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

↓

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

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

↓

(FPCore (x y) :precision binary64 (fma 1.5 x (* -0.5 y)))

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

↓

double code(double x, double y) {
	return fma(1.5, x, (-0.5 * y));
}

function code(x, y)
	return Float64(x + Float64(Float64(x - y) / 2.0))
end

↓

function code(x, y)
	return fma(1.5, x, Float64(-0.5 * y))
end

code[x_, y_] := N[(x + N[(N[(x - y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(1.5 * x + N[(-0.5 * y), $MachinePrecision]), $MachinePrecision]

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

↓

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

Original	0.1
Target	0.1
Herbie	0

Derivation

Initial program 0.1
\[x + \frac{x - y}{2} \]
Taylor expanded in x around 0 0.1
\[\leadsto \color{blue}{-0.5 \cdot y + 1.5 \cdot x} \]
Simplified0
\[\leadsto \color{blue}{\mathsf{fma}\left(1.5, x, -0.5 \cdot y\right)} \]
Proof
(fma.f64 3/2 x (*.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 3/2 x) (*.f64 -1/2 y))): 22 points increase in error, 0 points decrease in error
(+.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 1/2 1)) x) (*.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 1/2 x) x)) (*.f64 -1/2 y)): 0 points increase in error, 0 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/2 y) (+.f64 (*.f64 1/2 x) x))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 -1/2 y) (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 1/2 1) x))): 0 points increase in error, 0 points decrease in error
(+.f64 (*.f64 -1/2 y) (*.f64 (Rewrite=> metadata-eval 3/2) x)): 0 points increase in error, 0 points decrease in error
Final simplification0
\[\leadsto \mathsf{fma}\left(1.5, x, -0.5 \cdot y\right) \]

Alternatives

Alternative 1
Error	18.3
Cost	1248

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+48}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq -380000000000:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-64}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-143}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-101}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;1.5 \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+98}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1.5 \cdot x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	448

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

Alternative 3
Error	32.1
Cost	192

\[-0.5 \cdot y \]

Reproduce

herbie shell --seed 2022330 
(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

Alternatives

Error

Reproduce