Average Error: 0.0 → 0.0
Time: 20.0s
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 r19287609 = x;
        double r19287610 = r19287609 * r19287609;
        double r19287611 = r19287609 * r19287610;
        double r19287612 = r19287611 + r19287610;
        return r19287612;
}
↓
double f(double x) {
        double r19287613 = x;
        double r19287614 = fma(r19287613, r19287613, r19287613);
        double r19287615 = r19287613 * r19287614;
        return r19287615;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 +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)))