Average Error: 11.9 → 10.1
Time: 18.0s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.39273614561637652 \cdot 10^{98} \lor \neg \left(y \le 3.1039725660684998 \cdot 10^{119}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)\right)\right)\right)\\ \end{array}\]
\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}\;y \le -8.39273614561637652 \cdot 10^{98} \lor \neg \left(y \le 3.1039725660684998 \cdot 10^{119}\right):\\
\;\;\;\;\mathsf{fma}\left(t \cdot i - 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(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \mathsf{fma}\left(y, z, -a \cdot t\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 r972650 = x;
        double r972651 = y;
        double r972652 = z;
        double r972653 = r972651 * r972652;
        double r972654 = t;
        double r972655 = a;
        double r972656 = r972654 * r972655;
        double r972657 = r972653 - r972656;
        double r972658 = r972650 * r972657;
        double r972659 = b;
        double r972660 = c;
        double r972661 = r972660 * r972652;
        double r972662 = i;
        double r972663 = r972654 * r972662;
        double r972664 = r972661 - r972663;
        double r972665 = r972659 * r972664;
        double r972666 = r972658 - r972665;
        double r972667 = j;
        double r972668 = r972660 * r972655;
        double r972669 = r972651 * r972662;
        double r972670 = r972668 - r972669;
        double r972671 = r972667 * r972670;
        double r972672 = r972666 + r972671;
        return r972672;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r972673 = y;
        double r972674 = -8.392736145616377e+98;
        bool r972675 = r972673 <= r972674;
        double r972676 = 3.1039725660684998e+119;
        bool r972677 = r972673 <= r972676;
        double r972678 = !r972677;
        bool r972679 = r972675 || r972678;
        double r972680 = t;
        double r972681 = i;
        double r972682 = r972680 * r972681;
        double r972683 = c;
        double r972684 = z;
        double r972685 = r972683 * r972684;
        double r972686 = r972682 - r972685;
        double r972687 = b;
        double r972688 = x;
        double r972689 = r972688 * r972684;
        double r972690 = j;
        double r972691 = r972681 * r972690;
        double r972692 = r972689 - r972691;
        double r972693 = r972673 * r972692;
        double r972694 = a;
        double r972695 = r972688 * r972680;
        double r972696 = r972694 * r972695;
        double r972697 = r972693 - r972696;
        double r972698 = fma(r972686, r972687, r972697);
        double r972699 = r972683 * r972694;
        double r972700 = r972673 * r972681;
        double r972701 = r972699 - r972700;
        double r972702 = cbrt(r972688);
        double r972703 = r972702 * r972702;
        double r972704 = r972694 * r972680;
        double r972705 = -r972704;
        double r972706 = fma(r972673, r972684, r972705);
        double r972707 = r972702 * r972706;
        double r972708 = r972703 * r972707;
        double r972709 = fma(r972690, r972701, r972708);
        double r972710 = fma(r972686, r972687, r972709);
        double r972711 = r972679 ? r972698 : r972710;
        return r972711;
}

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

Original11.9
Target19.8
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -8.392736145616377e+98 or 3.1039725660684998e+119 < y

    1. Initial program 21.3

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

    2. Simplified21.3

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

    4. Applied fma-neg21.3

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

    5. Simplified21.3

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

    7. Applied add-cube-cbrt21.5

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

    8. Applied associate-*l*21.5

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

    9. Taylor expanded around inf 23.6

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

    10. Simplified11.9

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

    if -8.392736145616377e+98 < y < 3.1039725660684998e+119

    1. Initial program 9.3

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

    2. Simplified9.3

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

    4. Applied fma-neg9.3

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

    5. Simplified9.3

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

    7. Applied add-cube-cbrt9.6

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

    8. Applied associate-*l*9.6

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.39273614561637652 \cdot 10^{98} \lor \neg \left(y \le 3.1039725660684998 \cdot 10^{119}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, y \cdot \left(x \cdot z - i \cdot j\right) - a \cdot \left(x \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot i - c \cdot z, b, \mathsf{fma}\left(j, c \cdot a - y \cdot i, \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \mathsf{fma}\left(y, z, -a \cdot t\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (if (< x -1.469694296777705e-64) (+ (- (* x (- (* y z) (* t a))) (/ (* b (- (pow (* c z) 2) (pow (* t i) 2))) (+ (* 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) (pow (* t i) 2))) (+ (* c z) (* t i)))) (* j (- (* c a) (* y i))))))

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