Average Error: 0.0 → 0.0
Time: 18.4s
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 r18797879 = a;
        double r18797880 = b;
        double r18797881 = r18797879 + r18797880;
        double r18797882 = r18797881 * r18797881;
        return r18797882;
}
↓
double f(double a, double b) {
        double r18797883 = 2.0;
        double r18797884 = a;
        double r18797885 = b;
        double r18797886 = fma(r18797883, r18797884, r18797885);
        double r18797887 = r18797884 * r18797884;
        double r18797888 = fma(r18797886, r18797885, r18797887);
        return r18797888;
}

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 2019125 +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)))