Average Error: 0.0 → 0.0
Time: 3.8s
Precision: 64
\[5 \le a \le 10 \land 0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(b, 2, a\right), a, b \cdot b\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(\mathsf{fma}\left(b, 2, a\right), a, b \cdot b\right)
double f(double a, double b) {
        double r1233768 = a;
        double r1233769 = b;
        double r1233770 = r1233768 + r1233769;
        double r1233771 = r1233770 * r1233770;
        return r1233771;
}
↓
double f(double a, double b) {
        double r1233772 = b;
        double r1233773 = 2.0;
        double r1233774 = a;
        double r1233775 = fma(r1233772, r1233773, r1233774);
        double r1233776 = r1233772 * r1233772;
        double r1233777 = fma(r1233775, r1233774, r1233776);
        return r1233777;
}

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

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 +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)))