Average Error: 12.3 → 13.0
Time: 13.0s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.93411759412074228 \cdot 10^{-89} \lor \neg \left(x \le 1.65682922233939055 \cdot 10^{-226}\right):\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, 0 - \left(b \cdot \left(c \cdot z - t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right)\\ \end{array}\]
\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}\;x \le -2.93411759412074228 \cdot 10^{-89} \lor \neg \left(x \le 1.65682922233939055 \cdot 10^{-226}\right):\\
\;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, 0 - \left(b \cdot \left(c \cdot z - t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\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 r1167546 = x;
        double r1167547 = y;
        double r1167548 = z;
        double r1167549 = r1167547 * r1167548;
        double r1167550 = t;
        double r1167551 = a;
        double r1167552 = r1167550 * r1167551;
        double r1167553 = r1167549 - r1167552;
        double r1167554 = r1167546 * r1167553;
        double r1167555 = b;
        double r1167556 = c;
        double r1167557 = r1167556 * r1167548;
        double r1167558 = i;
        double r1167559 = r1167550 * r1167558;
        double r1167560 = r1167557 - r1167559;
        double r1167561 = r1167555 * r1167560;
        double r1167562 = r1167554 - r1167561;
        double r1167563 = j;
        double r1167564 = r1167556 * r1167551;
        double r1167565 = r1167547 * r1167558;
        double r1167566 = r1167564 - r1167565;
        double r1167567 = r1167563 * r1167566;
        double r1167568 = r1167562 + r1167567;
        return r1167568;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r1167569 = x;
        double r1167570 = -2.9341175941207423e-89;
        bool r1167571 = r1167569 <= r1167570;
        double r1167572 = 1.6568292223393905e-226;
        bool r1167573 = r1167569 <= r1167572;
        double r1167574 = !r1167573;
        bool r1167575 = r1167571 || r1167574;
        double r1167576 = c;
        double r1167577 = a;
        double r1167578 = r1167576 * r1167577;
        double r1167579 = y;
        double r1167580 = i;
        double r1167581 = r1167579 * r1167580;
        double r1167582 = r1167578 - r1167581;
        double r1167583 = j;
        double r1167584 = z;
        double r1167585 = r1167579 * r1167584;
        double r1167586 = t;
        double r1167587 = r1167586 * r1167577;
        double r1167588 = r1167585 - r1167587;
        double r1167589 = r1167569 * r1167588;
        double r1167590 = b;
        double r1167591 = cbrt(r1167590);
        double r1167592 = r1167591 * r1167591;
        double r1167593 = r1167576 * r1167584;
        double r1167594 = r1167586 * r1167580;
        double r1167595 = r1167593 - r1167594;
        double r1167596 = r1167591 * r1167595;
        double r1167597 = r1167592 * r1167596;
        double r1167598 = -r1167580;
        double r1167599 = r1167580 * r1167586;
        double r1167600 = fma(r1167598, r1167586, r1167599);
        double r1167601 = r1167590 * r1167600;
        double r1167602 = r1167597 + r1167601;
        double r1167603 = r1167589 - r1167602;
        double r1167604 = fma(r1167582, r1167583, r1167603);
        double r1167605 = 0.0;
        double r1167606 = r1167590 * r1167595;
        double r1167607 = r1167606 + r1167601;
        double r1167608 = r1167605 - r1167607;
        double r1167609 = fma(r1167582, r1167583, r1167608);
        double r1167610 = r1167575 ? r1167604 : r1167609;
        return r1167610;
}

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
Target19.9
Herbie13.0
\[\begin{array}{l} \mathbf{if}\;x \lt -1.46969429677770502 \cdot 10^{-64}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;x \lt 3.2113527362226803 \cdot 10^{-147}:\\ \;\;\;\;\left(b \cdot i - x \cdot a\right) \cdot t - \left(z \cdot \left(c \cdot b\right) - j \cdot \left(c \cdot a - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \frac{b \cdot \left({\left(c \cdot z\right)}^{2} - {\left(t \cdot i\right)}^{2}\right)}{c \cdot z + t \cdot i}\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.9341175941207423e-89 or 1.6568292223393905e-226 < x

    1. Initial program 10.4

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

    2. Simplified10.4

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

    4. Applied prod-diff10.4

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

    5. Applied distribute-lft-in10.4

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

    6. Simplified10.4

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

    8. Applied add-cube-cbrt10.7

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

    9. Applied associate-*l*10.7

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

    if -2.9341175941207423e-89 < x < 1.6568292223393905e-226

    1. Initial program 16.3

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

    2. Simplified16.3

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

    4. Applied prod-diff16.3

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

    5. Applied distribute-lft-in16.3

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

    6. Simplified16.3

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

    7. Taylor expanded around 0 18.0

      \[\leadsto \mathsf{fma}\left(c \cdot a - y \cdot i, j, \color{blue}{0} - \left(b \cdot \left(c \cdot z - t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification13.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.93411759412074228 \cdot 10^{-89} \lor \neg \left(x \le 1.65682922233939055 \cdot 10^{-226}\right):\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - t \cdot i\right)\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - y \cdot i, j, 0 - \left(b \cdot \left(c \cdot z - t \cdot i\right) + b \cdot \mathsf{fma}\left(-i, t, i \cdot t\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))