Average Error: 0.1 → 0
Time: 10.4s
Precision: 64
\[x + \frac{x - y}{2.0}\]
↓
\[\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)\]
x + \frac{x - y}{2.0}
↓
\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)
double f(double x, double y) {
        double r27671976 = x;
        double r27671977 = y;
        double r27671978 = r27671976 - r27671977;
        double r27671979 = 2.0;
        double r27671980 = r27671978 / r27671979;
        double r27671981 = r27671976 + r27671980;
        return r27671981;
}
↓
double f(double x, double y) {
        double r27671982 = 1.5;
        double r27671983 = x;
        double r27671984 = 0.5;
        double r27671985 = y;
        double r27671986 = r27671984 * r27671985;
        double r27671987 = -r27671986;
        double r27671988 = fma(r27671982, r27671983, r27671987);
        return r27671988;
}

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

    \[x + \frac{x - y}{2.0}\]

  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 2019163 +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)))