Average Error: 0.0 → 0.0
Time: 1.1s
Precision: binary64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
↓
\[x \cdot 2 + \mathsf{fma}\left(x, x, y \cdot y\right) \]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
x \cdot 2 + \mathsf{fma}\left(x, x, y \cdot y\right)

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))
↓
(FPCore (x y) :precision binary64 (+ (* x 2.0) (fma x x (* y y))))
double code(double x, double y) {
	return ((x * 2.0) + (x * x)) + (y * y);
}
↓
double code(double x, double y) {
	return (x * 2.0) + fma(x, x, (y * y));
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0.0
Herbie0.0
\[y \cdot y + \left(2 \cdot x + x \cdot x\right) \]

Derivation

  1. Initial program 0.0

    \[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]

  2. Applied associate-+l+_binary640.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto x \cdot 2 + \mathsf{fma}\left(x, x, y \cdot y\right) \]

Reproduce

herbie shell --seed 2021215 
(FPCore (x y)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, A"
  :precision binary64

  :herbie-target
  (+ (* y y) (+ (* 2.0 x) (* x x)))

  (+ (+ (* x 2.0) (* x x)) (* y y)))