Average Error: 0.0 → 0.0
Time: 3.3s
Precision: 64
\[5 \le a \le 10 \land 0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, \left(2 \cdot a\right) \cdot b\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, a, \left(2 \cdot a\right) \cdot b\right)\right)
double f(double a, double b) {
        double r2011258 = a;
        double r2011259 = b;
        double r2011260 = r2011258 + r2011259;
        double r2011261 = r2011260 * r2011260;
        return r2011261;
}
↓
double f(double a, double b) {
        double r2011262 = b;
        double r2011263 = a;
        double r2011264 = 2.0;
        double r2011265 = r2011264 * r2011263;
        double r2011266 = r2011265 * r2011262;
        double r2011267 = fma(r2011263, r2011263, r2011266);
        double r2011268 = fma(r2011262, r2011262, r2011267);
        return r2011268;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019152 +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)))