Average Error: 0.2 → 0.1
Time: 2.2s
Precision: binary64
\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
↓
\[\mathsf{fma}\left(3, x \cdot x, {x}^{3} \cdot -2\right) \]
\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)
↓
\mathsf{fma}\left(3, x \cdot x, {x}^{3} \cdot -2\right)

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))
↓
(FPCore (x) :precision binary64 (fma 3.0 (* x x) (* (pow x 3.0) -2.0)))
double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}
↓
double code(double x) {
	return fma(3.0, (x * x), (pow(x, 3.0) * -2.0));
}

Error

Bits error versus x

Target

Original0.2
Target0.2
Herbie0.1
\[x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded in x around 0 0.1

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022130 
(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))))