Average Error: 12.4 → 14.3
Time: 11.2s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.13021924517858906 \cdot 10^{149} \lor \neg \left(z \le 3.5058906695120028 \cdot 10^{174}\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot \left(\sqrt[3]{z \cdot c - a \cdot i} \cdot \sqrt[3]{z \cdot c - a \cdot i}\right)\right) \cdot \sqrt[3]{z \cdot c - a \cdot i} + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \end{array}\]
\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}\;z \le -3.13021924517858906 \cdot 10^{149} \lor \neg \left(z \le 3.5058906695120028 \cdot 10^{174}\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(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot \left(\sqrt[3]{z \cdot c - a \cdot i} \cdot \sqrt[3]{z \cdot c - a \cdot i}\right)\right) \cdot \sqrt[3]{z \cdot c - a \cdot i} + b \cdot \mathsf{fma}\left(-a, i, a \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 r143618 = x;
        double r143619 = y;
        double r143620 = z;
        double r143621 = r143619 * r143620;
        double r143622 = t;
        double r143623 = a;
        double r143624 = r143622 * r143623;
        double r143625 = r143621 - r143624;
        double r143626 = r143618 * r143625;
        double r143627 = b;
        double r143628 = c;
        double r143629 = r143628 * r143620;
        double r143630 = i;
        double r143631 = r143630 * r143623;
        double r143632 = r143629 - r143631;
        double r143633 = r143627 * r143632;
        double r143634 = r143626 - r143633;
        double r143635 = j;
        double r143636 = r143628 * r143622;
        double r143637 = r143630 * r143619;
        double r143638 = r143636 - r143637;
        double r143639 = r143635 * r143638;
        double r143640 = r143634 + r143639;
        return r143640;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r143641 = z;
        double r143642 = -3.130219245178589e+149;
        bool r143643 = r143641 <= r143642;
        double r143644 = 3.505890669512003e+174;
        bool r143645 = r143641 <= r143644;
        double r143646 = !r143645;
        bool r143647 = r143643 || r143646;
        double r143648 = a;
        double r143649 = i;
        double r143650 = b;
        double r143651 = r143649 * r143650;
        double r143652 = c;
        double r143653 = r143650 * r143652;
        double r143654 = x;
        double r143655 = t;
        double r143656 = r143654 * r143655;
        double r143657 = r143648 * r143656;
        double r143658 = fma(r143641, r143653, r143657);
        double r143659 = -r143658;
        double r143660 = fma(r143648, r143651, r143659);
        double r143661 = r143652 * r143655;
        double r143662 = y;
        double r143663 = r143649 * r143662;
        double r143664 = r143661 - r143663;
        double r143665 = j;
        double r143666 = r143662 * r143641;
        double r143667 = r143655 * r143648;
        double r143668 = r143666 - r143667;
        double r143669 = r143654 * r143668;
        double r143670 = r143641 * r143652;
        double r143671 = r143648 * r143649;
        double r143672 = r143670 - r143671;
        double r143673 = cbrt(r143672);
        double r143674 = r143673 * r143673;
        double r143675 = r143650 * r143674;
        double r143676 = r143675 * r143673;
        double r143677 = -r143648;
        double r143678 = fma(r143677, r143649, r143671);
        double r143679 = r143650 * r143678;
        double r143680 = r143676 + r143679;
        double r143681 = r143669 - r143680;
        double r143682 = fma(r143664, r143665, r143681);
        double r143683 = r143647 ? r143660 : r143682;
        return r143683;
}

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 z < -3.130219245178589e+149 or 3.505890669512003e+174 < z

    1. Initial program 23.0

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

    2. Simplified23.0

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

    3. Taylor expanded around inf 34.4

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

    4. Simplified34.4

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

    if -3.130219245178589e+149 < z < 3.505890669512003e+174

    1. Initial program 10.6

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

    2. Simplified10.6

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

    4. Applied prod-diff10.6

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

    5. Applied distribute-lft-in10.6

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

    6. Simplified10.6

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

    8. Applied add-cube-cbrt10.9

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

    9. Applied associate-*r*10.9

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

  4. Final simplification14.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.13021924517858906 \cdot 10^{149} \lor \neg \left(z \le 3.5058906695120028 \cdot 10^{174}\right):\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(b \cdot \left(\sqrt[3]{z \cdot c - a \cdot i} \cdot \sqrt[3]{z \cdot c - a \cdot i}\right)\right) \cdot \sqrt[3]{z \cdot c - a \cdot i} + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))