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

Average Error: 3.5 → 0.7

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -9.92985419778116282 \cdot 10^{-14} \lor \neg \left(t \le 891587068291242500\right):\\ \;\;\;\;\mathsf{fma}\left(x, 2, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

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 (((t <= -9.929854197781163e-14) || !(t <= 8.915870682912425e+17))) {
		temp = fma(x, 2.0, ((27.0 * (a * b)) - (9.0 * (t * (z * y)))));
	} else {
		temp = (((x * 2.0) - ((y * 9.0) * (z * t))) + ((a * 27.0) * b));
	}
	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	2.5
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 t < -9.929854197781163e-14 or 8.915870682912425e+17 < t

   1. Initial program 0.6

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

   3. Applied pow1 0.6

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot \color{blue}{{b}^{1}}\]

   4. Applied pow1 0.6

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot \color{blue}{{27}^{1}}\right) \cdot {b}^{1}\]

   5. Applied pow1 0.6

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(\color{blue}{{a}^{1}} \cdot {27}^{1}\right) \cdot {b}^{1}\]

   6. Applied pow-prod-down 0.6

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{{\left(a \cdot 27\right)}^{1}} \cdot {b}^{1}\]

   7. Applied pow-prod-down 0.6

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \color{blue}{{\left(\left(a \cdot 27\right) \cdot b\right)}^{1}}\]

   8. Simplified 0.6

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + {\color{blue}{\left(27 \cdot \left(a \cdot b\right)\right)}}^{1}\]
   9. Using strategy rm

   10. Applied associate-+l- 0.6

       \[\leadsto \color{blue}{x \cdot 2 - \left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot t - {\left(27 \cdot \left(a \cdot b\right)\right)}^{1}\right)}\]

   11. Simplified 7.6

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

   13. Applied fma-neg 7.6

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

   14. Simplified 0.6

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

   if -9.929854197781163e-14 < t < 8.915870682912425e+17

   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. Using strategy rm

   3. Applied associate-*l* 0.7

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.92985419778116282 \cdot 10^{-14} \lor \neg \left(t \le 891587068291242500\right):\\ \;\;\;\;\mathsf{fma}\left(x, 2, 27 \cdot \left(a \cdot b\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))