Expression 4, p15

Average Error: 0.0 → 0.0

Time: 3.5s

Precision: 64

\[5 \le a \le 10 \land 0 \le b \le 0.001\]

Math

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

↓

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

C

double f(double a, double b) {
        double r3640839 = a;
        double r3640840 = b;
        double r3640841 = r3640839 + r3640840;
        double r3640842 = r3640841 * r3640841;
        return r3640842;
}

↓

double f(double a, double b) {
        double r3640843 = b;
        double r3640844 = a;
        double r3640845 = 2.0;
        double r3640846 = r3640845 * r3640844;
        double r3640847 = r3640846 * r3640843;
        double r3640848 = fma(r3640844, r3640844, r3640847);
        double r3640849 = fma(r3640843, r3640843, r3640848);
        return r3640849;
}

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

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

3. Simplified 0.0

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

4. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5 a 10) (<= 0 b 0.001))

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

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