Average Error: 0.0 → 0.0
Time: 15.2s
Precision: 64
\[5.0 \le a \le 10.0 \land 0.0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(2, b \cdot a, \mathsf{fma}\left(a, a, b \cdot b\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(2, b \cdot a, \mathsf{fma}\left(a, a, b \cdot b\right)\right)
double f(double a, double b) {
        double r3507570 = a;
        double r3507571 = b;
        double r3507572 = r3507570 + r3507571;
        double r3507573 = r3507572 * r3507572;
        return r3507573;
}
↓
double f(double a, double b) {
        double r3507574 = 2.0;
        double r3507575 = b;
        double r3507576 = a;
        double r3507577 = r3507575 * r3507576;
        double r3507578 = r3507575 * r3507575;
        double r3507579 = fma(r3507576, r3507576, r3507578);
        double r3507580 = fma(r3507574, r3507577, r3507579);
        return r3507580;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019165 +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)))