Average Error: 0.1 → 0
Time: 13.8s
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 r31083611 = x;
        double r31083612 = y;
        double r31083613 = r31083611 - r31083612;
        double r31083614 = 2.0;
        double r31083615 = r31083613 / r31083614;
        double r31083616 = r31083611 + r31083615;
        return r31083616;
}
↓
double f(double x, double y) {
        double r31083617 = 1.5;
        double r31083618 = x;
        double r31083619 = 0.5;
        double r31083620 = y;
        double r31083621 = r31083619 * r31083620;
        double r31083622 = -r31083621;
        double r31083623 = fma(r31083617, r31083618, r31083622);
        return r31083623;
}

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 2019169 +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)))