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

Average Error: 3.5 → 0.7

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.5396561823137964 \cdot 10^{-72}:\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 3.2983755837402388 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\right) + b \cdot \left(a \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

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)) <= -3.5396561823137964e-72)) {
		VAR = ((double) (((double) (x * 2.0)) + ((double) (((double) (a * ((double) (27.0 * b)))) - ((double) (y * ((double) (((double) (9.0 * z)) * t))))))));
	} else {
		double VAR_1;
		if ((((double) (y * 9.0)) <= 3.298375583740239e-49)) {
			VAR_1 = ((double) (((double) (((double) (x * 2.0)) - ((double) (t * ((double) (((double) (y * 9.0)) * z)))))) + ((double) (b * ((double) (a * 27.0))))));
		} else {
			VAR_1 = ((double) (((double) (x * 2.0)) + ((double) (((double) (a * ((double) (27.0 * b)))) - ((double) (((double) (y * 9.0)) * ((double) (z * t))))))));
		}
		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	3.5
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 3 regimes

2. if (* y 9.0) < -3.5396561823137964e-72

   1. Initial program 6.4

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

   2. Simplified 1.0

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

   4. Applied associate-*r* 1.1

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

   if -3.5396561823137964e-72 < (* y 9.0) < 3.2983755837402388e-49

   1. Initial program 0.5

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

   if 3.2983755837402388e-49 < (* y 9.0)

   1. Initial program 5.7

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

   2. Simplified 0.7

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

   4. Applied associate-*r* 0.7

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.5396561823137964 \cdot 10^{-72}:\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 3.2983755837402388 \cdot 10^{-49}:\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(\left(y \cdot 9\right) \cdot z\right)\right) + b \cdot \left(a \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))