Expression 4, p15

Average Error: 0.0 → 0.0

Time: 4.8s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b) {
        double r65597 = a;
        double r65598 = b;
        double r65599 = r65597 + r65598;
        double r65600 = r65599 * r65599;
        return r65600;
}

↓

double f(double a, double b) {
        double r65601 = a;
        double r65602 = b;
        double r65603 = r65601 + r65602;
        double r65604 = r65603 * r65603;
        return r65604;
}

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 distribute-lft-in 0.0

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

4. Simplified 0.0

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

5. Simplified 0.0

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

7. Applied distribute-lft-in 0.0

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

8. Applied associate-+l+ 0.0

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

9. Simplified 0.0

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

10. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019297 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (<= 5 a 10) (<= 0.0 b 1e-3))

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

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