Linear.Matrix:det33 from linear-1.19.1.3

Average Error: 11.8 → 10.3

Time: 1.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le -1453916164036345344:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot x\right) - \left(i \cdot \left(b \cdot \left(-a\right)\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \mathbf{elif}\;i \le -3.93386905412778367989248227678079934917 \cdot 10^{-115}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{elif}\;i \le -9.704488884191774685973946287074410145642 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b \cdot \left(i \cdot a\right)\right) + c \cdot \left(b \cdot z\right)\right)\right) + \sqrt[3]{j \cdot \left(c \cdot t - y \cdot i\right)} \cdot \left(\sqrt[3]{j \cdot \left(c \cdot t - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - y \cdot i\right)}\right)\\ \mathbf{elif}\;i \le 1.392000762622429387159763147815204755346 \cdot 10^{-21}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}} \cdot \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}}\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot x\right) - \left(i \cdot \left(b \cdot \left(-a\right)\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r4329620 = x;
        double r4329621 = y;
        double r4329622 = z;
        double r4329623 = r4329621 * r4329622;
        double r4329624 = t;
        double r4329625 = a;
        double r4329626 = r4329624 * r4329625;
        double r4329627 = r4329623 - r4329626;
        double r4329628 = r4329620 * r4329627;
        double r4329629 = b;
        double r4329630 = c;
        double r4329631 = r4329630 * r4329622;
        double r4329632 = i;
        double r4329633 = r4329632 * r4329625;
        double r4329634 = r4329631 - r4329633;
        double r4329635 = r4329629 * r4329634;
        double r4329636 = r4329628 - r4329635;
        double r4329637 = j;
        double r4329638 = r4329630 * r4329624;
        double r4329639 = r4329632 * r4329621;
        double r4329640 = r4329638 - r4329639;
        double r4329641 = r4329637 * r4329640;
        double r4329642 = r4329636 + r4329641;
        return r4329642;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
        double r4329643 = i;
        double r4329644 = -1.4539161640363453e+18;
        bool r4329645 = r4329643 <= r4329644;
        double r4329646 = j;
        double r4329647 = c;
        double r4329648 = t;
        double r4329649 = r4329647 * r4329648;
        double r4329650 = y;
        double r4329651 = r4329650 * r4329643;
        double r4329652 = r4329649 - r4329651;
        double r4329653 = r4329646 * r4329652;
        double r4329654 = z;
        double r4329655 = r4329650 * r4329654;
        double r4329656 = a;
        double r4329657 = r4329648 * r4329656;
        double r4329658 = r4329655 - r4329657;
        double r4329659 = cbrt(r4329658);
        double r4329660 = r4329659 * r4329659;
        double r4329661 = x;
        double r4329662 = r4329660 * r4329661;
        double r4329663 = r4329659 * r4329662;
        double r4329664 = b;
        double r4329665 = -r4329656;
        double r4329666 = r4329664 * r4329665;
        double r4329667 = r4329643 * r4329666;
        double r4329668 = r4329664 * r4329654;
        double r4329669 = r4329647 * r4329668;
        double r4329670 = r4329667 + r4329669;
        double r4329671 = r4329663 - r4329670;
        double r4329672 = r4329653 + r4329671;
        double r4329673 = -3.9338690541277837e-115;
        bool r4329674 = r4329643 <= r4329673;
        double r4329675 = r4329661 * r4329655;
        double r4329676 = r4329661 * r4329648;
        double r4329677 = r4329676 * r4329656;
        double r4329678 = r4329675 - r4329677;
        double r4329679 = cbrt(r4329664);
        double r4329680 = r4329679 * r4329679;
        double r4329681 = r4329647 * r4329654;
        double r4329682 = r4329643 * r4329656;
        double r4329683 = r4329681 - r4329682;
        double r4329684 = r4329683 * r4329679;
        double r4329685 = r4329680 * r4329684;
        double r4329686 = r4329678 - r4329685;
        double r4329687 = r4329686 + r4329653;
        double r4329688 = -9.704488884191775e-196;
        bool r4329689 = r4329643 <= r4329688;
        double r4329690 = r4329658 * r4329661;
        double r4329691 = r4329664 * r4329682;
        double r4329692 = -r4329691;
        double r4329693 = r4329692 + r4329669;
        double r4329694 = r4329690 - r4329693;
        double r4329695 = cbrt(r4329653);
        double r4329696 = r4329695 * r4329695;
        double r4329697 = r4329695 * r4329696;
        double r4329698 = r4329694 + r4329697;
        double r4329699 = 1.3920007626224294e-21;
        bool r4329700 = r4329643 <= r4329699;
        double r4329701 = cbrt(r4329684);
        double r4329702 = r4329701 * r4329701;
        double r4329703 = r4329701 * r4329702;
        double r4329704 = r4329703 * r4329680;
        double r4329705 = r4329690 - r4329704;
        double r4329706 = r4329653 + r4329705;
        double r4329707 = r4329700 ? r4329706 : r4329672;
        double r4329708 = r4329689 ? r4329698 : r4329707;
        double r4329709 = r4329674 ? r4329687 : r4329708;
        double r4329710 = r4329645 ? r4329672 : r4329709;
        return r4329710;
}

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 i < -1.4539161640363453e+18 or 1.3920007626224294e-21 < i

   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. Using strategy rm

   3. Applied add-cube-cbrt 16.1

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

   4. Applied associate-*l* 16.1

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

   6. Applied sub-neg 16.1

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

   7. Applied distribute-lft-in 16.1

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

   8. Applied distribute-lft-in 16.1

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

   9. Simplified 16.0

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

   10. Simplified 15.8

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

   11. Taylor expanded around inf 15.9

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

   12. Simplified 11.9

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

   14. Applied add-cube-cbrt 12.1

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

   15. Applied associate-*r* 12.1

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

   if -1.4539161640363453e+18 < i < -3.9338690541277837e-115

   1. Initial program 9.0

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

   3. Applied add-cube-cbrt 9.3

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

   4. Applied associate-*l* 9.3

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

   5. Taylor expanded around inf 8.3

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

   if -3.9338690541277837e-115 < i < -9.704488884191775e-196

   1. Initial program 9.1

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

   3. Applied add-cube-cbrt 9.4

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

   4. Applied associate-*l* 9.3

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

   6. Applied sub-neg 9.3

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

   7. Applied distribute-lft-in 9.3

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

   8. Applied distribute-lft-in 9.3

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

   9. Simplified 8.5

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

   10. Simplified 8.4

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

   12. Applied add-cube-cbrt 8.7

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

   if -9.704488884191775e-196 < i < 1.3920007626224294e-21

   1. Initial program 9.3

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

   3. Applied add-cube-cbrt 9.6

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

   4. Applied associate-*l* 9.6

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

   6. Applied add-cube-cbrt 9.6

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le -1453916164036345344:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot x\right) - \left(i \cdot \left(b \cdot \left(-a\right)\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \mathbf{elif}\;i \le -3.93386905412778367989248227678079934917 \cdot 10^{-115}:\\ \;\;\;\;\left(\left(x \cdot \left(y \cdot z\right) - \left(x \cdot t\right) \cdot a\right) - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \left(\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}\right)\right) + j \cdot \left(c \cdot t - y \cdot i\right)\\ \mathbf{elif}\;i \le -9.704488884191774685973946287074410145642 \cdot 10^{-196}:\\ \;\;\;\;\left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\left(-b \cdot \left(i \cdot a\right)\right) + c \cdot \left(b \cdot z\right)\right)\right) + \sqrt[3]{j \cdot \left(c \cdot t - y \cdot i\right)} \cdot \left(\sqrt[3]{j \cdot \left(c \cdot t - y \cdot i\right)} \cdot \sqrt[3]{j \cdot \left(c \cdot t - y \cdot i\right)}\right)\\ \mathbf{elif}\;i \le 1.392000762622429387159763147815204755346 \cdot 10^{-21}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\left(y \cdot z - t \cdot a\right) \cdot x - \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}} \cdot \left(\sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}} \cdot \sqrt[3]{\left(c \cdot z - i \cdot a\right) \cdot \sqrt[3]{b}}\right)\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;j \cdot \left(c \cdot t - y \cdot i\right) + \left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \left(\left(\sqrt[3]{y \cdot z - t \cdot a} \cdot \sqrt[3]{y \cdot z - t \cdot a}\right) \cdot x\right) - \left(i \cdot \left(b \cdot \left(-a\right)\right) + c \cdot \left(b \cdot z\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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)))))