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

Average Error: 5.5 → 2.4

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.2535829321441683 \cdot 10^{79}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;z \le 7.07036476202182496 \cdot 10^{-23}:\\ \;\;\;\;\left(\left(\left(y \cdot \left(18 \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(\left(x \cdot 18\right) \cdot t\right)\right) \cdot z - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(j \cdot k\right)\\ \end{array}\]

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 ((z <= -1.2535829321441683e+79)) {
		VAR = fma(t, ((((x * 18.0) * y) * z) - (a * 4.0)), ((b * c) - fma(x, (4.0 * i), ((j * 27.0) * k))));
	} else {
		double VAR_1;
		if ((z <= 7.070364762021825e-23)) {
			VAR_1 = (((((y * (18.0 * (t * (x * z)))) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - (27.0 * (j * k)));
		} else {
			VAR_1 = ((((((y * ((x * 18.0) * t)) * z) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - (27.0 * (j * 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

Target

Original	5.5
Target	1.7
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -1.2535829321441683e+79

   1. Initial program 7.2

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

   2. Simplified 7.2

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

   if -1.2535829321441683e+79 < z < 7.070364762021825e-23

   1. Initial program 4.7

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

   3. Applied *-commutative 4.7

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

   4. Applied associate-*l* 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* 1.4

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

   7. Applied *-commutative 1.4

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

   8. Applied associate-*l* 1.3

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

   10. Applied *-commutative 1.3

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

   11. Applied associate-*l* 3.3

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

   12. Taylor expanded around inf 1.3

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

   if 7.070364762021825e-23 < z

   1. Initial program 6.5

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

   3. Applied *-commutative 6.5

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

   4. Applied associate-*l* 9.8

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

   5. Applied associate-*l* 8.9

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

   7. Applied *-commutative 8.9

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

   8. Applied associate-*l* 8.9

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

   10. Applied *-commutative 8.9

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

   11. Applied associate-*l* 6.2

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

   13. Applied *-commutative 6.2

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

   14. Applied associate-*r* 2.2

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2535829321441683 \cdot 10^{79}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;z \le 7.07036476202182496 \cdot 10^{-23}:\\ \;\;\;\;\left(\left(\left(y \cdot \left(18 \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(y \cdot \left(\left(x \cdot 18\right) \cdot t\right)\right) \cdot z - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(j \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))