Expression 4, p15

Average Error: 0.0 → 0.0

Time: 2.6s

Precision: binary64

\[\left(5 \leq a \land a \leq 10\right) \land \left(0 \leq b \land b \leq 0.001\right)\]

\[[a, b] = \mathsf{sort}([a, b]) \\]

Math

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

↓

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

FPCore

(FPCore (a b) :precision binary64 (* (+ a b) (+ a b)))

↓

(FPCore (a b) :precision binary64 (+ (* 2.0 (* a b)) (fma b b (* a a))))

C

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

↓

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

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 in a around 0 0.0

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

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Reproduce

herbie shell --seed 2022130 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (and (<= 5.0 a) (<= a 10.0)) (and (<= 0.0 b) (<= b 0.001)))

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

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