Average Error: 0.1 → 0
Time: 13.9s
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 r24711217 = x;
        double r24711218 = y;
        double r24711219 = r24711217 - r24711218;
        double r24711220 = 2.0;
        double r24711221 = r24711219 / r24711220;
        double r24711222 = r24711217 + r24711221;
        return r24711222;
}
↓
double f(double x, double y) {
        double r24711223 = 1.5;
        double r24711224 = x;
        double r24711225 = 0.5;
        double r24711226 = y;
        double r24711227 = r24711225 * r24711226;
        double r24711228 = -r24711227;
        double r24711229 = fma(r24711223, r24711224, r24711228);
        return r24711229;
}

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)))