot z\right) \cdot x\\ \mathbf{elif}\;j \le 4.076138196077605 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\left(j \cdot t\right), c, \left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot a\right) \cdot b\right)\right) + \left(-j\right) \cdot \left(i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b} \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b}\right) \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b}\right)\\ \end{array}\]
\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}\;j \le -3.1388759839959296 \cdot 10^{+83}:\\
\;\;\;\;\left(j \cdot \left(c \cdot t - i \cdot y\right) - \mathsf{fma}\left(t, \left(a \cdot x\right), \left(\left(c \cdot z - i \cdot a\right) \cdot b\right)\right)\right) + \left(y \cdot z\right) \cdot x\\

\mathbf{elif}\;j \le 4.076138196077605 \cdot 10^{-22}:\\
\;\;\;\;\mathsf{fma}\left(\left(j \cdot t\right), c, \left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot a\right) \cdot b\right)\right) + \left(-j\right) \cdot \left(i \cdot y\right)\\

\mathbf{else}:\\
\;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b} \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b}\right) \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r27059372 = x;
        double r27059373 = y;
        double r27059374 = z;
        double r27059375 = r27059373 * r27059374;
        double r27059376 = t;
        double r27059377 = a;
        double r27059378 = r27059376 * r27059377;
        double r27059379 = r27059375 - r27059378;
        double r27059380 = r27059372 * r27059379;
        double r27059381 = b;
        double r27059382 = c;
        double r27059383 = r27059382 * r27059374;
        double r27059384 = i;
        double r27059385 = r27059384 * r27059377;
        double r27059386 = r27059383 - r27059385;
        double r27059387 = r27059381 * r27059386;
        double r27059388 = r27059380 - r27059387;
        double r27059389 = j;
        double r27059390 = r27059382 * r27059376;
        double r27059391 = r27059384 * r27059373;
        double r27059392 = r27059390 - r27059391;
        double r27059393 = r27059389 * r27059392;
        double r27059394 = r27059388 + r27059393;
        return r27059394;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r27059395 = j;
        double r27059396 = -3.1388759839959296e+83;
        bool r27059397 = r27059395 <= r27059396;
        double r27059398 = c;
        double r27059399 = t;
        double r27059400 = r27059398 * r27059399;
        double r27059401 = i;
        double r27059402 = y;
        double r27059403 = r27059401 * r27059402;
        double r27059404 = r27059400 - r27059403;
        double r27059405 = r27059395 * r27059404;
        double r27059406 = a;
        double r27059407 = x;
        double r27059408 = r27059406 * r27059407;
        double r27059409 = z;
        double r27059410 = r27059398 * r27059409;
        double r27059411 = r27059401 * r27059406;
        double r27059412 = r27059410 - r27059411;
        double r27059413 = b;
        double r27059414 = r27059412 * r27059413;
        double r27059415 = fma(r27059399, r27059408, r27059414);
        double r27059416 = r27059405 - r27059415;
        double r27059417 = r27059402 * r27059409;
        double r27059418 = r27059417 * r27059407;
        double r27059419 = r27059416 + r27059418;
        double r27059420 = 4.076138196077605e-22;
        bool r27059421 = r27059395 <= r27059420;
        double r27059422 = r27059395 * r27059399;
        double r27059423 = r27059406 * r27059399;
        double r27059424 = r27059417 - r27059423;
        double r27059425 = r27059424 * r27059407;
        double r27059426 = r27059425 - r27059414;
        double r27059427 = fma(r27059422, r27059398, r27059426);
        double r27059428 = -r27059395;
        double r27059429 = r27059428 * r27059403;
        double r27059430 = r27059427 + r27059429;
        double r27059431 = cbrt(r27059414);
        double r27059432 = r27059431 * r27059431;
        double r27059433 = r27059432 * r27059431;
        double r27059434 = r27059425 - r27059433;
        double r27059435 = r27059405 + r27059434;
        double r27059436 = r27059421 ? r27059430 : r27059435;
        double r27059437 = r27059397 ? r27059419 : r27059436;
        return r27059437;
}

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 3 regimes

  2. if j < -3.1388759839959296e+83

    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. Using strategy rm

    3. Applied sub-neg6.6

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

    4. Applied distribute-lft-in6.6

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

    5. Applied associate--l+6.6

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

    6. Applied associate-+l+6.6

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

    7. Simplified8.1

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

    if -3.1388759839959296e+83 < j < 4.076138196077605e-22

    1. Initial program 13.4

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
    2. Using strategy rm

    3. Applied sub-neg13.4

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

    4. Applied distribute-lft-in13.4

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

    5. Applied associate-+r+13.4

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

    6. Simplified11.6

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

    if 4.076138196077605e-22 < j

    1. Initial program 7.2

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
    2. Using strategy rm

    3. Applied add-cube-cbrt7.4

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -3.1388759839959296 \cdot 10^{+83}:\\ \;\;\;\;\left(j \cdot \left(c \cdot t - i \cdot y\right) - \mathsf{fma}\left(t, \left(a \cdot x\right), \left(\left(c \cdot z - i \cdot a\right) \cdot b\right)\right)\right) + \left(y \cdot z\right) \cdot x\\ \mathbf{elif}\;j \le 4.076138196077605 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\left(j \cdot t\right), c, \left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot a\right) \cdot b\right)\right) + \left(-j\right) \cdot \left(i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - i \cdot y\right) + \left(\left(y \cdot z - a \cdot t\right) \cdot x - \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b} \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b}\right) \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +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)))))