Average Error: 16.0 → 0.0
Time: 2.5s
Precision: binary64
\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]
\[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
↓
\[\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right) \]
{\left(x + \varepsilon\right)}^{2} - {x}^{2}
↓
\varepsilon \cdot \mathsf{fma}\left(2, x, \varepsilon\right)

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

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Initial program 16.0

    \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2022125 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=2"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 2.0) (pow x 2.0)))