Average Error: 0.0 → 0.0
Time: 8.8s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 0.001000000000000000020816681711721685132943\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(2, b, a\right), a, b \cdot b\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(\mathsf{fma}\left(2, b, a\right), a, b \cdot b\right)
double f(double a, double b) {
        double r4155568 = a;
        double r4155569 = b;
        double r4155570 = r4155568 + r4155569;
        double r4155571 = r4155570 * r4155570;
        return r4155571;
}
↓
double f(double a, double b) {
        double r4155572 = 2.0;
        double r4155573 = b;
        double r4155574 = a;
        double r4155575 = fma(r4155572, r4155573, r4155574);
        double r4155576 = r4155573 * r4155573;
        double r4155577 = fma(r4155575, r4155574, r4155576);
        return r4155577;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5.0 a 10.0) (<= 0.0 b 0.001))

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

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