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

Average Error: 3.5 → 0.8

Time: 4.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -1778721528140901650000:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 260815090086.4133:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, \left(2 \cdot x - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right) + \left(\left(y \cdot 9\right) \cdot z\right) \cdot \left(\left(-t\right) + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right)\\ \end{array}\]

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 temp;
	if (((y * 9.0) <= -1.7787215281409016e+21)) {
		temp = fma(a, (27.0 * b), ((x * 2.0) - ((y * 9.0) * (z * t))));
	} else {
		double temp_1;
		if (((y * 9.0) <= 260815090086.4133)) {
			temp_1 = fma(a, (27.0 * b), (((2.0 * x) - (9.0 * (t * (z * y)))) + (((y * 9.0) * z) * (-t + t))));
		} else {
			temp_1 = fma(a, (27.0 * b), ((x * 2.0) - (y * (9.0 * (z * t)))));
		}
		temp = temp_1;
	}
	return temp;
}

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	0.8

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

Derivation

1. Split input into 3 regimes

2. if (* y 9.0) < -1.7787215281409016e+21

   1. Initial program 8.3

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

   2. Simplified 8.2

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

   4. Applied associate-*l* 1.0

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

   if -1.7787215281409016e+21 < (* y 9.0) < 260815090086.4133

   1. Initial program 0.7

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

   2. Simplified 0.6

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

   4. Applied prod-diff 0.6

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

   5. Simplified 0.6

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

   6. Simplified 0.6

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

   if 260815090086.4133 < (* y 9.0)

   1. Initial program 7.7

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

   2. Simplified 7.8

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

   4. Applied associate-*l* 1.1

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

   6. Applied associate-*l* 1.0

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -1778721528140901650000:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;y \cdot 9 \le 260815090086.4133:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, \left(2 \cdot x - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right) + \left(\left(y \cdot 9\right) \cdot z\right) \cdot \left(\left(-t\right) + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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)))