Expression 4, p15

Average Error: 0.0 → 0.0

Time: 3.9s

Precision: 64

\[5 \le a \le 10 \land 0 \le b \le 0.001\]

Math

\[\left(a + b\right) \cdot \left(a + b\right)\]

↓

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

C

double f(double a, double b) {
        double r2451566 = a;
        double r2451567 = b;
        double r2451568 = r2451566 + r2451567;
        double r2451569 = r2451568 * r2451568;
        return r2451569;
}

↓

double f(double a, double b) {
        double r2451570 = b;
        double r2451571 = a;
        double r2451572 = 2.0;
        double r2451573 = r2451572 * r2451571;
        double r2451574 = r2451573 * r2451570;
        double r2451575 = fma(r2451571, r2451571, r2451574);
        double r2451576 = fma(r2451570, r2451570, r2451575);
        return r2451576;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.0
Target	0.0
Herbie	0.0

\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a\]

Derivation

1. Initial program 0.0

   \[\left(a + b\right) \cdot \left(a + b\right)\]

2. Taylor expanded around -inf 0.0

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

3. Simplified 0.0

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

4. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019144 +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)))