Average Error: 0.0 → 0
Time: 597.0ms
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 r482164 = x;
        double r482165 = y;
        double r482166 = r482165 - r482164;
        double r482167 = 2.0;
        double r482168 = r482166 / r482167;
        double r482169 = r482164 + r482168;
        return r482169;
}
↓
double f(double x, double y) {
        double r482170 = 0.5;
        double r482171 = x;
        double r482172 = y;
        double r482173 = r482170 * r482172;
        double r482174 = fma(r482170, r482171, r482173);
        return r482174;
}

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 2020025 +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)))