Average Error: 0.0 → 0.0

Time: 1.4s

Precision: binary64

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

↓

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

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

↓

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

(FPCore (x y) :precision binary64 (* (+ x y) (+ x y)))

↓

(FPCore (x y) :precision binary64 (fma x (+ x y) (* y (+ x y))))

double code(double x, double y) {
	return (x + y) * (x + y);
}

↓

double code(double x, double y) {
	return fma(x, (x + y), (y * (x + y)));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

Initial program 0.0
\[\left(x + y\right) \cdot \left(x + y\right) \]
Applied distribute-rgt-in_binary640.0
\[\leadsto \color{blue}{x \cdot \left(x + y\right) + y \cdot \left(x + y\right)} \]
Applied fma-def_binary640.0
\[\leadsto \color{blue}{\mathsf{fma}\left(x, x + y, y \cdot \left(x + y\right)\right)} \]
Final simplification0.0
\[\leadsto \mathsf{fma}\left(x, x + y, y \cdot \left(x + y\right)\right) \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f3 from sbv-4.4"
  :precision binary64

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

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