Average Error: 11.4 → 11.1
Time: 1.3m
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -4.5347081220454865 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\sqrt[3]{x} \cdot \left(\left(z \cdot y - a \cdot t\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\\ \mathbf{elif}\;j \le 4.490984132478977 \cdot 10^{-224}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - i \cdot \left(j \cdot y\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\left(\sqrt[3]{z \cdot y - a \cdot t} \cdot x\right) \cdot \left(\sqrt[3]{z \cdot y - a \cdot t} \cdot \sqrt[3]{z \cdot y - a \cdot t}\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\\ \end{array}\]
\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 -4.5347081220454865 \cdot 10^{-240}:\\
\;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\sqrt[3]{x} \cdot \left(\left(z \cdot y - a \cdot t\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\\

\mathbf{elif}\;j \le 4.490984132478977 \cdot 10^{-224}:\\
\;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - i \cdot \left(j \cdot y\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r19551700 = x;
        double r19551701 = y;
        double r19551702 = z;
        double r19551703 = r19551701 * r19551702;
        double r19551704 = t;
        double r19551705 = a;
        double r19551706 = r19551704 * r19551705;
        double r19551707 = r19551703 - r19551706;
        double r19551708 = r19551700 * r19551707;
        double r19551709 = b;
        double r19551710 = c;
        double r19551711 = r19551710 * r19551702;
        double r19551712 = i;
        double r19551713 = r19551712 * r19551705;
        double r19551714 = r19551711 - r19551713;
        double r19551715 = r19551709 * r19551714;
        double r19551716 = r19551708 - r19551715;
        double r19551717 = j;
        double r19551718 = r19551710 * r19551704;
        double r19551719 = r19551712 * r19551701;
        double r19551720 = r19551718 - r19551719;
        double r19551721 = r19551717 * r19551720;
        double r19551722 = r19551716 + r19551721;
        return r19551722;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r19551723 = j;
        double r19551724 = -4.5347081220454865e-240;
        bool r19551725 = r19551723 <= r19551724;
        double r19551726 = t;
        double r19551727 = c;
        double r19551728 = r19551726 * r19551727;
        double r19551729 = i;
        double r19551730 = y;
        double r19551731 = r19551729 * r19551730;
        double r19551732 = r19551728 - r19551731;
        double r19551733 = x;
        double r19551734 = cbrt(r19551733);
        double r19551735 = z;
        double r19551736 = r19551735 * r19551730;
        double r19551737 = a;
        double r19551738 = r19551737 * r19551726;
        double r19551739 = r19551736 - r19551738;
        double r19551740 = r19551734 * r19551734;
        double r19551741 = r19551739 * r19551740;
        double r19551742 = r19551734 * r19551741;
        double r19551743 = fma(r19551732, r19551723, r19551742);
        double r19551744 = b;
        double r19551745 = r19551727 * r19551735;
        double r19551746 = r19551737 * r19551729;
        double r19551747 = r19551745 - r19551746;
        double r19551748 = r19551744 * r19551747;
        double r19551749 = r19551743 - r19551748;
        double r19551750 = 4.490984132478977e-224;
        bool r19551751 = r19551723 <= r19551750;
        double r19551752 = r19551733 * r19551739;
        double r19551753 = r19551723 * r19551730;
        double r19551754 = r19551729 * r19551753;
        double r19551755 = r19551752 - r19551754;
        double r19551756 = r19551755 - r19551748;
        double r19551757 = cbrt(r19551739);
        double r19551758 = r19551757 * r19551733;
        double r19551759 = r19551757 * r19551757;
        double r19551760 = r19551758 * r19551759;
        double r19551761 = fma(r19551732, r19551723, r19551760);
        double r19551762 = r19551761 - r19551748;
        double r19551763 = r19551751 ? r19551756 : r19551762;
        double r19551764 = r19551725 ? r19551749 : r19551763;
        return r19551764;
}

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 < -4.5347081220454865e-240

    1. Initial program 11.0

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

    2. Simplified11.0

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

    4. Applied add-cube-cbrt11.3

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

    5. Applied associate-*r*11.3

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

    if -4.5347081220454865e-240 < j < 4.490984132478977e-224

    1. Initial program 15.5

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

    2. Simplified15.5

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

    3. Taylor expanded around -inf 13.3

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

    4. Simplified11.9

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

    if 4.490984132478977e-224 < j

    1. Initial program 10.3

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

    2. Simplified10.3

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

    4. Applied add-cube-cbrt10.6

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

    5. Applied associate-*l*10.6

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -4.5347081220454865 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\sqrt[3]{x} \cdot \left(\left(z \cdot y - a \cdot t\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\\ \mathbf{elif}\;j \le 4.490984132478977 \cdot 10^{-224}:\\ \;\;\;\;\left(x \cdot \left(z \cdot y - a \cdot t\right) - i \cdot \left(j \cdot y\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t \cdot c - i \cdot y\right), j, \left(\left(\sqrt[3]{z \cdot y - a \cdot t} \cdot x\right) \cdot \left(\sqrt[3]{z \cdot y - a \cdot t} \cdot \sqrt[3]{z \cdot y - a \cdot t}\right)\right)\right) - b \cdot \left(c \cdot z - a \cdot i\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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)))))