Average Error: 0.1 → 0
Time: 4.1s
Precision: 64
\[x + \frac{x - y}{2}\]
↓
\[\mathsf{fma}\left(1.5, x, -y \cdot 0.5\right)\]
x + \frac{x - y}{2}
↓
\mathsf{fma}\left(1.5, x, -y \cdot 0.5\right)
double f(double x, double y) {
        double r659609 = x;
        double r659610 = y;
        double r659611 = r659609 - r659610;
        double r659612 = 2.0;
        double r659613 = r659611 / r659612;
        double r659614 = r659609 + r659613;
        return r659614;
}
↓
double f(double x, double y) {
        double r659615 = 1.5;
        double r659616 = x;
        double r659617 = y;
        double r659618 = 0.5;
        double r659619 = r659617 * r659618;
        double r659620 = -r659619;
        double r659621 = fma(r659615, r659616, r659620);
        return r659621;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.1
Herbie0
\[1.5 \cdot x - 0.5 \cdot y\]

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{1.5 \cdot x - 0.5 \cdot y}\]
  3. Using strategy rm

  4. Applied fma-neg0

    \[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)}\]

  5. Simplified0

    \[\leadsto \mathsf{fma}\left(1.5, x, \color{blue}{-y \cdot 0.5}\right)\]

  6. Final simplification0

    \[\leadsto \mathsf{fma}\left(1.5, x, -y \cdot 0.5\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"
  :precision binary64

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

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