Average Error: 0.0 → 0
Time: 6.7s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a, a, b \cdot \mathsf{fma}\left(a, 2, b\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(a, a, b \cdot \mathsf{fma}\left(a, 2, b\right)\right)
double f(double a, double b) {
        double r101102 = a;
        double r101103 = b;
        double r101104 = r101102 + r101103;
        double r101105 = r101104 * r101104;
        return r101105;
}
↓
double f(double a, double b) {
        double r101106 = a;
        double r101107 = b;
        double r101108 = 2.0;
        double r101109 = fma(r101106, r101108, r101107);
        double r101110 = r101107 * r101109;
        double r101111 = fma(r101106, r101106, r101110);
        return r101111;
}

Error

Bits error versus a

Bits error versus b

Target

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

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. Simplified0

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

  4. Final simplification0

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5.0 a 10.0) (<= 0.0 b 0.001))

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

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