Average Error: 0.0 → 0.0
Time: 1.6s
Precision: 64
\[0.0 \le x \le 2\]
\[x \cdot \left(x \cdot x\right) + x \cdot x\]
↓
\[\mathsf{fma}\left(x, x, {x}^{3}\right)\]
x \cdot \left(x \cdot x\right) + x \cdot x
↓
\mathsf{fma}\left(x, x, {x}^{3}\right)
double f(double x) {
        double r103089 = x;
        double r103090 = r103089 * r103089;
        double r103091 = r103089 * r103090;
        double r103092 = r103091 + r103090;
        return r103092;
}
↓
double f(double x) {
        double r103093 = x;
        double r103094 = 3.0;
        double r103095 = pow(r103093, r103094);
        double r103096 = fma(r103093, r103093, r103095);
        return r103096;
}

Error

Bits error versus x

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(1 + x\right) \cdot x\right) \cdot x\]

Derivation

  1. Initial program 0.0

    \[x \cdot \left(x \cdot x\right) + x \cdot x\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, {x}^{3}\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x, x, {x}^{3}\right)\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "Expression 3, p15"
  :precision binary64
  :pre (<= 0.0 x 2)

  :herbie-target
  (* (* (+ 1 x) x) x)

  (+ (* x (* x x)) (* x x)))