Average Error: 0.0 → 0.0
Time: 10.3s
Precision: 64
\[0 \le x \le 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[\mathsf{fma}\left(x \cdot x, x, x \cdot x\right)\]
x \cdot \left(x \cdot x\right) + x \cdot x
↓
\mathsf{fma}\left(x \cdot x, x, x \cdot x\right)
double f(double x) {
        double r1728156 = x;
        double r1728157 = r1728156 * r1728156;
        double r1728158 = r1728156 * r1728157;
        double r1728159 = r1728158 + r1728157;
        return r1728159;
}
↓
double f(double x) {
        double r1728160 = x;
        double r1728161 = r1728160 * r1728160;
        double r1728162 = fma(r1728161, r1728160, r1728161);
        return r1728162;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +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)))