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

Average Error: 3.6 → 1.2

Time: 6.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq -9.267786445046086 \cdot 10^{+306} \lor \neg \left(\left(y \cdot 9\right) \cdot z \leq 1080933396028.0312\right):\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \end{array}\]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))

↓

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* (* y 9.0) z) -9.267786445046086e+306)
         (not (<= (* (* y 9.0) z) 1080933396028.0312)))
   (+ (- (* x 2.0) (* (* y 9.0) (* z t))) (* (* a 27.0) b))
   (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b)))))

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 tmp;
	if ((((y * 9.0) * z) <= -9.267786445046086e+306) || !(((y * 9.0) * z) <= 1080933396028.0312)) {
		tmp = ((x * 2.0) - ((y * 9.0) * (z * t))) + ((a * 27.0) * b);
	} else {
		tmp = ((x * 2.0) - (((y * 9.0) * z) * t)) + (a * (27.0 * b));
	}
	return tmp;
}

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	3.6
Target	2.5
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y 9) z) < -9.2677864450460862e306 or 1080933396028.031 < (*.f64 (*.f64 y 9) z)

   1. Initial program 15.1

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

   3. Applied associate-*l*_binary64 4.1

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

   if -9.2677864450460862e306 < (*.f64 (*.f64 y 9) z) < 1080933396028.031

   1. Initial program 0.5

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

   3. Applied associate-*l*_binary64 0.4

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq -9.267786445046086 \cdot 10^{+306} \lor \neg \left(\left(y \cdot 9\right) \cdot z \leq 1080933396028.0312\right):\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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)))