Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.1 → 14.8

Time: 8.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.19802269886933 \cdot 10^{-44}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{\left(z \cdot b\right) \cdot c} \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c}\right) \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c} + \left(-b\right) \cdot \left(i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le -3.03250526217835203 \cdot 10^{-113}:\\ \;\;\;\;\left(t \cdot \left(j \cdot c\right) + a \cdot \left(i \cdot b\right)\right) - i \cdot \left(j \cdot y\right)\\ \mathbf{elif}\;x \le 8.1670837412048603 \cdot 10^{-187}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{\left(z \cdot b\right) \cdot c} \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c}\right) \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c} + \left(-b\right) \cdot \left(i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 6.994862030262357 \cdot 10^{141}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + 1 \cdot \left(-1 \cdot \left(a \cdot \left(i \cdot b\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 4.8310657242358433 \cdot 10^{287}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + 0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{\left(z \cdot b\right) \cdot c} \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c}\right) \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c} + \left(-b\right) \cdot \left(i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r138711 = x;
        double r138712 = y;
        double r138713 = z;
        double r138714 = r138712 * r138713;
        double r138715 = t;
        double r138716 = a;
        double r138717 = r138715 * r138716;
        double r138718 = r138714 - r138717;
        double r138719 = r138711 * r138718;
        double r138720 = b;
        double r138721 = c;
        double r138722 = r138721 * r138713;
        double r138723 = i;
        double r138724 = r138723 * r138716;
        double r138725 = r138722 - r138724;
        double r138726 = r138720 * r138725;
        double r138727 = r138719 - r138726;
        double r138728 = j;
        double r138729 = r138721 * r138715;
        double r138730 = r138723 * r138712;
        double r138731 = r138729 - r138730;
        double r138732 = r138728 * r138731;
        double r138733 = r138727 + r138732;
        return r138733;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r138734 = x;
        double r138735 = -2.198022698869332e-44;
        bool r138736 = r138734 <= r138735;
        double r138737 = y;
        double r138738 = z;
        double r138739 = r138737 * r138738;
        double r138740 = t;
        double r138741 = a;
        double r138742 = r138740 * r138741;
        double r138743 = r138739 - r138742;
        double r138744 = r138734 * r138743;
        double r138745 = b;
        double r138746 = r138738 * r138745;
        double r138747 = c;
        double r138748 = r138746 * r138747;
        double r138749 = cbrt(r138748);
        double r138750 = r138749 * r138749;
        double r138751 = r138750 * r138749;
        double r138752 = -r138745;
        double r138753 = i;
        double r138754 = r138753 * r138741;
        double r138755 = r138752 * r138754;
        double r138756 = r138751 + r138755;
        double r138757 = r138744 - r138756;
        double r138758 = j;
        double r138759 = r138747 * r138740;
        double r138760 = r138753 * r138737;
        double r138761 = r138759 - r138760;
        double r138762 = r138758 * r138761;
        double r138763 = r138757 + r138762;
        double r138764 = -3.032505262178352e-113;
        bool r138765 = r138734 <= r138764;
        double r138766 = r138758 * r138747;
        double r138767 = r138740 * r138766;
        double r138768 = r138753 * r138745;
        double r138769 = r138741 * r138768;
        double r138770 = r138767 + r138769;
        double r138771 = r138758 * r138737;
        double r138772 = r138753 * r138771;
        double r138773 = r138770 - r138772;
        double r138774 = 8.16708374120486e-187;
        bool r138775 = r138734 <= r138774;
        double r138776 = 6.994862030262357e+141;
        bool r138777 = r138734 <= r138776;
        double r138778 = r138745 * r138747;
        double r138779 = r138738 * r138778;
        double r138780 = 1.0;
        double r138781 = -1.0;
        double r138782 = r138781 * r138769;
        double r138783 = r138780 * r138782;
        double r138784 = r138779 + r138783;
        double r138785 = r138744 - r138784;
        double r138786 = r138785 + r138762;
        double r138787 = 4.831065724235843e+287;
        bool r138788 = r138734 <= r138787;
        double r138789 = r138747 * r138738;
        double r138790 = r138789 - r138754;
        double r138791 = r138745 * r138790;
        double r138792 = r138744 - r138791;
        double r138793 = 0.0;
        double r138794 = r138792 + r138793;
        double r138795 = r138788 ? r138794 : r138763;
        double r138796 = r138777 ? r138786 : r138795;
        double r138797 = r138775 ? r138763 : r138796;
        double r138798 = r138765 ? r138773 : r138797;
        double r138799 = r138736 ? r138763 : r138798;
        return r138799;
}

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

Derivation

1. Split input into 4 regimes

2. if x < -2.198022698869332e-44 or -3.032505262178352e-113 < x < 8.16708374120486e-187 or 4.831065724235843e+287 < x

   1. Initial program 13.3

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

   3. Applied add-cube-cbrt 13.6

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

   4. Applied associate-*l* 13.6

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

   6. Applied sub-neg 13.6

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

   7. Applied distribute-lft-in 13.6

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

   8. Applied distribute-lft-in 13.6

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

   9. Simplified 14.0

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

   10. Simplified 13.8

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

   12. Applied associate-*r* 13.8

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

   14. Applied add-cube-cbrt 13.9

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

   if -2.198022698869332e-44 < x < -3.032505262178352e-113

   1. Initial program 11.9

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

   2. Taylor expanded around inf 33.3

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

   if 8.16708374120486e-187 < x < 6.994862030262357e+141

   1. Initial program 11.0

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

   3. Applied add-cube-cbrt 11.3

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

   4. Applied associate-*l* 11.3

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

   6. Applied sub-neg 11.3

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

   7. Applied distribute-lft-in 11.3

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

   8. Applied distribute-lft-in 11.3

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

   9. Simplified 11.6

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

   10. Simplified 11.4

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

   12. Applied *-un-lft-identity 11.4

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{\left(1 \cdot \left(-b\right)\right)} \cdot \left(i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   13. Applied associate-*l* 11.4

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + \color{blue}{1 \cdot \left(\left(-b\right) \cdot \left(i \cdot a\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   14. Simplified 11.6

       \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + 1 \cdot \color{blue}{\left(-1 \cdot \left(a \cdot \left(i \cdot b\right)\right)\right)}\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\]

   if 6.994862030262357e+141 < x < 4.831065724235843e+287

   1. Initial program 7.3

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

   2. Taylor expanded around 0 17.0

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \color{blue}{0}\]
3. Recombined 4 regimes into one program.

4. Final simplification 14.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.19802269886933 \cdot 10^{-44}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{\left(z \cdot b\right) \cdot c} \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c}\right) \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c} + \left(-b\right) \cdot \left(i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le -3.03250526217835203 \cdot 10^{-113}:\\ \;\;\;\;\left(t \cdot \left(j \cdot c\right) + a \cdot \left(i \cdot b\right)\right) - i \cdot \left(j \cdot y\right)\\ \mathbf{elif}\;x \le 8.1670837412048603 \cdot 10^{-187}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{\left(z \cdot b\right) \cdot c} \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c}\right) \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c} + \left(-b\right) \cdot \left(i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 6.994862030262357 \cdot 10^{141}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(z \cdot \left(b \cdot c\right) + 1 \cdot \left(-1 \cdot \left(a \cdot \left(i \cdot b\right)\right)\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 4.8310657242358433 \cdot 10^{287}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + 0\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - \left(\left(\sqrt[3]{\left(z \cdot b\right) \cdot c} \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c}\right) \cdot \sqrt[3]{\left(z \cdot b\right) \cdot c} + \left(-b\right) \cdot \left(i \cdot a\right)\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))