Average Error: 11.9 → 12.0
Time: 30.4s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le -3.2644788703811975 \cdot 10^{-276}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b\right) \cdot \left(i \cdot a\right) + z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-y, i, y \cdot i\right) \cdot j + j \cdot \left(t \cdot c - y \cdot i\right)\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(z \cdot c\right) \cdot b + \left(-\left(b \cdot a\right) \cdot i\right)\right)\right)\\ \end{array}\]
\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}\;c \le -3.2644788703811975 \cdot 10^{-276}:\\
\;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b\right) \cdot \left(i \cdot a\right) + z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\mathsf{fma}\left(-y, i, y \cdot i\right) \cdot j + j \cdot \left(t \cdot c - y \cdot i\right)\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(z \cdot c\right) \cdot b + \left(-\left(b \cdot a\right) \cdot i\right)\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 r3889516 = x;
        double r3889517 = y;
        double r3889518 = z;
        double r3889519 = r3889517 * r3889518;
        double r3889520 = t;
        double r3889521 = a;
        double r3889522 = r3889520 * r3889521;
        double r3889523 = r3889519 - r3889522;
        double r3889524 = r3889516 * r3889523;
        double r3889525 = b;
        double r3889526 = c;
        double r3889527 = r3889526 * r3889518;
        double r3889528 = i;
        double r3889529 = r3889528 * r3889521;
        double r3889530 = r3889527 - r3889529;
        double r3889531 = r3889525 * r3889530;
        double r3889532 = r3889524 - r3889531;
        double r3889533 = j;
        double r3889534 = r3889526 * r3889520;
        double r3889535 = r3889528 * r3889517;
        double r3889536 = r3889534 - r3889535;
        double r3889537 = r3889533 * r3889536;
        double r3889538 = r3889532 + r3889537;
        return r3889538;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r3889539 = c;
        double r3889540 = -3.2644788703811975e-276;
        bool r3889541 = r3889539 <= r3889540;
        double r3889542 = y;
        double r3889543 = z;
        double r3889544 = r3889542 * r3889543;
        double r3889545 = t;
        double r3889546 = a;
        double r3889547 = r3889545 * r3889546;
        double r3889548 = r3889544 - r3889547;
        double r3889549 = x;
        double r3889550 = r3889548 * r3889549;
        double r3889551 = b;
        double r3889552 = -r3889551;
        double r3889553 = i;
        double r3889554 = r3889553 * r3889546;
        double r3889555 = r3889552 * r3889554;
        double r3889556 = r3889551 * r3889539;
        double r3889557 = r3889543 * r3889556;
        double r3889558 = r3889555 + r3889557;
        double r3889559 = r3889550 - r3889558;
        double r3889560 = j;
        double r3889561 = r3889545 * r3889539;
        double r3889562 = r3889542 * r3889553;
        double r3889563 = r3889561 - r3889562;
        double r3889564 = r3889560 * r3889563;
        double r3889565 = r3889559 + r3889564;
        double r3889566 = -r3889542;
        double r3889567 = fma(r3889566, r3889553, r3889562);
        double r3889568 = r3889567 * r3889560;
        double r3889569 = r3889568 + r3889564;
        double r3889570 = r3889543 * r3889539;
        double r3889571 = r3889570 * r3889551;
        double r3889572 = r3889551 * r3889546;
        double r3889573 = r3889572 * r3889553;
        double r3889574 = -r3889573;
        double r3889575 = r3889571 + r3889574;
        double r3889576 = r3889550 - r3889575;
        double r3889577 = r3889569 + r3889576;
        double r3889578 = r3889541 ? r3889565 : r3889577;
        return r3889578;
}

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 c < -3.2644788703811975e-276

    1. Initial program 11.9

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

    3. Applied sub-neg11.9

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

    4. Applied distribute-lft-in11.9

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

    6. Applied associate-*r*12.0

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

    if -3.2644788703811975e-276 < c

    1. Initial program 11.9

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

    3. Applied sub-neg11.9

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

    4. Applied distribute-lft-in11.9

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

    6. Applied prod-diff12.0

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

    7. Applied distribute-rgt-in12.0

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

    8. Simplified12.0

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

    9. Taylor expanded around inf 12.1

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

    10. Simplified12.1

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -3.2644788703811975 \cdot 10^{-276}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b\right) \cdot \left(i \cdot a\right) + z \cdot \left(b \cdot c\right)\right)\right) + j \cdot \left(t \cdot c - y \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-y, i, y \cdot i\right) \cdot j + j \cdot \left(t \cdot c - y \cdot i\right)\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(z \cdot c\right) \cdot b + \left(-\left(b \cdot a\right) \cdot i\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))))