Average Error: 0.1 → 0.1
Time: 3.9s
Precision: 64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y\]
↓
\[\left(x \cdot x + \mathsf{fma}\left(y, y, y \cdot y\right)\right) + y \cdot y\]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\left(x \cdot x + \mathsf{fma}\left(y, y, y \cdot y\right)\right) + y \cdot y
double code(double x, double y) {
	return ((double) (((double) (((double) (((double) (x * x)) + ((double) (y * y)))) + ((double) (y * y)))) + ((double) (y * y))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) (x * x)) + ((double) fma(y, y, ((double) (y * y)))))) + ((double) (y * y))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied associate-+l+0.1

    \[\leadsto \color{blue}{\left(x \cdot x + \left(y \cdot y + y \cdot y\right)\right)} + y \cdot y\]

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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)))