Average Error: 0.0 → 0.0
Time: 7.5s
Precision: 64
\[0 \le x \le 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[\mathsf{fma}\left(\left(x \cdot x\right), x, \left(x \cdot x\right)\right)\]
x \cdot \left(x \cdot x\right) + x \cdot x
↓
\mathsf{fma}\left(\left(x \cdot x\right), x, \left(x \cdot x\right)\right)
double f(double x) {
        double r2480024 = x;
        double r2480025 = r2480024 * r2480024;
        double r2480026 = r2480024 * r2480025;
        double r2480027 = r2480026 + r2480025;
        return r2480027;
}
↓
double f(double x) {
        double r2480028 = x;
        double r2480029 = r2480028 * r2480028;
        double r2480030 = fma(r2480029, r2480028, r2480029);
        return r2480030;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019132 +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)))