Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1

Average Error: 5.3 → 6.0

Time: 13.8s

Precision: 64

Math

\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

↓

\[t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r91670 = x;
        double r91671 = 18.0;
        double r91672 = r91670 * r91671;
        double r91673 = y;
        double r91674 = r91672 * r91673;
        double r91675 = z;
        double r91676 = r91674 * r91675;
        double r91677 = t;
        double r91678 = r91676 * r91677;
        double r91679 = a;
        double r91680 = 4.0;
        double r91681 = r91679 * r91680;
        double r91682 = r91681 * r91677;
        double r91683 = r91678 - r91682;
        double r91684 = b;
        double r91685 = c;
        double r91686 = r91684 * r91685;
        double r91687 = r91683 + r91686;
        double r91688 = r91670 * r91680;
        double r91689 = i;
        double r91690 = r91688 * r91689;
        double r91691 = r91687 - r91690;
        double r91692 = j;
        double r91693 = 27.0;
        double r91694 = r91692 * r91693;
        double r91695 = k;
        double r91696 = r91694 * r91695;
        double r91697 = r91691 - r91696;
        return r91697;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r91698 = t;
        double r91699 = x;
        double r91700 = 18.0;
        double r91701 = r91699 * r91700;
        double r91702 = y;
        double r91703 = z;
        double r91704 = r91702 * r91703;
        double r91705 = r91701 * r91704;
        double r91706 = a;
        double r91707 = 4.0;
        double r91708 = r91706 * r91707;
        double r91709 = r91705 - r91708;
        double r91710 = r91698 * r91709;
        double r91711 = b;
        double r91712 = c;
        double r91713 = r91711 * r91712;
        double r91714 = r91699 * r91707;
        double r91715 = i;
        double r91716 = r91714 * r91715;
        double r91717 = j;
        double r91718 = 27.0;
        double r91719 = k;
        double r91720 = r91718 * r91719;
        double r91721 = r91717 * r91720;
        double r91722 = r91716 + r91721;
        double r91723 = r91713 - r91722;
        double r91724 = r91710 + r91723;
        return r91724;
}

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

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if x < -1.9520536695656065e-73

   1. Initial program 9.4

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   2. Using strategy rm

   3. Applied associate-*l* 9.3

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{j \cdot \left(27 \cdot k\right)}\]
   4. Using strategy rm

   5. Applied associate-*l* 9.3

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \color{blue}{a \cdot \left(4 \cdot t\right)}\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]
   6. Using strategy rm

   7. Applied associate-*l* 6.3

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right)} \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   if -1.9520536695656065e-73 < x < 53501025862446216.0

   1. Initial program 1.4

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   2. Using strategy rm

   3. Applied associate-*l* 1.4

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{j \cdot \left(27 \cdot k\right)}\]
   4. Using strategy rm

   5. Applied associate-*l* 1.3

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \color{blue}{a \cdot \left(4 \cdot t\right)}\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]
   6. Using strategy rm

   7. Applied associate-*l* 1.5

      \[\leadsto \left(\left(\left(\left(\color{blue}{\left(x \cdot \left(18 \cdot y\right)\right)} \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 1.5

      \[\leadsto \left(\left(\left(\color{blue}{\left(\sqrt[3]{\left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z\right) \cdot t} \cdot \sqrt[3]{\left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z\right) \cdot t}\right) \cdot \sqrt[3]{\left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z\right) \cdot t}} - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   if 53501025862446216.0 < x

   1. Initial program 12.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   2. Using strategy rm

   3. Applied associate-*l* 12.4

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{j \cdot \left(27 \cdot k\right)}\]
   4. Using strategy rm

   5. Applied associate-*l* 12.4

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \color{blue}{a \cdot \left(4 \cdot t\right)}\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]
   6. Using strategy rm

   7. Applied associate-*l* 12.3

      \[\leadsto \left(\left(\left(\left(\color{blue}{\left(x \cdot \left(18 \cdot y\right)\right)} \cdot z\right) \cdot t - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]
   8. Using strategy rm

   9. Applied associate-*l* 8.8

      \[\leadsto \left(\left(\left(\color{blue}{\left(x \cdot \left(18 \cdot y\right)\right) \cdot \left(z \cdot t\right)} - a \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.0

   \[\leadsto t \cdot \left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))