Expanding a square

Average Error: 38.9 → 0

Time: 17.9s

Precision: 64

Math

\[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

↓

\[\mathsf{fma}\left(x, x, x \cdot 2\right)\]

C

double f(double x) {
        double r364962 = x;
        double r364963 = 1.0;
        double r364964 = r364962 + r364963;
        double r364965 = r364964 * r364964;
        double r364966 = r364965 - r364963;
        return r364966;
}

↓

double f(double x) {
        double r364967 = x;
        double r364968 = 2.0;
        double r364969 = r364967 * r364968;
        double r364970 = fma(r364967, r364967, r364969);
        return r364970;
}

Error

Bits error versus x

Derivation

1. Initial program 38.9

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1\]

2. Taylor expanded around 0 0.0

   \[\leadsto \color{blue}{2 \cdot x + {x}^{2}}\]

3. Simplified 0.0

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

4. Taylor expanded around 0 0.0

   \[\leadsto \color{blue}{2 \cdot x + {x}^{2}}\]

5. Simplified 0

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

6. Final simplification 0

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "Expanding a square"
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))