Average Error: 0.1 → 0
Time: 11.1s
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 r359059 = x;
        double r359060 = y;
        double r359061 = r359059 - r359060;
        double r359062 = 2.0;
        double r359063 = r359061 / r359062;
        double r359064 = r359059 + r359063;
        return r359064;
}
↓
double f(double x, double y) {
        double r359065 = 1.5;
        double r359066 = x;
        double r359067 = y;
        double r359068 = 0.5;
        double r359069 = r359067 * r359068;
        double r359070 = -r359069;
        double r359071 = fma(r359065, r359066, r359070);
        return r359071;
}

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 prod-diff0

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

  5. Simplified0

    \[\leadsto \mathsf{fma}\left(1.5, x, -y \cdot 0.5\right) + \color{blue}{0}\]

  6. Final simplification0

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

Reproduce

herbie shell --seed 2019306 +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)))