Expression 4, p15

Average Error: 0.0 → 0.0

Time: 1.3s

Precision: 64

\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]

Math

\[\left(a + b\right) \cdot \left(a + b\right)\]

↓

\[\mathsf{fma}\left(\mathsf{fma}\left(a, 2, b\right), b, {a}^{2}\right)\]

C

double code(double a, double b) {
	return ((a + b) * (a + b));
}

↓

double code(double a, double b) {
	return fma(fma(a, 2.0, b), b, pow(a, 2.0));
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.0
Target	0.0
Herbie	0.0

\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a\]

Derivation

1. Initial program 0.0

   \[\left(a + b\right) \cdot \left(a + b\right)\]

2. Taylor expanded around 0 0.0

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

3. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, 2, b\right), b, {a}^{2}\right)}\]

4. Final simplification 0.0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, 2, b\right), b, {a}^{2}\right)\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (<= 5 a 10) (<= 0.0 b 0.001))

  :herbie-target
  (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a))

  (* (+ a b) (+ a b)))