Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2

Average Error: 0.2 → 0.1

Time: 2.0s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))

↓

(FPCore (x) :precision binary64 (fma 3.0 (pow x 2.0) (* -2.0 (pow x 3.0))))

C

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

↓

double code(double x) {
	return fma(3.0, pow(x, 2.0), (-2.0 * 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. Simplified 0.2

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

3. Taylor expanded in x around 0 0.1

   \[\leadsto \color{blue}{3 \cdot {x}^{2} - 2 \cdot {x}^{3}} \]

4. Applied fma-neg_binary64 0.1

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

5. Simplified 0.1

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

6. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3.0 (* x 2.0))))

  (* (* x x) (- 3.0 (* x 2.0))))