Average Error: 0.0 → 0
Time: 713.0ms
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 r424190 = x;
        double r424191 = y;
        double r424192 = r424191 - r424190;
        double r424193 = 2.0;
        double r424194 = r424192 / r424193;
        double r424195 = r424190 + r424194;
        return r424195;
}
↓
double f(double x, double y) {
        double r424196 = 0.5;
        double r424197 = x;
        double r424198 = y;
        double r424199 = r424196 * r424198;
        double r424200 = fma(r424196, r424197, r424199);
        return r424200;
}

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 2020001 +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)))