Average Error: 0.0 → 0
Time: 5.9s
Precision: 64
\[5 \le a \le 10 \land 0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a, a, \mathsf{fma}\left(a \cdot 2, b, b \cdot b\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(a, a, \mathsf{fma}\left(a \cdot 2, b, b \cdot b\right)\right)
double f(double a, double b) {
        double r3172679 = a;
        double r3172680 = b;
        double r3172681 = r3172679 + r3172680;
        double r3172682 = r3172681 * r3172681;
        return r3172682;
}
↓
double f(double a, double b) {
        double r3172683 = a;
        double r3172684 = 2.0;
        double r3172685 = r3172683 * r3172684;
        double r3172686 = b;
        double r3172687 = r3172686 * r3172686;
        double r3172688 = fma(r3172685, r3172686, r3172687);
        double r3172689 = fma(r3172683, r3172683, r3172688);
        return r3172689;
}

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. 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. Taylor expanded around 0 0.0

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

  5. Simplified0

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

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019162 +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)))