Average Error: 11.6 → 9.3
Time: 21.8s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.564188589531598 \cdot 10^{-162}:\\ \;\;\;\;\left(\mathsf{fma}\left(y, z, \left(-t\right) \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(a \cdot \left(c \cdot j\right) - i \cdot \left(y \cdot j\right)\right)\\ \mathbf{elif}\;x \le 4.766395936578848 \cdot 10^{-177}:\\ \;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y - a \cdot \left(t \cdot x\right)\right) - \left(\left(b \cdot c\right) \cdot z - \left(i \cdot b\right) \cdot t\right)\right) + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(y, z, \left(-t\right) \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(a \cdot \left(c \cdot j\right) - i \cdot \left(y \cdot j\right)\right)\\ \end{array}\]
\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}\;x \le -3.564188589531598 \cdot 10^{-162}:\\
\;\;\;\;\left(\mathsf{fma}\left(y, z, \left(-t\right) \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(a \cdot \left(c \cdot j\right) - i \cdot \left(y \cdot j\right)\right)\\

\mathbf{elif}\;x \le 4.766395936578848 \cdot 10^{-177}:\\
\;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y - a \cdot \left(t \cdot x\right)\right) - \left(\left(b \cdot c\right) \cdot z - \left(i \cdot b\right) \cdot t\right)\right) + \left(c \cdot a - i \cdot y\right) \cdot j\\

\mathbf{else}:\\
\;\;\;\;\left(\mathsf{fma}\left(y, z, \left(-t\right) \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(a \cdot \left(c \cdot j\right) - i \cdot \left(y \cdot j\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 r16981666 = x;
        double r16981667 = y;
        double r16981668 = z;
        double r16981669 = r16981667 * r16981668;
        double r16981670 = t;
        double r16981671 = a;
        double r16981672 = r16981670 * r16981671;
        double r16981673 = r16981669 - r16981672;
        double r16981674 = r16981666 * r16981673;
        double r16981675 = b;
        double r16981676 = c;
        double r16981677 = r16981676 * r16981668;
        double r16981678 = i;
        double r16981679 = r16981670 * r16981678;
        double r16981680 = r16981677 - r16981679;
        double r16981681 = r16981675 * r16981680;
        double r16981682 = r16981674 - r16981681;
        double r16981683 = j;
        double r16981684 = r16981676 * r16981671;
        double r16981685 = r16981667 * r16981678;
        double r16981686 = r16981684 - r16981685;
        double r16981687 = r16981683 * r16981686;
        double r16981688 = r16981682 + r16981687;
        return r16981688;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r16981689 = x;
        double r16981690 = -3.564188589531598e-162;
        bool r16981691 = r16981689 <= r16981690;
        double r16981692 = y;
        double r16981693 = z;
        double r16981694 = t;
        double r16981695 = -r16981694;
        double r16981696 = a;
        double r16981697 = r16981695 * r16981696;
        double r16981698 = fma(r16981692, r16981693, r16981697);
        double r16981699 = r16981698 * r16981689;
        double r16981700 = b;
        double r16981701 = c;
        double r16981702 = r16981693 * r16981701;
        double r16981703 = i;
        double r16981704 = r16981703 * r16981694;
        double r16981705 = r16981702 - r16981704;
        double r16981706 = r16981700 * r16981705;
        double r16981707 = r16981699 - r16981706;
        double r16981708 = j;
        double r16981709 = r16981701 * r16981708;
        double r16981710 = r16981696 * r16981709;
        double r16981711 = r16981692 * r16981708;
        double r16981712 = r16981703 * r16981711;
        double r16981713 = r16981710 - r16981712;
        double r16981714 = r16981707 + r16981713;
        double r16981715 = 4.766395936578848e-177;
        bool r16981716 = r16981689 <= r16981715;
        double r16981717 = r16981693 * r16981689;
        double r16981718 = r16981717 * r16981692;
        double r16981719 = r16981694 * r16981689;
        double r16981720 = r16981696 * r16981719;
        double r16981721 = r16981718 - r16981720;
        double r16981722 = r16981700 * r16981701;
        double r16981723 = r16981722 * r16981693;
        double r16981724 = r16981703 * r16981700;
        double r16981725 = r16981724 * r16981694;
        double r16981726 = r16981723 - r16981725;
        double r16981727 = r16981721 - r16981726;
        double r16981728 = r16981701 * r16981696;
        double r16981729 = r16981703 * r16981692;
        double r16981730 = r16981728 - r16981729;
        double r16981731 = r16981730 * r16981708;
        double r16981732 = r16981727 + r16981731;
        double r16981733 = r16981716 ? r16981732 : r16981714;
        double r16981734 = r16981691 ? r16981714 : r16981733;
        return r16981734;
}

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

Original11.6
Target18.6
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;x \lt -1.469694296777705 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.564188589531598e-162 or 4.766395936578848e-177 < x

    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. Using strategy rm

    3. Applied fma-neg9.3

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

    4. Taylor expanded around inf 9.5

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

    if -3.564188589531598e-162 < x < 4.766395936578848e-177

    1. Initial program 17.2

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
    2. Using strategy rm

    3. Applied fma-neg17.2

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

    4. Taylor expanded around inf 13.5

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

    6. Applied associate-*r*10.3

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

    7. Taylor expanded around inf 8.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.564188589531598 \cdot 10^{-162}:\\ \;\;\;\;\left(\mathsf{fma}\left(y, z, \left(-t\right) \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(a \cdot \left(c \cdot j\right) - i \cdot \left(y \cdot j\right)\right)\\ \mathbf{elif}\;x \le 4.766395936578848 \cdot 10^{-177}:\\ \;\;\;\;\left(\left(\left(z \cdot x\right) \cdot y - a \cdot \left(t \cdot x\right)\right) - \left(\left(b \cdot c\right) \cdot z - \left(i \cdot b\right) \cdot t\right)\right) + \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(y, z, \left(-t\right) \cdot a\right) \cdot x - b \cdot \left(z \cdot c - i \cdot t\right)\right) + \left(a \cdot \left(c \cdot j\right) - i \cdot \left(y \cdot j\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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) (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)))))