Average Error: 11.7 → 12.9
Time: 19.2s
Precision: 64
\[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.3603574852745576 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \mathbf{elif}\;x \le 6.4460055181613243 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \end{array}\]
\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}\;x \le -2.3603574852745576 \cdot 10^{-289}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\

\mathbf{elif}\;x \le 6.4460055181613243 \cdot 10^{-280}:\\
\;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\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 r578677 = x;
        double r578678 = y;
        double r578679 = z;
        double r578680 = r578678 * r578679;
        double r578681 = t;
        double r578682 = a;
        double r578683 = r578681 * r578682;
        double r578684 = r578680 - r578683;
        double r578685 = r578677 * r578684;
        double r578686 = b;
        double r578687 = c;
        double r578688 = r578687 * r578679;
        double r578689 = i;
        double r578690 = r578689 * r578682;
        double r578691 = r578688 - r578690;
        double r578692 = r578686 * r578691;
        double r578693 = r578685 - r578692;
        double r578694 = j;
        double r578695 = r578687 * r578681;
        double r578696 = r578689 * r578678;
        double r578697 = r578695 - r578696;
        double r578698 = r578694 * r578697;
        double r578699 = r578693 + r578698;
        return r578699;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r578700 = x;
        double r578701 = -2.3603574852745576e-289;
        bool r578702 = r578700 <= r578701;
        double r578703 = c;
        double r578704 = t;
        double r578705 = r578703 * r578704;
        double r578706 = i;
        double r578707 = y;
        double r578708 = r578706 * r578707;
        double r578709 = r578705 - r578708;
        double r578710 = j;
        double r578711 = z;
        double r578712 = r578707 * r578711;
        double r578713 = r578700 * r578712;
        double r578714 = a;
        double r578715 = r578704 * r578714;
        double r578716 = -r578715;
        double r578717 = r578700 * r578716;
        double r578718 = r578713 + r578717;
        double r578719 = b;
        double r578720 = cbrt(r578719);
        double r578721 = r578720 * r578720;
        double r578722 = r578703 * r578711;
        double r578723 = r578706 * r578714;
        double r578724 = r578722 - r578723;
        double r578725 = r578720 * r578724;
        double r578726 = r578721 * r578725;
        double r578727 = -r578714;
        double r578728 = r578714 * r578706;
        double r578729 = fma(r578727, r578706, r578728);
        double r578730 = r578719 * r578729;
        double r578731 = r578726 + r578730;
        double r578732 = r578718 - r578731;
        double r578733 = fma(r578709, r578710, r578732);
        double r578734 = 6.446005518161324e-280;
        bool r578735 = r578700 <= r578734;
        double r578736 = r578706 * r578719;
        double r578737 = r578719 * r578703;
        double r578738 = r578700 * r578704;
        double r578739 = r578714 * r578738;
        double r578740 = fma(r578711, r578737, r578739);
        double r578741 = -r578740;
        double r578742 = fma(r578714, r578736, r578741);
        double r578743 = sqrt(r578700);
        double r578744 = r578712 - r578715;
        double r578745 = r578743 * r578744;
        double r578746 = r578743 * r578745;
        double r578747 = r578719 * r578724;
        double r578748 = r578747 + r578730;
        double r578749 = r578746 - r578748;
        double r578750 = fma(r578709, r578710, r578749);
        double r578751 = r578735 ? r578742 : r578750;
        double r578752 = r578702 ? r578733 : r578751;
        return r578752;
}

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

Original11.7
Target15.8
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;t \lt -8.1209789191959122 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt -4.7125538182184851 \cdot 10^{-169}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{elif}\;t \lt -7.63353334603158369 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{elif}\;t \lt 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.3603574852745576e-289

    1. Initial program 11.6

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

    2. Simplified11.6

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

    4. Applied prod-diff11.6

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

    5. Applied distribute-lft-in11.6

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

    6. Simplified11.6

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

    8. Applied add-cube-cbrt11.9

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

    9. Applied associate-*l*11.9

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

    11. Applied sub-neg11.9

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

    12. Applied distribute-lft-in11.9

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

    if -2.3603574852745576e-289 < x < 6.446005518161324e-280

    1. Initial program 17.7

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

    2. Simplified17.6

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

    3. Taylor expanded around inf 36.1

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

    4. Simplified36.1

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

    if 6.446005518161324e-280 < x

    1. Initial program 11.2

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

    2. Simplified11.2

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

    4. Applied prod-diff11.2

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

    5. Applied distribute-lft-in11.2

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

    6. Simplified11.2

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

    8. Applied add-sqr-sqrt11.3

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

    9. Applied associate-*l*11.3

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.3603574852745576 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \left(x \cdot \left(y \cdot z\right) + x \cdot \left(-t \cdot a\right)\right) - \left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \left(c \cdot z - i \cdot a\right)\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \mathbf{elif}\;x \le 6.4460055181613243 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(a, i \cdot b, -\mathsf{fma}\left(z, b \cdot c, a \cdot \left(x \cdot t\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot t - i \cdot y, j, \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(y \cdot z - t \cdot a\right)\right) - \left(b \cdot \left(c \cdot z - i \cdot a\right) + b \cdot \mathsf{fma}\left(-a, i, a \cdot i\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -8.120978919195912e-33) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -4.712553818218485e-169) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -7.633533346031584e-308) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 1.0535888557455487e-139) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))))

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