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

Average Error: 3.7 → 0.8

Time: 4.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z = -\infty:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot t\right) \cdot \left(9 \cdot z\right)\right)\\ \mathbf{elif}\;\left(y \cdot 9\right) \cdot z \le 3.5007333173844521 \cdot 10^{56}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 + \left(-\left(\left(z \cdot t\right) \cdot 9\right) \cdot y\right)\right)\\ \end{array}\]

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 VAR;
	if ((((y * 9.0) * z) <= -inf.0)) {
		VAR = fma(a, (27.0 * b), ((x * 2.0) - ((y * t) * (9.0 * z))));
	} else {
		double VAR_1;
		if ((((y * 9.0) * z) <= 3.500733317384452e+56)) {
			VAR_1 = fma(x, 2.0, ((27.0 * (a * b)) - (9.0 * (t * (z * y)))));
		} else {
			VAR_1 = fma(a, (27.0 * b), ((x * 2.0) + -(((z * t) * 9.0) * y)));
		}
		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	3.7
Target	2.5
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) z) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied associate-*l* 63.0

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

   5. Applied associate-*l* 0.7

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

   7. Applied *-commutative 0.7

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

   8. Applied associate-*r* 0.6

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

   if -inf.0 < (* (* y 9.0) z) < 3.500733317384452e+56

   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. Simplified 0.5

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

   4. Applied sub-neg 0.5

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

   5. Simplified 3.5

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

   6. Taylor expanded around inf 0.4

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

   7. Simplified 0.4

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

   if 3.500733317384452e+56 < (* (* y 9.0) z)

   1. Initial program 11.7

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

   2. Simplified 11.8

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

   4. Applied sub-neg 11.8

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

   5. Simplified 3.2

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

   7. Applied *-commutative 3.2

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

   8. Applied associate-*r* 3.1

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z = -\infty:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot t\right) \cdot \left(9 \cdot z\right)\right)\\ \mathbf{elif}\;\left(y \cdot 9\right) \cdot z \le 3.5007333173844521 \cdot 10^{56}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 + \left(-\left(\left(z \cdot t\right) \cdot 9\right) \cdot y\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +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)))