Average Error: 0.0 → 0
Time: 688.0ms
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 r1184760 = x;
        double r1184761 = y;
        double r1184762 = r1184761 - r1184760;
        double r1184763 = 2.0;
        double r1184764 = r1184762 / r1184763;
        double r1184765 = r1184760 + r1184764;
        return r1184765;
}
↓
double f(double x, double y) {
        double r1184766 = 0.5;
        double r1184767 = x;
        double r1184768 = y;
        double r1184769 = r1184766 * r1184768;
        double r1184770 = fma(r1184766, r1184767, r1184769);
        return r1184770;
}

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 2020034 +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)))