Expression 4, p15

Average Error: 0.0 → 0.0

Time: 5.2s

Precision: 64

\[5 \le a \le 10 \land 0.0 \le b \le 0.001000000000000000020816681711721685132943\]

Math

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

↓

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

C

double f(double a, double b) {
        double r4067945 = a;
        double r4067946 = b;
        double r4067947 = r4067945 + r4067946;
        double r4067948 = r4067947 * r4067947;
        return r4067948;
}

↓

double f(double a, double b) {
        double r4067949 = a;
        double r4067950 = b;
        double r4067951 = 2.0;
        double r4067952 = r4067950 * r4067951;
        double r4067953 = r4067950 * r4067950;
        double r4067954 = fma(r4067949, r4067949, r4067953);
        double r4067955 = fma(r4067949, r4067952, r4067954);
        return r4067955;
}

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

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

4. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5.0 a 10.0) (<= 0.0 b 0.001))

  :herbie-target
  (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a))

  (* (+ a b) (+ a b)))