Average Error: 39.3 → 0
Time: 1.9s
Precision: binary64
\[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
↓
\[\mathsf{fma}\left(x, x, x \cdot 2\right) \]
\left(x + 1\right) \cdot \left(x + 1\right) - 1
↓
\mathsf{fma}\left(x, x, x \cdot 2\right)

(FPCore (x) :precision binary64 (- (* (+ x 1.0) (+ x 1.0)) 1.0))
↓
(FPCore (x) :precision binary64 (fma x x (* x 2.0)))
double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}
↓
double code(double x) {
	return fma(x, x, (x * 2.0));
}

Error

Bits error versus x

Derivation

  1. Initial program 39.3

    \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

  2. Taylor expanded in x around 0 0.0

    \[\leadsto \color{blue}{{x}^{2} + 2 \cdot x} \]

  3. Applied egg-rr0

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

  4. Final simplification0

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

Reproduce

herbie shell --seed 2022130 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))