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

Average Error: 3.5 → 0.7

Time: 11.6s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -0.0014544364767717878 \lor \neg \left(y \cdot 9 \le 8.3602843622211546 \cdot 10^{-101}\right):\\ \;\;\;\;\left(2 \cdot x + 27 \cdot \left(a \cdot b\right)\right) - \left(\left(9 \cdot t\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x + 27 \cdot \left(a \cdot b\right)\right) - \left(9 \cdot t\right) \cdot \left(z \cdot y\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((((double) (y * 9.0)) <= -0.0014544364767717878) || !(((double) (y * 9.0)) <= 8.360284362221155e-101))) {
		VAR = ((double) (((double) (((double) (2.0 * x)) + ((double) (27.0 * ((double) (a * b)))))) - ((double) (((double) (((double) (9.0 * t)) * z)) * y))));
	} else {
		VAR = ((double) (((double) (((double) (2.0 * x)) + ((double) (27.0 * ((double) (a * b)))))) - ((double) (((double) (9.0 * t)) * ((double) (z * y))))));
	}
	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	3.5
Target	2.6
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (* y 9.0) < -0.0014544364767717878 or 8.3602843622211546e-101 < (* y 9.0)

   1. Initial program 6.3

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

   2. Taylor expanded around inf 6.1

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

   4. Applied associate-*r* 1.0

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

   6. Applied associate-*r* 1.0

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

   8. Applied associate-*r* 1.0

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

   if -0.0014544364767717878 < (* y 9.0) < 8.3602843622211546e-101

   1. Initial program 0.4

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

   2. Taylor expanded around inf 0.4

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

   4. Applied associate-*r* 0.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -0.0014544364767717878 \lor \neg \left(y \cdot 9 \le 8.3602843622211546 \cdot 10^{-101}\right):\\ \;\;\;\;\left(2 \cdot x + 27 \cdot \left(a \cdot b\right)\right) - \left(\left(9 \cdot t\right) \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x + 27 \cdot \left(a \cdot b\right)\right) - \left(9 \cdot t\right) \cdot \left(z \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))