Expression 4, p15

Average Error: 0.0 → 0.0

Time: 6.2s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b) {
        double r3987225 = a;
        double r3987226 = b;
        double r3987227 = r3987225 + r3987226;
        double r3987228 = r3987227 * r3987227;
        return r3987228;
}

↓

double f(double a, double b) {
        double r3987229 = b;
        double r3987230 = a;
        double r3987231 = r3987229 * r3987230;
        double r3987232 = r3987231 + r3987231;
        double r3987233 = fma(r3987230, r3987230, r3987232);
        double r3987234 = fma(r3987229, r3987229, r3987233);
        return r3987234;
}

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 0 0.0

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

3. Simplified 0.0

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

4. Taylor expanded around 0 0.0

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

5. Simplified 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019164 +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)))