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

Average Error: 5.6 → 2.2

Time: 25.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.36644240597627353 \cdot 10^{69}:\\ \;\;\;\;\left(\left(\left(\left(y \cdot 18\right) \cdot \left(\left(x \cdot z\right) \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;y \le 9.1086999467185389 \cdot 10^{-123}:\\ \;\;\;\;\left(\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(y \cdot 18\right) \cdot \left(z \cdot \left(x \cdot t\right)\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return ((double) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (x * 18.0)) * y)) * z)) * t)) - ((double) (((double) (a * 4.0)) * t)))) + ((double) (b * c)))) - ((double) (((double) (x * 4.0)) * i)))) - ((double) (((double) (j * 27.0)) * k))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double VAR;
	if ((y <= -4.3664424059762735e+69)) {
		VAR = ((double) (((double) (((double) (((double) (((double) (((double) (y * 18.0)) * ((double) (((double) (x * z)) * t)))) - ((double) (((double) (a * 1.0)) * ((double) (4.0 * t)))))) + ((double) (b * c)))) - ((double) (((double) (x * 4.0)) * i)))) - ((double) (j * ((double) (27.0 * k))))));
	} else {
		double VAR_1;
		if ((y <= 9.108699946718539e-123)) {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (18.0 * ((double) (t * ((double) (x * ((double) (z * y)))))))) - ((double) (((double) (a * 1.0)) * ((double) (4.0 * t)))))) + ((double) (b * c)))) - ((double) (((double) (x * 4.0)) * i)))) - ((double) (((double) (j * 27.0)) * k))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) (((double) (((double) (y * 18.0)) * ((double) (z * ((double) (x * t)))))) - ((double) (((double) (a * 1.0)) * ((double) (4.0 * t)))))) + ((double) (b * c)))) - ((double) (((double) (x * 4.0)) * i)))) - ((double) (((double) (j * 27.0)) * k))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 y < -4.3664424059762735e+69

   1. Initial program 12.9

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

   3. Applied *-commutative 12.9

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

   4. Applied associate-*l* 8.0

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

   5. Applied associate-*l* 2.3

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

   7. Applied *-un-lft-identity 2.3

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \left(a \cdot \color{blue}{\left(1 \cdot 4\right)}\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   8. Applied associate-*r* 2.3

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \color{blue}{\left(\left(a \cdot 1\right) \cdot 4\right)} \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   9. Applied associate-*l* 2.3

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \color{blue}{\left(a \cdot 1\right) \cdot \left(4 \cdot t\right)}\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   10. Using strategy rm

   11. Applied *-commutative 2.3

       \[\leadsto \left(\left(\left(y \cdot \left(\left(\color{blue}{\left(18 \cdot x\right)} \cdot z\right) \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   12. Applied associate-*l* 2.3

       \[\leadsto \left(\left(\left(y \cdot \left(\color{blue}{\left(18 \cdot \left(x \cdot z\right)\right)} \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   13. Applied associate-*l* 2.2

       \[\leadsto \left(\left(\left(y \cdot \color{blue}{\left(18 \cdot \left(\left(x \cdot z\right) \cdot t\right)\right)} - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   14. Applied associate-*r* 2.5

       \[\leadsto \left(\left(\left(\color{blue}{\left(y \cdot 18\right) \cdot \left(\left(x \cdot z\right) \cdot t\right)} - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   15. Using strategy rm

   16. Applied associate-*l* 2.3

       \[\leadsto \left(\left(\left(\left(y \cdot 18\right) \cdot \left(\left(x \cdot z\right) \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{j \cdot \left(27 \cdot k\right)}\]

   if -4.3664424059762735e+69 < y < 9.108699946718539e-123

   1. Initial program 1.7

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

   3. Applied *-commutative 1.7

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

   4. Applied associate-*l* 4.8

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

   5. Applied associate-*l* 7.1

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

   7. Applied *-un-lft-identity 7.1

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \left(a \cdot \color{blue}{\left(1 \cdot 4\right)}\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   8. Applied associate-*r* 7.1

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \color{blue}{\left(\left(a \cdot 1\right) \cdot 4\right)} \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   9. Applied associate-*l* 7.1

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \color{blue}{\left(a \cdot 1\right) \cdot \left(4 \cdot t\right)}\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   10. Using strategy rm

   11. Applied *-commutative 7.1

       \[\leadsto \left(\left(\left(y \cdot \left(\left(\color{blue}{\left(18 \cdot x\right)} \cdot z\right) \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   12. Applied associate-*l* 7.1

       \[\leadsto \left(\left(\left(y \cdot \left(\color{blue}{\left(18 \cdot \left(x \cdot z\right)\right)} \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   13. Applied associate-*l* 7.1

       \[\leadsto \left(\left(\left(y \cdot \color{blue}{\left(18 \cdot \left(\left(x \cdot z\right) \cdot t\right)\right)} - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   14. Applied associate-*r* 7.0

       \[\leadsto \left(\left(\left(\color{blue}{\left(y \cdot 18\right) \cdot \left(\left(x \cdot z\right) \cdot t\right)} - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   15. Taylor expanded around inf 1.5

       \[\leadsto \left(\left(\left(\color{blue}{18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)} - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   if 9.108699946718539e-123 < y

   1. Initial program 8.0

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

   3. Applied *-commutative 8.0

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

   4. Applied associate-*l* 5.8

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

   5. Applied associate-*l* 3.3

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

   7. Applied *-un-lft-identity 3.3

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \left(a \cdot \color{blue}{\left(1 \cdot 4\right)}\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   8. Applied associate-*r* 3.3

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \color{blue}{\left(\left(a \cdot 1\right) \cdot 4\right)} \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   9. Applied associate-*l* 3.4

      \[\leadsto \left(\left(\left(y \cdot \left(\left(\left(x \cdot 18\right) \cdot z\right) \cdot t\right) - \color{blue}{\left(a \cdot 1\right) \cdot \left(4 \cdot t\right)}\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   10. Using strategy rm

   11. Applied *-commutative 3.4

       \[\leadsto \left(\left(\left(y \cdot \left(\left(\color{blue}{\left(18 \cdot x\right)} \cdot z\right) \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   12. Applied associate-*l* 3.4

       \[\leadsto \left(\left(\left(y \cdot \left(\color{blue}{\left(18 \cdot \left(x \cdot z\right)\right)} \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   13. Applied associate-*l* 3.3

       \[\leadsto \left(\left(\left(y \cdot \color{blue}{\left(18 \cdot \left(\left(x \cdot z\right) \cdot t\right)\right)} - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   14. Applied associate-*r* 3.4

       \[\leadsto \left(\left(\left(\color{blue}{\left(y \cdot 18\right) \cdot \left(\left(x \cdot z\right) \cdot t\right)} - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
   15. Using strategy rm

   16. Applied *-commutative 3.4

       \[\leadsto \left(\left(\left(\left(y \cdot 18\right) \cdot \left(\color{blue}{\left(z \cdot x\right)} \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   17. Applied associate-*l* 3.3

       \[\leadsto \left(\left(\left(\left(y \cdot 18\right) \cdot \color{blue}{\left(z \cdot \left(x \cdot t\right)\right)} - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.36644240597627353 \cdot 10^{69}:\\ \;\;\;\;\left(\left(\left(\left(y \cdot 18\right) \cdot \left(\left(x \cdot z\right) \cdot t\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;y \le 9.1086999467185389 \cdot 10^{-123}:\\ \;\;\;\;\left(\left(\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(y \cdot 18\right) \cdot \left(z \cdot \left(x \cdot t\right)\right) - \left(a \cdot 1\right) \cdot \left(4 \cdot t\right)\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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)))