Average Error: 0.1 → 0
Time: 28.9s
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 r28357199 = x;
        double r28357200 = y;
        double r28357201 = r28357199 - r28357200;
        double r28357202 = 2.0;
        double r28357203 = r28357201 / r28357202;
        double r28357204 = r28357199 + r28357203;
        return r28357204;
}
↓
double f(double x, double y) {
        double r28357205 = 1.5;
        double r28357206 = x;
        double r28357207 = 0.5;
        double r28357208 = y;
        double r28357209 = r28357207 * r28357208;
        double r28357210 = -r28357209;
        double r28357211 = fma(r28357205, r28357206, r28357210);
        return r28357211;
}

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 2019168 +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)))