Average Error: 0.0 → 0.0
Time: 7.7s
Precision: 64
\[5 \le a \le 10 \land 0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(b + a, a, \left(b + a\right) \cdot b\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(b + a, a, \left(b + a\right) \cdot b\right)
double f(double a, double b) {
        double r3238537 = a;
        double r3238538 = b;
        double r3238539 = r3238537 + r3238538;
        double r3238540 = r3238539 * r3238539;
        return r3238540;
}
↓
double f(double a, double b) {
        double r3238541 = b;
        double r3238542 = a;
        double r3238543 = r3238541 + r3238542;
        double r3238544 = r3238543 * r3238541;
        double r3238545 = fma(r3238543, r3238542, r3238544);
        return r3238545;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a\]

Derivation

  1. Initial program 0.0

    \[\left(a + b\right) \cdot \left(a + b\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019168 +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)))