Expression 4, p15

Average Error: 0.0 → 0

Time: 6.2s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b) {
        double r2479988 = a;
        double r2479989 = b;
        double r2479990 = r2479988 + r2479989;
        double r2479991 = r2479990 * r2479990;
        return r2479991;
}

↓

double f(double a, double b) {
        double r2479992 = a;
        double r2479993 = b;
        double r2479994 = 2.0;
        double r2479995 = fma(r2479992, r2479994, r2479993);
        double r2479996 = r2479993 * r2479995;
        double r2479997 = fma(r2479992, r2479992, r2479996);
        return r2479997;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.0
Target	0.0
Herbie	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

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

4. Taylor expanded around -inf 0.0

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

5. Simplified 0

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

6. Final simplification 0

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

Reproduce

herbie shell --seed 2019134 +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)))