Average Error: 0.1 → 0
Time: 15.7s
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 r24183908 = x;
        double r24183909 = y;
        double r24183910 = r24183908 - r24183909;
        double r24183911 = 2.0;
        double r24183912 = r24183910 / r24183911;
        double r24183913 = r24183908 + r24183912;
        return r24183913;
}
↓
double f(double x, double y) {
        double r24183914 = 1.5;
        double r24183915 = x;
        double r24183916 = 0.5;
        double r24183917 = y;
        double r24183918 = r24183916 * r24183917;
        double r24183919 = -r24183918;
        double r24183920 = fma(r24183914, r24183915, r24183919);
        return r24183920;
}

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 2019172 +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)))