Expression 4, p15

Average Error: 0.0 → 0.0

Time: 1.7s

Precision: 64

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

Math

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

↓

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

C

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

↓

double code(double a, double b) {
	return ((double) (((double) (b * ((double) (((double) (a + b)) + a)))) + ((double) (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. Using strategy rm

3. Applied +-commutative 0.0

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

4. Applied distribute-lft-in 0.0

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

5. Simplified 0.0

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

6. Simplified 0.0

   \[\leadsto b \cdot \left(a + b\right) + \color{blue}{a \cdot \left(a + b\right)}\]
7. Using strategy rm

8. Applied +-commutative 0.0

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

9. Applied distribute-lft-in 0.0

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

10. Applied associate-+r+ 0.0

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

11. Simplified 0.0

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

12. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020113 
(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)))