Average Error: 0.0 → 0
Time: 805.0ms
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 r421446 = x;
        double r421447 = y;
        double r421448 = r421447 - r421446;
        double r421449 = 2.0;
        double r421450 = r421448 / r421449;
        double r421451 = r421446 + r421450;
        return r421451;
}
↓
double f(double x, double y) {
        double r421452 = 0.5;
        double r421453 = x;
        double r421454 = y;
        double r421455 = r421452 * r421454;
        double r421456 = fma(r421452, r421453, r421455);
        return r421456;
}

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 2020036 +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)))