Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.2 → 0.1

Time: 2.8s

Precision: 64

Math

\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\]

↓

\[\mathsf{fma}\left(x \cdot 1, x \cdot 3, -2 \cdot {x}^{3}\right)\]

C

double code(double x) {
	return ((double) (((double) (x * x)) * ((double) (3.0 - ((double) (x * 2.0))))));
}

↓

double code(double x) {
	return ((double) fma(((double) (x * 1.0)), ((double) (x * 3.0)), ((double) -(((double) (2.0 * ((double) pow(x, 3.0))))))));
}

Error

Bits error versus x

Target

Original	0.2
Target	0.2
Herbie	0.1

\[x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right)\]

Derivation

1. Initial program 0.2

   \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)\]
2. Using strategy rm

3. Applied sub-neg 0.2

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(3 + \left(-x \cdot 2\right)\right)}\]

4. Applied distribute-lft-in 0.2

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 3 + \left(x \cdot x\right) \cdot \left(-x \cdot 2\right)}\]

5. Simplified 0.1

   \[\leadsto \left(x \cdot x\right) \cdot 3 + \color{blue}{\left(-2 \cdot {x}^{3}\right)}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.1

   \[\leadsto \left(x \cdot \color{blue}{\left(1 \cdot x\right)}\right) \cdot 3 + \left(-2 \cdot {x}^{3}\right)\]

8. Applied associate-*r* 0.1

   \[\leadsto \color{blue}{\left(\left(x \cdot 1\right) \cdot x\right)} \cdot 3 + \left(-2 \cdot {x}^{3}\right)\]

9. Applied associate-*l* 0.1

   \[\leadsto \color{blue}{\left(x \cdot 1\right) \cdot \left(x \cdot 3\right)} + \left(-2 \cdot {x}^{3}\right)\]

10. Applied fma-def 0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 1, x \cdot 3, -2 \cdot {x}^{3}\right)}\]

11. Final simplification 0.1

    \[\leadsto \mathsf{fma}\left(x \cdot 1, x \cdot 3, -2 \cdot {x}^{3}\right)\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3 (* x 2))))

  (* (* x x) (- 3 (* x 2))))