Expression 4, p15

Average Error: 0.0 → 0

Time: 21.1s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b) {
        double r2269820 = a;
        double r2269821 = b;
        double r2269822 = r2269820 + r2269821;
        double r2269823 = r2269822 * r2269822;
        return r2269823;
}

↓

double f(double a, double b) {
        double r2269824 = a;
        double r2269825 = b;
        double r2269826 = 2.0;
        double r2269827 = r2269824 * r2269826;
        double r2269828 = r2269825 + r2269827;
        double r2269829 = r2269825 * r2269828;
        double r2269830 = fma(r2269824, r2269824, r2269829);
        return r2269830;
}

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. Using strategy rm

3. Applied distribute-lft-in 0.0

   \[\leadsto \color{blue}{\left(a + b\right) \cdot a + \left(a + b\right) \cdot b}\]
4. Using strategy rm

5. Applied fma-def 0.0

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

6. Taylor expanded around inf 0.0

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

7. Simplified 0

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

8. Final simplification 0

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

Reproduce

herbie shell --seed 2019139 +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)))