Average Error: 0.0 → 0.0
Time: 3.6s
Precision: 64
\[0.0 \le x \le 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[\mathsf{fma}\left(x, x, {x}^{3}\right)\]
x \cdot \left(x \cdot x\right) + x \cdot x
↓
\mathsf{fma}\left(x, x, {x}^{3}\right)
double f(double x) {
        double r54833 = x;
        double r54834 = r54833 * r54833;
        double r54835 = r54833 * r54834;
        double r54836 = r54835 + r54834;
        return r54836;
}
↓
double f(double x) {
        double r54837 = x;
        double r54838 = 3.0;
        double r54839 = pow(r54837, r54838);
        double r54840 = fma(r54837, r54837, r54839);
        return r54840;
}

Error

Bits error versus x

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(1 + x\right) \cdot x\right) \cdot x\]

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, x, {x}^{3}\right)\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x)
  :name "Expression 3, p15"
  :precision binary64
  :pre (<= 0.0 x 2)

  :herbie-target
  (* (* (+ 1 x) x) x)

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