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

↓

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

(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	20.6
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(x + 1\right)\\ \mathbf{if}\;y \leq -1188055.149700893:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.9705343794959834 \cdot 10^{-152}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.158584108004265 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.286880263735206 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	20.2
Cost	716

\[\begin{array}{l} t_0 := y \cdot \left(x + 1\right)\\ \mathbf{if}\;y \leq 2.9705343794959834 \cdot 10^{-152}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \mathbf{elif}\;y \leq 4.158584108004265 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.286880263735206 \cdot 10^{-85}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	25.1
Cost	592

\[\begin{array}{l} \mathbf{if}\;y \leq -1188055.149700893:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 2.9705343794959834 \cdot 10^{-152}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.158584108004265 \cdot 10^{-112}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.429036703158842 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4
Error	29.1
Cost	460

\[\begin{array}{l} \mathbf{if}\;y \leq 2.9705343794959834 \cdot 10^{-152}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.158584108004265 \cdot 10^{-112}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.429036703158842 \cdot 10^{-59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Error	0.0
Cost	448

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

Alternative 6
Error	35.8
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022308 
(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