Average Error: 0.0 → 0.0
Time: 6.9s
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 r71376 = a;
        double r71377 = b;
        double r71378 = r71376 + r71377;
        double r71379 = r71378 * r71378;
        return r71379;
}
↓
double f(double a, double b) {
        double r71380 = b;
        double r71381 = a;
        double r71382 = 2.0;
        double r71383 = fma(r71381, r71382, r71380);
        double r71384 = r71381 * r71381;
        double r71385 = fma(r71380, r71383, r71384);
        return r71385;
}

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 2019326 +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)))