Expression 4, p15

Average Error: 0.0 → 0.0

Time: 3.6s

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 r1595224 = a;
        double r1595225 = b;
        double r1595226 = r1595224 + r1595225;
        double r1595227 = r1595226 * r1595226;
        return r1595227;
}

↓

double f(double a, double b) {
        double r1595228 = b;
        double r1595229 = a;
        double r1595230 = 2.0;
        double r1595231 = r1595230 * r1595229;
        double r1595232 = r1595231 * r1595228;
        double r1595233 = fma(r1595229, r1595229, r1595232);
        double r1595234 = fma(r1595228, r1595228, r1595233);
        return r1595234;
}

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 2019142 +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)))