Expression 4, p15

Average Error: 0.0 → 0.0

Time: 9.2s

Precision: 64

\[5 \le a \le 10 \land 0.0 \le b \le 0.001000000000000000020816681711721685132943\]

Math

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

↓

\[\left(2 \cdot a\right) \cdot b + \left(b \cdot b + a \cdot a\right)\]

C

double f(double a, double b) {
        double r48999 = a;
        double r49000 = b;
        double r49001 = r48999 + r49000;
        double r49002 = r49001 * r49001;
        return r49002;
}

↓

double f(double a, double b) {
        double r49003 = 2.0;
        double r49004 = a;
        double r49005 = r49003 * r49004;
        double r49006 = b;
        double r49007 = r49005 * r49006;
        double r49008 = r49006 * r49006;
        double r49009 = r49004 * r49004;
        double r49010 = r49008 + r49009;
        double r49011 = r49007 + r49010;
        return r49011;
}

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

   \[\leadsto \color{blue}{b \cdot \left(2 \cdot a + b\right) + a \cdot a}\]
4. Using strategy rm

5. Applied distribute-rgt-in 0.0

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

6. Applied associate-+l+ 0.0

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

7. Final simplification 0.0

   \[\leadsto \left(2 \cdot a\right) \cdot b + \left(b \cdot b + a \cdot a\right)\]

Reproduce

herbie shell --seed 2019326 
(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)))