Average Error: 0.0 → 0
Time: 5.5s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 0.001000000000000000020816681711721685132943\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 2, b\right) \cdot b\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(a, a, \mathsf{fma}\left(a, 2, b\right) \cdot b\right)
double f(double a, double b) {
        double r61594 = a;
        double r61595 = b;
        double r61596 = r61594 + r61595;
        double r61597 = r61596 * r61596;
        return r61597;
}
↓
double f(double a, double b) {
        double r61598 = a;
        double r61599 = 2.0;
        double r61600 = b;
        double r61601 = fma(r61598, r61599, r61600);
        double r61602 = r61601 * r61600;
        double r61603 = fma(r61598, r61598, r61602);
        return r61603;
}

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. Simplified0.0

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0

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

  5. Final simplification0

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5.0 a 10.0) (<= 0.0 b 0.001))

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

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