Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 12.3 → 11.9

Time: 24.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.125714986291682273805372429336846047599 \cdot 10^{-87}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + \left(i \cdot \left(b \cdot a\right) - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;x \le -5.851479915422898385518905209125160674093 \cdot 10^{-184}:\\ \;\;\;\;\left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\left(\sqrt[3]{a} \cdot i\right) \cdot b\right) - z \cdot \left(b \cdot c\right)\right) + \left(\left(\left(y \cdot j\right) \cdot \left(-i\right) + \left(t \cdot j\right) \cdot c\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;x \le 1.078548306168217375586414278153095298333 \cdot 10^{-171}:\\ \;\;\;\;\left(\left(i \cdot b\right) \cdot a - z \cdot \left(b \cdot c\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 3.351495691018441958957028519658107878312 \cdot 10^{104}:\\ \;\;\;\;\left(\left(i \cdot b\right) \cdot a - b \cdot \left(c \cdot z\right)\right) + \left(\left(y \cdot \left(i \cdot \left(-j\right)\right) + t \cdot \left(j \cdot c\right)\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + \left(i \cdot \left(b \cdot a\right) - z \cdot \left(b \cdot c\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r83581 = x;
        double r83582 = y;
        double r83583 = z;
        double r83584 = r83582 * r83583;
        double r83585 = t;
        double r83586 = a;
        double r83587 = r83585 * r83586;
        double r83588 = r83584 - r83587;
        double r83589 = r83581 * r83588;
        double r83590 = b;
        double r83591 = c;
        double r83592 = r83591 * r83583;
        double r83593 = i;
        double r83594 = r83593 * r83586;
        double r83595 = r83592 - r83594;
        double r83596 = r83590 * r83595;
        double r83597 = r83589 - r83596;
        double r83598 = j;
        double r83599 = r83591 * r83585;
        double r83600 = r83593 * r83582;
        double r83601 = r83599 - r83600;
        double r83602 = r83598 * r83601;
        double r83603 = r83597 + r83602;
        return r83603;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r83604 = x;
        double r83605 = -9.125714986291682e-87;
        bool r83606 = r83604 <= r83605;
        double r83607 = y;
        double r83608 = z;
        double r83609 = r83607 * r83608;
        double r83610 = t;
        double r83611 = a;
        double r83612 = r83610 * r83611;
        double r83613 = r83609 - r83612;
        double r83614 = r83604 * r83613;
        double r83615 = j;
        double r83616 = c;
        double r83617 = r83616 * r83610;
        double r83618 = i;
        double r83619 = r83618 * r83607;
        double r83620 = r83617 - r83619;
        double r83621 = r83615 * r83620;
        double r83622 = r83614 + r83621;
        double r83623 = b;
        double r83624 = r83623 * r83611;
        double r83625 = r83618 * r83624;
        double r83626 = r83623 * r83616;
        double r83627 = r83608 * r83626;
        double r83628 = r83625 - r83627;
        double r83629 = r83622 + r83628;
        double r83630 = -5.851479915422898e-184;
        bool r83631 = r83604 <= r83630;
        double r83632 = cbrt(r83611);
        double r83633 = r83632 * r83632;
        double r83634 = r83632 * r83618;
        double r83635 = r83634 * r83623;
        double r83636 = r83633 * r83635;
        double r83637 = r83636 - r83627;
        double r83638 = r83607 * r83615;
        double r83639 = -r83618;
        double r83640 = r83638 * r83639;
        double r83641 = r83610 * r83615;
        double r83642 = r83641 * r83616;
        double r83643 = r83640 + r83642;
        double r83644 = r83643 + r83614;
        double r83645 = r83637 + r83644;
        double r83646 = 1.0785483061682174e-171;
        bool r83647 = r83604 <= r83646;
        double r83648 = r83618 * r83623;
        double r83649 = r83648 * r83611;
        double r83650 = r83649 - r83627;
        double r83651 = r83650 + r83621;
        double r83652 = 3.351495691018442e+104;
        bool r83653 = r83604 <= r83652;
        double r83654 = r83616 * r83608;
        double r83655 = r83623 * r83654;
        double r83656 = r83649 - r83655;
        double r83657 = -r83615;
        double r83658 = r83618 * r83657;
        double r83659 = r83607 * r83658;
        double r83660 = r83615 * r83616;
        double r83661 = r83610 * r83660;
        double r83662 = r83659 + r83661;
        double r83663 = r83662 + r83614;
        double r83664 = r83656 + r83663;
        double r83665 = r83653 ? r83664 : r83629;
        double r83666 = r83647 ? r83651 : r83665;
        double r83667 = r83631 ? r83645 : r83666;
        double r83668 = r83606 ? r83629 : r83667;
        return r83668;
}

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 < -9.125714986291682e-87 or 3.351495691018442e+104 < x

   1. Initial program 7.9

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

   2. Simplified 7.9

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

   3. Taylor expanded around inf 9.6

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

   4. Simplified 9.4

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

   5. Taylor expanded around inf 9.6

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

   6. Simplified 9.6

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

   8. Applied add-cube-cbrt 9.8

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

   9. Applied associate-*l* 9.8

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

   10. Simplified 8.7

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

   11. Taylor expanded around inf 9.6

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

   12. Simplified 8.7

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

   if -9.125714986291682e-87 < x < -5.851479915422898e-184

   1. Initial program 15.8

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

   2. Simplified 15.8

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

   3. Taylor expanded around inf 15.9

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

   4. Simplified 16.1

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

   5. Taylor expanded around inf 15.9

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

   6. Simplified 15.9

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

   8. Applied add-cube-cbrt 16.0

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

   9. Applied associate-*l* 16.0

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

   10. Simplified 15.3

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

   12. Applied sub-neg 15.3

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

   13. Applied distribute-lft-in 15.3

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

   14. Simplified 15.6

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

   15. Simplified 15.0

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

   if -5.851479915422898e-184 < x < 1.0785483061682174e-171

   1. Initial program 17.8

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

   2. Simplified 17.8

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

   3. Taylor expanded around inf 16.7

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

   4. Simplified 17.7

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

   5. Taylor expanded around inf 16.7

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

   6. Simplified 16.7

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

   7. Taylor expanded around 0 15.7

      \[\leadsto \left(a \cdot \left(b \cdot i\right) - z \cdot \left(c \cdot b\right)\right) + \left(\color{blue}{0} + j \cdot \left(t \cdot c - i \cdot y\right)\right)\]

   if 1.0785483061682174e-171 < x < 3.351495691018442e+104

   1. Initial program 11.7

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

   2. Simplified 11.7

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

   3. Taylor expanded around inf 11.5

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

   4. Simplified 12.1

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

   6. Applied sub-neg 12.1

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

   7. Applied distribute-lft-in 12.1

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

   8. Simplified 12.1

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

   9. Simplified 11.5

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

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.125714986291682273805372429336846047599 \cdot 10^{-87}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + \left(i \cdot \left(b \cdot a\right) - z \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;x \le -5.851479915422898385518905209125160674093 \cdot 10^{-184}:\\ \;\;\;\;\left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \left(\left(\sqrt[3]{a} \cdot i\right) \cdot b\right) - z \cdot \left(b \cdot c\right)\right) + \left(\left(\left(y \cdot j\right) \cdot \left(-i\right) + \left(t \cdot j\right) \cdot c\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{elif}\;x \le 1.078548306168217375586414278153095298333 \cdot 10^{-171}:\\ \;\;\;\;\left(\left(i \cdot b\right) \cdot a - z \cdot \left(b \cdot c\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right)\\ \mathbf{elif}\;x \le 3.351495691018441958957028519658107878312 \cdot 10^{104}:\\ \;\;\;\;\left(\left(i \cdot b\right) \cdot a - b \cdot \left(c \cdot z\right)\right) + \left(\left(y \cdot \left(i \cdot \left(-j\right)\right) + t \cdot \left(j \cdot c\right)\right) + x \cdot \left(y \cdot z - t \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(y \cdot z - t \cdot a\right) + j \cdot \left(c \cdot t - i \cdot y\right)\right) + \left(i \cdot \left(b \cdot a\right) - z \cdot \left(b \cdot c\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))