Average Error: 0.0 → 0
Time: 585.0ms
Precision: 64
\[x + \frac{y - x}{2}\]
↓
\[\mathsf{fma}\left(0.5, x, 0.5 \cdot y\right)\]
x + \frac{y - x}{2}
↓
\mathsf{fma}\left(0.5, x, 0.5 \cdot y\right)
double f(double x, double y) {
        double r417233 = x;
        double r417234 = y;
        double r417235 = r417234 - r417233;
        double r417236 = 2.0;
        double r417237 = r417235 / r417236;
        double r417238 = r417233 + r417237;
        return r417238;
}
↓
double f(double x, double y) {
        double r417239 = 0.5;
        double r417240 = x;
        double r417241 = y;
        double r417242 = r417239 * r417241;
        double r417243 = fma(r417239, r417240, r417242);
        return r417243;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.0
Target0
Herbie0
\[0.5 \cdot \left(x + y\right)\]

Derivation

  1. Initial program 0.0

    \[x + \frac{y - x}{2}\]

  2. Taylor expanded around 0 0

    \[\leadsto \color{blue}{0.5 \cdot x + 0.5 \cdot y}\]

  3. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x, 0.5 \cdot y\right)}\]

  4. Final simplification0

    \[\leadsto \mathsf{fma}\left(0.5, x, 0.5 \cdot y\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y)
  :name "Numeric.Interval.Internal:bisect from intervals-0.7.1, A"
  :precision binary64

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

  (+ x (/ (- y x) 2)))