Expression 4, p15

Average Error: 0.0 → 0

Time: 11.0s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b) {
        double r2227413 = a;
        double r2227414 = b;
        double r2227415 = r2227413 + r2227414;
        double r2227416 = r2227415 * r2227415;
        return r2227416;
}

↓

double f(double a, double b) {
        double r2227417 = a;
        double r2227418 = b;
        double r2227419 = 2.0;
        double r2227420 = fma(r2227417, r2227419, r2227418);
        double r2227421 = r2227418 * r2227420;
        double r2227422 = fma(r2227417, r2227417, r2227421);
        return r2227422;
}

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 0 0.0

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

3. Simplified 0.0

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

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

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

6. Final simplification 0

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

Reproduce

herbie shell --seed 2019132 +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)))