Average Error: 11.7 → 11.3
Time: 26.8s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;c \le 6.14458466408961470336255307627453402921 \cdot 10^{94} \lor \neg \left(c \le 3.235357331789103476780750730187488778281 \cdot 10^{269}\right):\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, i \cdot a - c \cdot z, t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, c \cdot \left(t \cdot j - z \cdot b\right) - i \cdot \left(j \cdot y\right)\right)\\ \end{array}\]
\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}\;c \le 6.14458466408961470336255307627453402921 \cdot 10^{94} \lor \neg \left(c \le 3.235357331789103476780750730187488778281 \cdot 10^{269}\right):\\
\;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, i \cdot a - c \cdot z, t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, c \cdot \left(t \cdot j - z \cdot b\right) - i \cdot \left(j \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 r119520 = x;
        double r119521 = y;
        double r119522 = z;
        double r119523 = r119521 * r119522;
        double r119524 = t;
        double r119525 = a;
        double r119526 = r119524 * r119525;
        double r119527 = r119523 - r119526;
        double r119528 = r119520 * r119527;
        double r119529 = b;
        double r119530 = c;
        double r119531 = r119530 * r119522;
        double r119532 = i;
        double r119533 = r119532 * r119525;
        double r119534 = r119531 - r119533;
        double r119535 = r119529 * r119534;
        double r119536 = r119528 - r119535;
        double r119537 = j;
        double r119538 = r119530 * r119524;
        double r119539 = r119532 * r119521;
        double r119540 = r119538 - r119539;
        double r119541 = r119537 * r119540;
        double r119542 = r119536 + r119541;
        return r119542;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r119543 = c;
        double r119544 = 6.144584664089615e+94;
        bool r119545 = r119543 <= r119544;
        double r119546 = 3.2353573317891035e+269;
        bool r119547 = r119543 <= r119546;
        double r119548 = !r119547;
        bool r119549 = r119545 || r119548;
        double r119550 = x;
        double r119551 = y;
        double r119552 = z;
        double r119553 = r119551 * r119552;
        double r119554 = t;
        double r119555 = a;
        double r119556 = r119554 * r119555;
        double r119557 = r119553 - r119556;
        double r119558 = b;
        double r119559 = i;
        double r119560 = r119559 * r119555;
        double r119561 = r119543 * r119552;
        double r119562 = r119560 - r119561;
        double r119563 = j;
        double r119564 = r119563 * r119543;
        double r119565 = r119554 * r119564;
        double r119566 = r119559 * r119563;
        double r119567 = r119566 * r119551;
        double r119568 = -r119567;
        double r119569 = r119565 + r119568;
        double r119570 = fma(r119558, r119562, r119569);
        double r119571 = fma(r119550, r119557, r119570);
        double r119572 = r119554 * r119563;
        double r119573 = r119552 * r119558;
        double r119574 = r119572 - r119573;
        double r119575 = r119543 * r119574;
        double r119576 = r119563 * r119551;
        double r119577 = r119559 * r119576;
        double r119578 = r119575 - r119577;
        double r119579 = fma(r119550, r119557, r119578);
        double r119580 = r119549 ? r119571 : r119579;
        return r119580;
}

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 < 6.144584664089615e+94 or 3.2353573317891035e+269 < c

    1. Initial program 10.9

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

    2. Simplified10.9

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

    4. Applied add-cube-cbrt11.2

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

    5. Applied associate-*l*11.2

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

    7. Applied sub-neg11.2

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

    8. Applied distribute-lft-in11.2

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

    9. Applied distribute-lft-in11.2

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

    10. Simplified11.2

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

    11. Simplified11.2

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

    13. Applied associate-*r*11.2

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

    if 6.144584664089615e+94 < c < 3.2353573317891035e+269

    1. Initial program 19.3

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

    2. Simplified19.3

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

    4. Applied add-cube-cbrt19.6

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

    5. Applied associate-*l*19.6

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

    6. Taylor expanded around inf 25.1

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

    7. Simplified12.0

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 6.14458466408961470336255307627453402921 \cdot 10^{94} \lor \neg \left(c \le 3.235357331789103476780750730187488778281 \cdot 10^{269}\right):\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, \mathsf{fma}\left(b, i \cdot a - c \cdot z, t \cdot \left(j \cdot c\right) + \left(-\left(i \cdot j\right) \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot z - t \cdot a, c \cdot \left(t \cdot j - z \cdot b\right) - i \cdot \left(j \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)))))