Average Error: 5.1 → 5.1
Time: 11.3s
Precision: 64
\[x \cdot \left(1 + y \cdot y\right)\]
↓
\[\mathsf{fma}\left(y, y, 1\right) \cdot x\]
x \cdot \left(1 + y \cdot y\right)
↓
\mathsf{fma}\left(y, y, 1\right) \cdot x
double f(double x, double y) {
        double r18730978 = x;
        double r18730979 = 1.0;
        double r18730980 = y;
        double r18730981 = r18730980 * r18730980;
        double r18730982 = r18730979 + r18730981;
        double r18730983 = r18730978 * r18730982;
        return r18730983;
}
↓
double f(double x, double y) {
        double r18730984 = y;
        double r18730985 = 1.0;
        double r18730986 = fma(r18730984, r18730984, r18730985);
        double r18730987 = x;
        double r18730988 = r18730986 * r18730987;
        return r18730988;
}

Error

Bits error versus x

Bits error versus y

Target

Original5.1
Target0.1
Herbie5.1
\[x + \left(x \cdot y\right) \cdot y\]

Derivation

  1. Initial program 5.1

    \[x \cdot \left(1 + y \cdot y\right)\]

  2. Simplified5.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, 1\right) \cdot x}\]

  3. Final simplification5.1

    \[\leadsto \mathsf{fma}\left(y, y, 1\right) \cdot x\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Integration.TanhSinh:everywhere from integration-0.2.1"

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

  (* x (+ 1.0 (* y y))))