Average Error: 0.1 → 0.1
Time: 1.4s
Precision: binary64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(y, y \cdot 3, x \cdot x\right) \]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\mathsf{fma}\left(y, y \cdot 3, x \cdot x\right)

(FPCore (x y) :precision binary64 (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))
↓
(FPCore (x y) :precision binary64 (fma y (* y 3.0) (* x x)))
double code(double x, double y) {
	return (((x * x) + (y * y)) + (y * y)) + (y * y);
}
↓
double code(double x, double y) {
	return fma(y, (y * 3.0), (x * x));
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.1
Herbie0.1
\[x \cdot x + y \cdot \left(y + \left(y + y\right)\right) \]

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2021275 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :herbie-target
  (+ (* x x) (* y (+ y (+ y y))))

  (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))