Average Error: 0.1 → 0
Time: 15.6s
Precision: 64
\[x + \frac{x - y}{2}\]
↓
\[\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\]
x + \frac{x - y}{2}
↓
\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)
double f(double x, double y) {
        double r22955659 = x;
        double r22955660 = y;
        double r22955661 = r22955659 - r22955660;
        double r22955662 = 2.0;
        double r22955663 = r22955661 / r22955662;
        double r22955664 = r22955659 + r22955663;
        return r22955664;
}
↓
double f(double x, double y) {
        double r22955665 = 1.5;
        double r22955666 = x;
        double r22955667 = 0.5;
        double r22955668 = y;
        double r22955669 = r22955667 * r22955668;
        double r22955670 = -r22955669;
        double r22955671 = fma(r22955665, r22955666, r22955670);
        return r22955671;
}

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. Final simplification0

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"

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

  (+ x (/ (- x y) 2.0)))