Average Error: 0.0 → 0.0
Time: 6.3s
Precision: binary64
\[\left(x \cdot 2 + x \cdot x\right) + y \cdot y \]
↓
\[\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, x \cdot 2\right)\right) \]
\left(x \cdot 2 + x \cdot x\right) + y \cdot y
↓
\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, x \cdot 2\right)\right)

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

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. Simplified0.0

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

  3. Taylor expanded in x around 0 0.0

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

  4. Simplified0.0

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

  5. Taylor expanded in x around 0 0.0

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

  6. Applied unpow2_binary640.0

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

  7. Applied fma-def_binary640.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2022024 
(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)))