Average Error: 0.1 → 0
Time: 3.0s
Precision: binary64
\[x + \frac{x - y}{2} \]
↓
\[\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right) \]
x + \frac{x - y}{2}
↓
\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))
↓
(FPCore (x y) :precision binary64 (fma x 1.5 (* y -0.5)))
double code(double x, double y) {
	return x + ((x - y) / 2.0);
}
↓
double code(double x, double y) {
	return fma(x, 1.5, (y * -0.5));
}

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. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, 0.5, x\right)} \]

  3. Applied add-cube-cbrt_binary641.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x - y, 0.5, x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x - y, 0.5, x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x - y, 0.5, x\right)}} \]

  4. Applied pow3_binary641.3

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x - y, 0.5, x\right)}\right)}^{3}} \]

  5. Taylor expanded in y around 0 0.3

    \[\leadsto \color{blue}{x \cdot {3.375}^{0.3333333333333333} - 0.3333333333333333 \cdot \left(y \cdot {3.375}^{0.3333333333333333}\right)} \]

  6. Simplified0.8

    \[\leadsto \color{blue}{\sqrt[3]{3.375} \cdot \mathsf{fma}\left(y, -0.3333333333333333, x\right)} \]

  7. Taylor expanded in y around 0 0.1

    \[\leadsto \color{blue}{1.5 \cdot x - 0.5 \cdot y} \]

  8. Simplified0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1.5, y \cdot -0.5\right)} \]

  9. Final simplification0

    \[\leadsto \mathsf{fma}\left(x, 1.5, y \cdot -0.5\right) \]

Reproduce

herbie shell --seed 2021313 
(FPCore (x y)
  :name "Graphics.Rendering.Chart.Axis.Types:hBufferRect from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- (* 1.5 x) (* 0.5 y))

  (+ x (/ (- x y) 2.0)))