Average Error: 0.0 → 0.0
Time: 5.7s
Precision: 64
\[0 \le x \le 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[x \cdot \mathsf{fma}\left(x, x, x\right)\]
x \cdot \left(x \cdot x\right) + x \cdot x
↓
x \cdot \mathsf{fma}\left(x, x, x\right)
double f(double x) {
        double r2545545 = x;
        double r2545546 = r2545545 * r2545545;
        double r2545547 = r2545545 * r2545546;
        double r2545548 = r2545547 + r2545546;
        return r2545548;
}
↓
double f(double x) {
        double r2545549 = x;
        double r2545550 = fma(r2545549, r2545549, r2545549);
        double r2545551 = r2545549 * r2545550;
        return r2545551;
}

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. Taylor expanded around -inf 0.0

    \[\leadsto \color{blue}{{x}^{3} + {x}^{2}}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019129 +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)))