Average Error: 0.0 → 0.0
Time: 7.7s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 0.001000000000000000020816681711721685132943\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a + b, a, \left(a + b\right) \cdot b\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(a + b, a, \left(a + b\right) \cdot b\right)
double f(double a, double b) {
        double r70201 = a;
        double r70202 = b;
        double r70203 = r70201 + r70202;
        double r70204 = r70203 * r70203;
        return r70204;
}
↓
double f(double a, double b) {
        double r70205 = a;
        double r70206 = b;
        double r70207 = r70205 + r70206;
        double r70208 = r70207 * r70206;
        double r70209 = fma(r70207, r70205, r70208);
        return r70209;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a\]

Derivation

  1. Initial program 0.0

    \[\left(a + b\right) \cdot \left(a + b\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  5. Applied fma-def0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (<= 5 a 10) (<= 0.0 b 1e-3))

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

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