Expression 4, p15

Average Error: 0.0 → 0.0

Time: 6.6s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b) {
        double r82012 = a;
        double r82013 = b;
        double r82014 = r82012 + r82013;
        double r82015 = r82014 * r82014;
        return r82015;
}

↓

double f(double a, double b) {
        double r82016 = b;
        double r82017 = a;
        double r82018 = 2.0;
        double r82019 = r82017 * r82016;
        double r82020 = r82018 * r82019;
        double r82021 = fma(r82017, r82017, r82020);
        double r82022 = fma(r82016, r82016, r82021);
        return r82022;
}

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. 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. Simplified 0.0

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

5. Taylor expanded around 0 0.0

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

6. Simplified 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (<= 5 a 10) (<= 0.0 b 0.001))

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

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