Expression 4, p15

Average Error: 0.0 → 0.0

Time: 8.9s

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 r56953 = a;
        double r56954 = b;
        double r56955 = r56953 + r56954;
        double r56956 = r56955 * r56955;
        return r56956;
}

↓

double f(double a, double b) {
        double r56957 = 2.0;
        double r56958 = a;
        double r56959 = r56957 * r56958;
        double r56960 = b;
        double r56961 = r56959 * r56960;
        double r56962 = r56960 * r56960;
        double r56963 = r56958 * r56958;
        double r56964 = r56962 + r56963;
        double r56965 = r56961 + r56964;
        return r56965;
}

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)))