Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[0.0 \le x \le 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[\mathsf{fma}\left(x, x \cdot x, x \cdot x\right)\]
x \cdot \left(x \cdot x\right) + x \cdot x
↓
\mathsf{fma}\left(x, x \cdot x, x \cdot x\right)
double f(double x) {
        double r2946749 = x;
        double r2946750 = r2946749 * r2946749;
        double r2946751 = r2946749 * r2946750;
        double r2946752 = r2946751 + r2946750;
        return r2946752;
}
↓
double f(double x) {
        double r2946753 = x;
        double r2946754 = r2946753 * r2946753;
        double r2946755 = fma(r2946753, r2946754, r2946754);
        return r2946755;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Expression 3, p15"
  :pre (<= 0.0 x 2.0)

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

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