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

Average Error: 5.5 → 2.0

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.3949608434743329 \cdot 10^{62} \lor \neg \left(t \le 1.5985230459207121 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(\left(x \cdot 18\right) \cdot z\right)\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot \left(z \cdot \left(\left(x \cdot 18\right) \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot k\right) \cdot j\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return ((((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((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 (((t <= -1.394960843474333e+62) || !(t <= 1.598523045920712e-19))) {
		VAR = ((((((y * ((x * 18.0) * z)) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - (27.0 * (j * k)));
	} else {
		VAR = (((((y * (z * ((x * 18.0) * t))) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((27.0 * k) * j));
	}
	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 2 regimes

2. if t < -1.394960843474333e+62 or 1.598523045920712e-19 < t

   1. Initial program 1.9

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

   3. Applied *-commutative 1.9

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

   4. Applied associate-*l* 1.8

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

   5. Applied associate-*l* 6.3

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

   7. Applied *-commutative 6.3

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

   8. Applied associate-*l* 6.3

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

   10. Applied associate-*r* 1.7

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

   if -1.394960843474333e+62 < t < 1.598523045920712e-19

   1. Initial program 7.3

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

   3. Applied *-commutative 7.3

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

   4. Applied associate-*l* 7.3

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

   5. Applied associate-*l* 4.1

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

   7. Applied *-commutative 4.1

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

   8. Applied associate-*l* 4.0

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

   10. Applied *-commutative 4.0

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

   11. Applied associate-*l* 2.0

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

   13. Applied *-commutative 2.0

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

   14. Applied associate-*r* 2.1

       \[\leadsto \left(\left(\left(y \cdot \left(z \cdot \left(\left(x \cdot 18\right) \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{\left(27 \cdot k\right) \cdot j}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.3949608434743329 \cdot 10^{62} \lor \neg \left(t \le 1.5985230459207121 \cdot 10^{-19}\right):\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(\left(x \cdot 18\right) \cdot z\right)\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot \left(z \cdot \left(\left(x \cdot 18\right) \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot k\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(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)))