Average Error: 0.0 → 0.0
Time: 10.5s
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 r1829285 = x;
        double r1829286 = r1829285 * r1829285;
        double r1829287 = r1829285 * r1829286;
        double r1829288 = r1829287 + r1829286;
        return r1829288;
}
↓
double f(double x) {
        double r1829289 = x;
        double r1829290 = r1829289 * r1829289;
        double r1829291 = fma(r1829290, r1829289, r1829290);
        return r1829291;
}

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 2019153 +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)))