Average Error: 0.0 → 0
Time: 4.4s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a, a, b \cdot \left(2 \cdot a + b\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(a, a, b \cdot \left(2 \cdot a + b\right)\right)
double f(double a, double b) {
        double r136481 = a;
        double r136482 = b;
        double r136483 = r136481 + r136482;
        double r136484 = r136483 * r136483;
        return r136484;
}
↓
double f(double a, double b) {
        double r136485 = a;
        double r136486 = b;
        double r136487 = 2.0;
        double r136488 = r136487 * r136485;
        double r136489 = r136488 + r136486;
        double r136490 = r136486 * r136489;
        double r136491 = fma(r136485, r136485, r136490);
        return r136491;
}

Error

Bits error versus a

Bits error versus b

Target

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

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. Simplified0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Simplified0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2020045 +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)))