Average Error: 0.0 → 0.0
Time: 7.2s
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 r66748 = a;
        double r66749 = b;
        double r66750 = r66748 + r66749;
        double r66751 = r66750 * r66750;
        return r66751;
}
↓
double f(double a, double b) {
        double r66752 = a;
        double r66753 = b;
        double r66754 = r66752 + r66753;
        double r66755 = r66754 * r66753;
        double r66756 = fma(r66754, r66752, r66755);
        return r66756;
}

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)))