Average Error: 0.0 → 0.0
Time: 21.6s
Precision: 64
\[5 \le a \le 10 \land 0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(\left(\mathsf{fma}\left(2, a, b\right)\right), b, \left(a \cdot a\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(\left(\mathsf{fma}\left(2, a, b\right)\right), b, \left(a \cdot a\right)\right)
double f(double a, double b) {
        double r6515671 = a;
        double r6515672 = b;
        double r6515673 = r6515671 + r6515672;
        double r6515674 = r6515673 * r6515673;
        return r6515674;
}
↓
double f(double a, double b) {
        double r6515675 = 2.0;
        double r6515676 = a;
        double r6515677 = b;
        double r6515678 = fma(r6515675, r6515676, r6515677);
        double r6515679 = r6515676 * r6515676;
        double r6515680 = fma(r6515678, r6515677, r6515679);
        return r6515680;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +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)))