Expression 4, p15

Average Error: 0.0 → 0.0

Time: 6.7s

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 r2404997 = a;
        double r2404998 = b;
        double r2404999 = r2404997 + r2404998;
        double r2405000 = r2404999 * r2404999;
        return r2405000;
}

↓

double f(double a, double b) {
        double r2405001 = b;
        double r2405002 = a;
        double r2405003 = r2405001 * r2405002;
        double r2405004 = r2405003 + r2405003;
        double r2405005 = fma(r2405002, r2405002, r2405004);
        double r2405006 = fma(r2405001, r2405001, r2405005);
        return r2405006;
}

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 2019158 +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)))