\[x + \frac{x - y}{2} \]

↓

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

x + \frac{x - y}{2}

↓

\mathsf{fma}\left(-0.5 \cdot x, -3, -0.5 \cdot y\right)

(FPCore (x y) :precision binary64 (+ x (/ (- x y) 2.0)))

↓

(FPCore (x y) :precision binary64 (fma (* -0.5 x) -3.0 (* -0.5 y)))

double code(double x, double y) {
	return x + ((x - y) / 2.0);
}

↓

double code(double x, double y) {
	return fma((-0.5 * x), -3.0, (-0.5 * y));
}

Original	0.1
Target	0.1
Herbie	0

Derivation

Initial program 0.1
\[x + \frac{x - y}{2} \]
Simplified0.1
\[\leadsto \color{blue}{\mathsf{fma}\left(x - y, 0.5, x\right)} \]
Applied add-cube-cbrt_binary641.3
\[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(x - y, 0.5, x\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x - y, 0.5, x\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x - y, 0.5, x\right)}} \]
Applied pow3_binary641.3
\[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x - y, 0.5, x\right)}\right)}^{3}} \]
Taylor expanded in y around inf 64.0
\[\leadsto \color{blue}{{\left(e^{0.3333333333333333 \cdot \left(\log -0.5 - \log \left(\frac{1}{y}\right)\right)}\right)}^{3} - 3 \cdot \frac{{\left(e^{0.3333333333333333 \cdot \left(\log -0.5 - \log \left(\frac{1}{y}\right)\right)}\right)}^{3} \cdot x}{y}} \]
Simplified11.6
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \left(y \cdot -0.5\right)}{y}, -3, y \cdot -0.5\right)} \]
Taylor expanded in x around 0 0
\[\leadsto \mathsf{fma}\left(\color{blue}{-0.5 \cdot x}, -3, y \cdot -0.5\right) \]
Final simplification0
\[\leadsto \mathsf{fma}\left(-0.5 \cdot x, -3, -0.5 \cdot y\right) \]

Reproduce

herbie shell --seed 2022068 
(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)))

Error

Target

Derivation

Reproduce