Average Error: 12.3 → 10.4
Time: 15.0s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\ \end{array}\]
\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}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\
\;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\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 r103681 = x;
        double r103682 = y;
        double r103683 = z;
        double r103684 = r103682 * r103683;
        double r103685 = t;
        double r103686 = a;
        double r103687 = r103685 * r103686;
        double r103688 = r103684 - r103687;
        double r103689 = r103681 * r103688;
        double r103690 = b;
        double r103691 = c;
        double r103692 = r103691 * r103683;
        double r103693 = i;
        double r103694 = r103693 * r103686;
        double r103695 = r103692 - r103694;
        double r103696 = r103690 * r103695;
        double r103697 = r103689 - r103696;
        double r103698 = j;
        double r103699 = r103691 * r103685;
        double r103700 = r103693 * r103682;
        double r103701 = r103699 - r103700;
        double r103702 = r103698 * r103701;
        double r103703 = r103697 + r103702;
        return r103703;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r103704 = y;
        double r103705 = -2.690019024448134e+149;
        bool r103706 = r103704 <= r103705;
        double r103707 = 6.851977512719908e+134;
        bool r103708 = r103704 <= r103707;
        double r103709 = !r103708;
        bool r103710 = r103706 || r103709;
        double r103711 = i;
        double r103712 = a;
        double r103713 = r103711 * r103712;
        double r103714 = c;
        double r103715 = z;
        double r103716 = r103714 * r103715;
        double r103717 = r103713 - r103716;
        double r103718 = b;
        double r103719 = x;
        double r103720 = r103719 * r103715;
        double r103721 = j;
        double r103722 = r103711 * r103721;
        double r103723 = r103720 - r103722;
        double r103724 = r103704 * r103723;
        double r103725 = t;
        double r103726 = r103719 * r103725;
        double r103727 = r103712 * r103726;
        double r103728 = r103724 - r103727;
        double r103729 = fma(r103717, r103718, r103728);
        double r103730 = r103714 * r103725;
        double r103731 = r103711 * r103704;
        double r103732 = r103730 - r103731;
        double r103733 = cbrt(r103719);
        double r103734 = r103733 * r103733;
        double r103735 = r103704 * r103715;
        double r103736 = r103725 * r103712;
        double r103737 = r103735 - r103736;
        double r103738 = r103733 * r103737;
        double r103739 = r103734 * r103738;
        double r103740 = fma(r103721, r103732, r103739);
        double r103741 = fma(r103717, r103718, r103740);
        double r103742 = r103710 ? r103729 : r103741;
        return r103742;
}

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 y < -2.690019024448134e+149 or 6.851977512719908e+134 < y

    1. Initial program 23.4

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

    2. Simplified23.4

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

    4. Applied add-cube-cbrt23.6

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

    5. Applied associate-*l*23.6

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

    6. Taylor expanded around inf 25.0

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

    7. Simplified10.5

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

    if -2.690019024448134e+149 < y < 6.851977512719908e+134

    1. Initial program 10.1

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

    2. Simplified10.1

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

    4. Applied add-cube-cbrt10.4

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

    5. Applied associate-*l*10.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.69001902444813421 \cdot 10^{149} \lor \neg \left(y \le 6.8519775127199077 \cdot 10^{134}\right):\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i \cdot a - c \cdot z, b, \mathsf{fma}\left(j, c \cdot t - i \cdot y, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \left(y \cdot z - t \cdot a\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))))