Expression 4, p15

Average Error: 0.0 → 0

Time: 10.2s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b) {
        double r103032 = a;
        double r103033 = b;
        double r103034 = r103032 + r103033;
        double r103035 = r103034 * r103034;
        return r103035;
}

↓

double f(double a, double b) {
        double r103036 = a;
        double r103037 = b;
        double r103038 = 2.0;
        double r103039 = fma(r103036, r103038, r103037);
        double r103040 = r103037 * r103039;
        double r103041 = fma(r103036, r103036, r103040);
        return r103041;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.0
Target	0.0
Herbie	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. Taylor expanded around 0 0.0

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

5. Simplified 0

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

6. Final simplification 0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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)))