Average Error: 0.1 → 0
Time: 1.2s
Precision: binary64
\[x + \frac{x - y}{2} \]
↓
\[\mathsf{fma}\left(1.5, x, y \cdot -0.5\right) \]
x + \frac{x - y}{2}
↓
\mathsf{fma}\left(1.5, x, y \cdot -0.5\right)

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))
↓
(FPCore (x y) :precision binary64 (fma 1.5 x (* y -0.5)))
double code(double x, double y) {
	return x + ((x - y) / 2.0);
}
↓
double code(double x, double y) {
	return fma(1.5, x, (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. Taylor expanded in x around 0 0.1

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

  4. Simplified0.1

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

  5. Taylor expanded in y around 0 0.1

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

  6. Simplified0

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

  7. Final simplification0

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

Reproduce

herbie shell --seed 2022076 
(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)))