Expression 4, p15

Average Error: 0.0 → 0.0

Time: 8.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

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

C

double f(double a, double b) {
        double r74038 = a;
        double r74039 = b;
        double r74040 = r74038 + r74039;
        double r74041 = r74040 * r74040;
        return r74041;
}

↓

double f(double a, double b) {
        double r74042 = b;
        double r74043 = a;
        double r74044 = 2.0;
        double r74045 = fma(r74043, r74044, r74042);
        double r74046 = r74043 * r74043;
        double r74047 = fma(r74042, r74045, r74046);
        return r74047;
}

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

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

3. Simplified 0.0

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

4. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019350 +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)))