Average Error: 0.0 → 0.0
Time: 1.3s
Precision: binary64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(x, 2, x \cdot x\right) + y \cdot y \]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
\mathsf{fma}\left(x, 2, x \cdot x\right) + y \cdot y

(FPCore (x y) :precision binary64 (+ (+ (* x 2.0) (* x x)) (* y y)))
↓
(FPCore (x y) :precision binary64 (+ (fma x 2.0 (* 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 fma(x, 2.0, (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 fma-def_binary640.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022081 
(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)))