\[\left(x \cdot y + x\right) + y \]

↓

\[x + \mathsf{fma}\left(x, y, y\right) \]

(FPCore (x y) :precision binary64 (+ (+ (* x y) x) y))

↓

(FPCore (x y) :precision binary64 (+ x (fma x y y)))

double code(double x, double y) {
	return ((x * y) + x) + y;
}

↓

double code(double x, double y) {
	return x + fma(x, y, y);
}

function code(x, y)
	return Float64(Float64(Float64(x * y) + x) + y)
end

↓

function code(x, y)
	return Float64(x + fma(x, y, y))
end

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

↓

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

\left(x \cdot y + x\right) + y

↓

x + \mathsf{fma}\left(x, y, y\right)

Derivation

Initial program 0.0
\[\left(x \cdot y + x\right) + y \]
Simplified0.0
\[\leadsto \color{blue}{x + \mathsf{fma}\left(x, y, y\right)} \]
Proof
(+.f64 x (fma.f64 x y y)): 0 points increase in error, 0 points decrease in error
(+.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x y) y))): 1 points increase in error, 0 points decrease in error
(Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 x (*.f64 x y)) y)): 0 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x y) x)) y): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto x + \mathsf{fma}\left(x, y, y\right) \]

Alternatives

Alternative 1
Error	23.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -4.963643335303846 \cdot 10^{+19}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 9.751138413741878 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+93}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+110}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	18.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -4.963643335303846 \cdot 10^{+19}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 5.1883375074604343 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 3
Error	18.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 5.1883375074604343 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 1\right)\\ \end{array} \]

Alternative 4
Error	0.0
Cost	448

\[y + \left(x + x \cdot y\right) \]

Alternative 5
Error	27.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 9.751138413741878 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Error	36.3
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y)
  :name "Numeric.Log:$cexpm1 from log-domain-0.10.2.1, B"
  :precision binary64
  (+ (+ (* x y) x) y))

Error

Derivation

Alternatives

Error

Reproduce