Average Error: 0.0 → 0.0
Time: 9.0s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 0.001000000000000000020816681711721685132943\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(b, \mathsf{fma}\left(a, 2, b\right), a \cdot a\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(b, \mathsf{fma}\left(a, 2, b\right), a \cdot a\right)
double f(double a, double b) {
        double r103422 = a;
        double r103423 = b;
        double r103424 = r103422 + r103423;
        double r103425 = r103424 * r103424;
        return r103425;
}
↓
double f(double a, double b) {
        double r103426 = b;
        double r103427 = a;
        double r103428 = 2.0;
        double r103429 = fma(r103427, r103428, r103426);
        double r103430 = r103427 * r103427;
        double r103431 = fma(r103426, r103429, r103430);
        return r103431;
}

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

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +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)))