Average Error: 0.0 → 0.0
Time: 12.3s
Precision: 64
\[0 \le x \le 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[\mathsf{fma}\left(\left(x \cdot x\right), x, \left(x \cdot x\right)\right)\]
x \cdot \left(x \cdot x\right) + x \cdot x
↓
\mathsf{fma}\left(\left(x \cdot x\right), x, \left(x \cdot x\right)\right)
double f(double x) {
        double r2848441 = x;
        double r2848442 = r2848441 * r2848441;
        double r2848443 = r2848441 * r2848442;
        double r2848444 = r2848443 + r2848442;
        return r2848444;
}
↓
double f(double x) {
        double r2848445 = x;
        double r2848446 = r2848445 * r2848445;
        double r2848447 = fma(r2848446, r2848445, r2848446);
        return r2848447;
}

Error

Bits error versus x

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(1.0 + x\right) \cdot x\right) \cdot x\]

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "Expression 3, p15"
  :pre (<= 0 x 2)

  :herbie-target
  (* (* (+ 1.0 x) x) x)

  (+ (* x (* x x)) (* x x)))