Average Error: 0.0 → 0.0
Time: 9.3s
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 r1427931 = x;
        double r1427932 = r1427931 * r1427931;
        double r1427933 = r1427931 * r1427932;
        double r1427934 = r1427933 + r1427932;
        return r1427934;
}
↓
double f(double x) {
        double r1427935 = x;
        double r1427936 = fma(r1427935, r1427935, r1427935);
        double r1427937 = r1427935 * r1427936;
        return r1427937;
}

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. Using strategy rm

  3. Applied distribute-lft-out0.0

    \[\leadsto \color{blue}{x \cdot \left(x \cdot x + x\right)}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019155 +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)))