Average Error: 0.0 → 0
Time: 4.8s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 0.001000000000000000020816681711721685132943\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a, a, b \cdot \mathsf{fma}\left(a, 2, b\right)\right)\]
\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 r109786 = a;
        double r109787 = b;
        double r109788 = r109786 + r109787;
        double r109789 = r109788 * r109788;
        return r109789;
}
↓
double f(double a, double b) {
        double r109790 = a;
        double r109791 = b;
        double r109792 = 2.0;
        double r109793 = fma(r109790, r109792, r109791);
        double r109794 = r109791 * r109793;
        double r109795 = fma(r109790, r109790, r109794);
        return r109795;
}

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 2019209 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (<= 5 a 10) (<= 0.0 b 1e-3))

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

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