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

Average Error: 3.8 → 1.2

Time: 3.5s

Precision: 64

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.68846225951938851 \cdot 10^{-59}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(9 \cdot t\right) \cdot y\right) \cdot z\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;z \le -7.36829316768083315 \cdot 10^{-306}:\\ \;\;\;\;\left(x \cdot 2 - \left(9 \cdot t\right) \cdot \left(z \cdot y\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;z \le 1.24771948413011858 \cdot 10^{-68}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;z \le 1.6726317168319845 \cdot 10^{257}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(9 \cdot t\right) \cdot y\right) \cdot z\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(9 \cdot t\right) \cdot \left(z \cdot y\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((z <= -2.6884622595193885e-59)) {
		VAR = (((x * 2.0) - (((9.0 * t) * y) * z)) + ((a * 27.0) * b));
	} else {
		double VAR_1;
		if ((z <= -7.368293167680833e-306)) {
			VAR_1 = (((x * 2.0) - ((9.0 * t) * (z * y))) + ((a * 27.0) * b));
		} else {
			double VAR_2;
			if ((z <= 1.2477194841301186e-68)) {
				VAR_2 = (((x * 2.0) - ((y * 9.0) * (z * t))) + (a * (27.0 * b)));
			} else {
				double VAR_3;
				if ((z <= 1.6726317168319845e+257)) {
					VAR_3 = (((x * 2.0) - (((9.0 * t) * y) * z)) + ((a * 27.0) * b));
				} else {
					VAR_3 = (((x * 2.0) - ((9.0 * t) * (z * y))) + ((a * 27.0) * b));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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

Target

Original	3.8
Target	2.7
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -2.6884622595193885e-59 or 1.2477194841301186e-68 < z < 1.6726317168319845e+257

   1. Initial program 5.9

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied *-commutative 5.9

      \[\leadsto \left(x \cdot 2 - \left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   4. Applied associate-*l* 5.9

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   5. Applied associate-*l* 5.9

      \[\leadsto \left(x \cdot 2 - \color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

   6. Simplified 5.9

      \[\leadsto \left(x \cdot 2 - 9 \cdot \color{blue}{\left(t \cdot \left(z \cdot y\right)\right)}\right) + \left(a \cdot 27\right) \cdot b\]
   7. Using strategy rm

   8. Applied associate-*r* 5.9

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(9 \cdot t\right) \cdot \left(z \cdot y\right)}\right) + \left(a \cdot 27\right) \cdot b\]
   9. Using strategy rm

   10. Applied *-commutative 5.9

       \[\leadsto \left(x \cdot 2 - \left(9 \cdot t\right) \cdot \color{blue}{\left(y \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b\]

   11. Applied associate-*r* 0.8

       \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(9 \cdot t\right) \cdot y\right) \cdot z}\right) + \left(a \cdot 27\right) \cdot b\]

   if -2.6884622595193885e-59 < z < -7.368293167680833e-306 or 1.6726317168319845e+257 < z

   1. Initial program 2.5

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied *-commutative 2.5

      \[\leadsto \left(x \cdot 2 - \left(\color{blue}{\left(9 \cdot y\right)} \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   4. Applied associate-*l* 2.4

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(9 \cdot \left(y \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   5. Applied associate-*l* 2.4

      \[\leadsto \left(x \cdot 2 - \color{blue}{9 \cdot \left(\left(y \cdot z\right) \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

   6. Simplified 2.4

      \[\leadsto \left(x \cdot 2 - 9 \cdot \color{blue}{\left(t \cdot \left(z \cdot y\right)\right)}\right) + \left(a \cdot 27\right) \cdot b\]
   7. Using strategy rm

   8. Applied associate-*r* 2.4

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(9 \cdot t\right) \cdot \left(z \cdot y\right)}\right) + \left(a \cdot 27\right) \cdot b\]

   if -7.368293167680833e-306 < z < 1.2477194841301186e-68

   1. Initial program 0.9

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied associate-*l* 0.8

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{a \cdot \left(27 \cdot b\right)}\]
   4. Using strategy rm

   5. Applied associate-*l* 0.7

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + a \cdot \left(27 \cdot b\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.68846225951938851 \cdot 10^{-59}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(9 \cdot t\right) \cdot y\right) \cdot z\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;z \le -7.36829316768083315 \cdot 10^{-306}:\\ \;\;\;\;\left(x \cdot 2 - \left(9 \cdot t\right) \cdot \left(z \cdot y\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;z \le 1.24771948413011858 \cdot 10^{-68}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{elif}\;z \le 1.6726317168319845 \cdot 10^{257}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(9 \cdot t\right) \cdot y\right) \cdot z\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(9 \cdot t\right) \cdot \left(z \cdot y\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))