Average Error: 12.3 → 12.3
Time: 33.3s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.304458212992853717965315340630575782583 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(t \cdot i - c \cdot z, b, \left(x \cdot \sqrt[3]{y \cdot z - a \cdot t}\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\\ \mathbf{elif}\;b \le 1.231745421381366465829001065133154069762 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, z \cdot \left(x \cdot y - c \cdot b\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{a \cdot c - y \cdot i} \cdot \left(\left(\sqrt[3]{\sqrt[3]{a \cdot c - y \cdot i}} \cdot \sqrt[3]{\sqrt[3]{a \cdot c - y \cdot i}}\right) \cdot \sqrt[3]{\sqrt[3]{a \cdot c - y \cdot i}}\right)\right) \cdot \left(\sqrt[3]{a \cdot c - y \cdot i} \cdot j\right) + \mathsf{fma}\left(t \cdot i - c \cdot z, b, \left(y \cdot z - a \cdot t\right) \cdot x\right)\\ \end{array}\]
\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)
↓
\begin{array}{l}
\mathbf{if}\;b \le -7.304458212992853717965315340630575782583 \cdot 10^{-17}:\\
\;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(t \cdot i - c \cdot z, b, \left(x \cdot \sqrt[3]{y \cdot z - a \cdot t}\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\\

\mathbf{elif}\;b \le 1.231745421381366465829001065133154069762 \cdot 10^{-306}:\\
\;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, z \cdot \left(x \cdot y - c \cdot b\right) - a \cdot \left(x \cdot t\right)\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r32780639 = x;
        double r32780640 = y;
        double r32780641 = z;
        double r32780642 = r32780640 * r32780641;
        double r32780643 = t;
        double r32780644 = a;
        double r32780645 = r32780643 * r32780644;
        double r32780646 = r32780642 - r32780645;
        double r32780647 = r32780639 * r32780646;
        double r32780648 = b;
        double r32780649 = c;
        double r32780650 = r32780649 * r32780641;
        double r32780651 = i;
        double r32780652 = r32780643 * r32780651;
        double r32780653 = r32780650 - r32780652;
        double r32780654 = r32780648 * r32780653;
        double r32780655 = r32780647 - r32780654;
        double r32780656 = j;
        double r32780657 = r32780649 * r32780644;
        double r32780658 = r32780640 * r32780651;
        double r32780659 = r32780657 - r32780658;
        double r32780660 = r32780656 * r32780659;
        double r32780661 = r32780655 + r32780660;
        return r32780661;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r32780662 = b;
        double r32780663 = -7.304458212992854e-17;
        bool r32780664 = r32780662 <= r32780663;
        double r32780665 = a;
        double r32780666 = c;
        double r32780667 = r32780665 * r32780666;
        double r32780668 = y;
        double r32780669 = i;
        double r32780670 = r32780668 * r32780669;
        double r32780671 = r32780667 - r32780670;
        double r32780672 = j;
        double r32780673 = t;
        double r32780674 = r32780673 * r32780669;
        double r32780675 = z;
        double r32780676 = r32780666 * r32780675;
        double r32780677 = r32780674 - r32780676;
        double r32780678 = x;
        double r32780679 = r32780668 * r32780675;
        double r32780680 = r32780665 * r32780673;
        double r32780681 = r32780679 - r32780680;
        double r32780682 = cbrt(r32780681);
        double r32780683 = r32780678 * r32780682;
        double r32780684 = r32780682 * r32780682;
        double r32780685 = r32780683 * r32780684;
        double r32780686 = fma(r32780677, r32780662, r32780685);
        double r32780687 = fma(r32780671, r32780672, r32780686);
        double r32780688 = 1.2317454213813665e-306;
        bool r32780689 = r32780662 <= r32780688;
        double r32780690 = r32780678 * r32780668;
        double r32780691 = r32780666 * r32780662;
        double r32780692 = r32780690 - r32780691;
        double r32780693 = r32780675 * r32780692;
        double r32780694 = r32780678 * r32780673;
        double r32780695 = r32780665 * r32780694;
        double r32780696 = r32780693 - r32780695;
        double r32780697 = fma(r32780671, r32780672, r32780696);
        double r32780698 = cbrt(r32780671);
        double r32780699 = cbrt(r32780698);
        double r32780700 = r32780699 * r32780699;
        double r32780701 = r32780700 * r32780699;
        double r32780702 = r32780698 * r32780701;
        double r32780703 = r32780698 * r32780672;
        double r32780704 = r32780702 * r32780703;
        double r32780705 = r32780681 * r32780678;
        double r32780706 = fma(r32780677, r32780662, r32780705);
        double r32780707 = r32780704 + r32780706;
        double r32780708 = r32780689 ? r32780697 : r32780707;
        double r32780709 = r32780664 ? r32780687 : r32780708;
        return r32780709;
}

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

Target

Original12.3
Target19.2
Herbie12.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705016266218530347997287942 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.21135273622268028942701600607048800714 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.304458212992854e-17

    1. Initial program 7.5

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

    2. Simplified7.5

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

    4. Applied add-cube-cbrt7.7

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

    5. Applied associate-*l*7.7

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

    if -7.304458212992854e-17 < b < 1.2317454213813665e-306

    1. Initial program 16.5

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

    2. Simplified16.5

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

    3. Taylor expanded around inf 16.3

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

    4. Simplified15.6

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

    if 1.2317454213813665e-306 < b

    1. Initial program 11.9

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

    2. Simplified11.9

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

    4. Applied fma-udef11.9

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

    6. Applied add-cube-cbrt12.2

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

    7. Applied associate-*l*12.2

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

    9. Applied add-cube-cbrt12.3

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.304458212992853717965315340630575782583 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, \mathsf{fma}\left(t \cdot i - c \cdot z, b, \left(x \cdot \sqrt[3]{y \cdot z - a \cdot t}\right) \cdot \left(\sqrt[3]{y \cdot z - a \cdot t} \cdot \sqrt[3]{y \cdot z - a \cdot t}\right)\right)\right)\\ \mathbf{elif}\;b \le 1.231745421381366465829001065133154069762 \cdot 10^{-306}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot c - y \cdot i, j, z \cdot \left(x \cdot y - c \cdot b\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{a \cdot c - y \cdot i} \cdot \left(\left(\sqrt[3]{\sqrt[3]{a \cdot c - y \cdot i}} \cdot \sqrt[3]{\sqrt[3]{a \cdot c - y \cdot i}}\right) \cdot \sqrt[3]{\sqrt[3]{a \cdot c - y \cdot i}}\right)\right) \cdot \left(\sqrt[3]{a \cdot c - y \cdot i} \cdot j\right) + \mathsf{fma}\left(t \cdot i - c \cdot z, b, \left(y \cdot z - a \cdot t\right) \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i)))) (if (< x 3.2113527362226803e-147) (- (* (- (* b i) (* x a)) t) (- (* z (* c b)) (* j (- (* c a) (* y i))))) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2.0) (pow (* t i) 2.0))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))