Average Error: 0.1 → 0
Time: 15.9s
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 r27115451 = x;
        double r27115452 = y;
        double r27115453 = r27115451 - r27115452;
        double r27115454 = 2.0;
        double r27115455 = r27115453 / r27115454;
        double r27115456 = r27115451 + r27115455;
        return r27115456;
}
↓
double f(double x, double y) {
        double r27115457 = 1.5;
        double r27115458 = x;
        double r27115459 = 0.5;
        double r27115460 = y;
        double r27115461 = r27115459 * r27115460;
        double r27115462 = -r27115461;
        double r27115463 = fma(r27115457, r27115458, r27115462);
        return r27115463;
}

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 2019179 +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)))