Average Error: 0.1 → 0
Time: 16.0s
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 r416169 = x;
        double r416170 = y;
        double r416171 = r416169 - r416170;
        double r416172 = 2.0;
        double r416173 = r416171 / r416172;
        double r416174 = r416169 + r416173;
        return r416174;
}
↓
double f(double x, double y) {
        double r416175 = 1.5;
        double r416176 = x;
        double r416177 = y;
        double r416178 = 0.5;
        double r416179 = r416177 * r416178;
        double r416180 = -r416179;
        double r416181 = fma(r416175, r416176, r416180);
        return r416181;
}

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 2019198 +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)))